  
  [1X11  [33X[0;0Y[5XGAP[105X[101X[1X  Computations  Concerning  Probabilistic Generation of Finite Simple[101X
  [1XGroups[133X[101X
  
  [33X[0;0YDate: March 28th, 2012[133X
  
  [33X[0;0YThis  is  a collection of examples showing how the [5XGAP[105X system [GAP21] can be
  used  to  compute  information  about the probabilistic generation of finite
  almost  simple  groups.  It  includes  all examples that were needed for the
  computational results in [BGK08].[133X
  
  [33X[0;0YThe  purpose  of  this writeup is twofold. On the one hand, the computations
  are  documented  this  way.  On  the  other hand, the [5XGAP[105X code shown for the
  examples  can  be  used as test input for automatic checking of the data and
  the functions used.[133X
  
  [33X[0;0YA  first  version  of this document, which was based on [5XGAP[105X 4.4.10, had been
  accessible  in  the  web since April 2006 and is available in the arXiv (no.
  0710.3267) since October 2007. The differences between that document and the
  current version are as follows.[133X
  
  [30X    [33X[0;6YThe  format of the [5XGAP[105X output was adjusted to the changed behaviour of
        [5XGAP[105X  until  version 4.10.  This affects mainly the way how [5XGAP[105X records
        are printed.[133X
  
  [30X    [33X[0;6YSeveral  computations  are now easier because more character tables of
        almost  simple  groups  and  maximal  subgroups  of  such  groups  are
        available  in  the  [5XGAP[105X  Character  Table  Library. (The more involved
        computations from the original version have been kept in the file.)[133X
  
  [30X    [33X[0;6YThe  computation  of all conjugacy classes of a subgroup of [22XPΩ^+(12,3)[122X
        has  been  replaced  by  the  computation  of the conjugacy classes of
        elements of prime order in this subgroup.[133X
  
  [30X    [33X[0;6YThe  irreducible  element  chosen  in  the simple group [22XPΩ^-(10,3)[122X has
        order [22X61[122X not [22X122[122X.[133X
  
  
  [1X11.1 [33X[0;0YOverview[133X[101X
  
  [33X[0;0YThe  main purpose of this note is to document the [5XGAP[105X computations that were
  carried out in order to obtain the computational results in [BGK08]. Table I
  lists  the  simple  groups  among these examples. The first column gives the
  group  names,  the second and third columns contain a plus sign [22X+[122X or a minus
  sign [22X-[122X, depending on whether the quantities [22Xσ(G,s)[122X and [22XP(G,s)[122X, respectively,
  are  less  than  [22X1/3[122X. The fourth column lists the orders of elements [22Xs[122X which
  either prove the [22X+[122X signs or cover most of the cases for proving these signs.
  The  fifth  column  lists  the  sections  in  this note where the example is
  treated.  The  rows of the table are ordered alphabetically w.r.t. the group
  names.[133X
  
  [33X[0;0YIn  order  to  keep  this  note self-contained, we first describe the theory
  needed,  in  Section [14X11.2[114X. The translation of the relevant formulae into [5XGAP[105X
  functions  can  be  found  in  Section [14X11.3[114X. Then Section [14X11.4[114X describes the
  computations  that  only  require  (ordinary)  character  tables  in the [5XGAP[105X
  Character  Table  Library [Bre24].  Computations  using  also the groups are
  shown  in  Section [14X11.5[114X.  In each of the last two sections, the examples are
  ordered alphabetically w.r.t. the names of the simple groups.[133X
  
      ┌───────────┬────────────┬────────────┬──────┬─────────────────┐
      │ [22XG[122X         │ [22Xσ < frac13[122X │ [22XP < frac13[122X │  [22X|s|[122X │             see │ 
      ├───────────┼────────────┼────────────┼──────┼─────────────────┤
      ├───────────┼────────────┼────────────┼──────┼─────────────────┤
      │ [22XA_5[122X       │     [22X-[122X      │     [22X-[122X      │    [22X5[122X │          [14X11.5-2[114X │ 
      │ [22XA_6[122X       │     [22X-[122X      │     [22X-[122X      │    [22X4[122X │          [14X11.5-3[114X │ 
      │ [22XA_7[122X       │     [22X-[122X      │     [22X-[122X      │    [22X7[122X │          [14X11.5-4[114X │ 
      │ [22XA_8[122X       │     [22X+[122X      │            │   [22X15[122X │  [14X11.4-3[114X, [14X11.5-5[114X │ 
      │ [22XA_9[122X       │     [22X+[122X      │            │    [22X9[122X │  [14X11.4-3[114X, [14X11.5-1[114X │ 
      │ [22XA_11[122X      │     [22X+[122X      │            │   [22X11[122X │  [14X11.4-3[114X, [14X11.5-1[114X │ 
      │ [22XA_13[122X      │     [22X+[122X      │            │   [22X13[122X │  [14X11.4-3[114X, [14X11.5-1[114X │ 
      │ [22XA_15[122X      │     [22X+[122X      │            │   [22X15[122X │          [14X11.5-1[114X │ 
      │ [22XA_17[122X      │     [22X+[122X      │            │   [22X17[122X │          [14X11.5-1[114X │ 
      │ [22XA_19[122X      │     [22X+[122X      │            │   [22X19[122X │          [14X11.5-1[114X │ 
      │ [22XA_21[122X      │     [22X+[122X      │            │   [22X21[122X │          [14X11.5-1[114X │ 
      │ [22XA_23[122X      │     [22X+[122X      │            │   [22X23[122X │          [14X11.5-1[114X │ 
      │ [22XL_3(2)[122X    │     [22X+[122X      │            │    [22X7[122X │ [14X11.4-3[114X, [14X11.4-4[114X, │ 
      │           │            │            │      │  [14X11.5-5[114X, [14X11.5-8[114X │ 
      │ [22XL_3(3)[122X    │     [22X+[122X      │            │   [22X13[122X │ [14X11.4-3[114X, [14X11.4-4[114X, │ 
      │           │            │            │      │          [14X11.5-5[114X │ 
      │ [22XL_3(4)[122X    │     [22X+[122X      │            │    [22X7[122X │  [14X11.4-3[114X, [14X11.4-4[114X │ 
      │ [22XL_4(3)[122X    │     [22X+[122X      │            │   [22X20[122X │  [14X11.4-3[114X, [14X11.5-5[114X │ 
      │ [22XL_4(4)[122X    │     [22X+[122X      │            │   [22X85[122X │          [14X11.5-5[114X │ 
      │ [22XL_6(2)[122X    │     [22X+[122X      │            │   [22X63[122X │          [14X11.5-5[114X │ 
      │ [22XL_6(3)[122X    │     [22X+[122X      │            │  [22X182[122X │          [14X11.5-5[114X │ 
      │ [22XL_6(4)[122X    │     [22X+[122X      │            │  [22X455[122X │          [14X11.5-5[114X │ 
      │ [22XL_6(5)[122X    │     [22X+[122X      │            │ [22X1953[122X │          [14X11.5-5[114X │ 
      │ [22XL_8(2)[122X    │     [22X+[122X      │            │  [22X255[122X │          [14X11.5-5[114X │ 
      │ [22XL_10(2)[122X   │     [22X+[122X      │            │ [22X1023[122X │          [14X11.5-5[114X │ 
      ├───────────┼────────────┼────────────┼──────┼─────────────────┤
      │ [22XM_11[122X      │     [22X-[122X      │     [22X-[122X      │   [22X11[122X │          [14X11.5-9[114X │ 
      │ [22XM_12[122X      │     [22X-[122X      │     [22X+[122X      │   [22X10[122X │ [14X11.4-2[114X, [14X11.5-10[114X │ 
      ├───────────┼────────────┼────────────┼──────┼─────────────────┤
      │ [22XO^+_8(2)[122X  │     [22X-[122X      │     [22X-[122X      │   [22X15[122X │         [14X11.5-12[114X │ 
      │ [22XO^+_8(3)[122X  │     [22X-[122X      │     [22X-[122X      │   [22X20[122X │         [14X11.5-13[114X │ 
      │ [22XO^+_8(4)[122X  │     [22X+[122X      │            │   [22X65[122X │         [14X11.5-14[114X │ 
      │ [22XO^+_10(2)[122X │     [22X+[122X      │            │   [22X45[122X │          [14X11.4-6[114X │ 
      │ [22XO^+_12(2)[122X │     [22X+[122X      │            │   [22X85[122X │          [14X11.4-8[114X │ 
      │ [22XO^+_12(3)[122X │     [22X+[122X      │            │  [22X205[122X │         [14X11.5-18[114X │ 
      │ [22XO^-_8(2)[122X  │     [22X+[122X      │            │   [22X17[122X │          [14X11.4-3[114X │ 
      │ [22XO^-_8(3)[122X  │     [22X+[122X      │            │   [22X41[122X │          [14X11.4-5[114X │ 
      │ [22XO^-_10(2)[122X │     [22X+[122X      │            │   [22X33[122X │          [14X11.4-7[114X │ 
      │ [22XO^-_10(3)[122X │     [22X+[122X      │            │  [22X122[122X │         [14X11.5-16[114X │ 
      │ [22XO^-_12(2)[122X │     [22X+[122X      │            │   [22X65[122X │          [14X11.4-9[114X │ 
      │ [22XO^-_14(2)[122X │     [22X+[122X      │            │  [22X129[122X │         [14X11.5-17[114X │ 
      │ [22XO_7(3)[122X    │     [22X-[122X      │     [22X-[122X      │   [22X14[122X │         [14X11.5-11[114X │ 
      ├───────────┼────────────┼────────────┼──────┼─────────────────┤
      │ [22XS_4(4)[122X    │     [22X+[122X      │            │   [22X17[122X │  [14X11.4-3[114X, [14X11.4-4[114X │ 
      │ [22XS_6(2)[122X    │     [22X-[122X      │     [22X-[122X      │    [22X9[122X │         [14X11.5-20[114X │ 
      │ [22XS_6(3)[122X    │     [22X+[122X      │            │   [22X14[122X │  [14X11.4-3[114X, [14X11.4-4[114X │ 
      │ [22XS_6(4)[122X    │     [22X+[122X      │            │   [22X65[122X │         [14X11.4-10[114X │ 
      │ [22XS_8(2)[122X    │     [22X-[122X      │     [22X-[122X      │   [22X17[122X │         [14X11.5-21[114X │ 
      │ [22XS_8(3)[122X    │     [22X+[122X      │            │   [22X41[122X │         [14X11.4-12[114X │ 
      ├───────────┼────────────┼────────────┼──────┼─────────────────┤
      │ [22XU_3(3)[122X    │     [22X+[122X      │            │    [22X6[122X │  [14X11.4-3[114X, [14X11.4-4[114X │ 
      │ [22XU_3(5)[122X    │     [22X+[122X      │            │   [22X10[122X │  [14X11.4-3[114X, [14X11.4-4[114X │ 
      │ [22XU_4(2)[122X    │     [22X-[122X      │     [22X-[122X      │    [22X9[122X │         [14X11.5-23[114X │ 
      │ [22XU_4(3)[122X    │     [22X-[122X      │     [22X+[122X      │    [22X7[122X │         [14X11.5-24[114X │ 
      │ [22XU_4(4)[122X    │     [22X+[122X      │            │   [22X65[122X │         [14X11.4-13[114X │ 
      │ [22XU_5(2)[122X    │     [22X+[122X      │            │   [22X11[122X │          [14X11.4-3[114X │ 
      │ [22XU_6(2)[122X    │     [22X+[122X      │            │   [22X11[122X │         [14X11.4-14[114X │ 
      │ [22XU_6(3)[122X    │     [22X+[122X      │            │  [22X122[122X │         [14X11.5-25[114X │ 
      │ [22XU_8(2)[122X    │     [22X+[122X      │            │  [22X129[122X │         [14X11.5-26[114X │ 
      └───────────┴────────────┴────────────┴──────┴─────────────────┘
  
       [1XTable:[101X Table I: Computations needed in [BGK08]
  
  
  [33X[0;0YContrary  to [BGK08],  [5XAtlas[105X  notation is used throughout this note, because
  the identifiers used for character tables in the [5XGAP[105X Character Table Library
  follow mainly the [5XAtlas[105X [CCN+85]. For example, we write [22XL_d(q)[122X for [22XPSL(d,q)[122X,
  [22XS_d(q)[122X   for  [22XPSp(d,q)[122X,  [22XU_d(q)[122X  for  [22XPSU(d,q)[122X,  and  [22XO^+_2d(q)[122X,  [22XO^-_2d(q)[122X,
  [22XO_2d+1(q)[122X for [22XPΩ^+(2d,q)[122X, [22XPΩ^-(2d,q)[122X, [22XPΩ(2d+1,q)[122X, respectively.[133X
  
  [33X[0;0YFurthermore,  in  the  case of classical groups, the character tables of the
  (almost)  [13Xsimple[113X  groups  are considered not the tables of the matrix groups
  (which  are in fact often not available in the [5XGAP[105X Character Table Library).
  Consequently,  also  element orders and the description of maximal subgroups
  refer to the (almost) simple groups not to the matrix groups.[133X
  
  [33X[0;0YThis  note contains also several examples that are not needed for the proofs
  in [BGK08].  Besides  several small simple groups [22XG[122X whose character table is
  contained   in  the  [5XGAP[105X  Character  Table  Library  and  for  which  enough
  information  is  available for computing [22Xσ(G)[122X, in Section [14X11.4-3[114X, a few such
  examples  appear  in  individual  sections.  In  the  table of contents, the
  section  headers  of the latter kind of examples are marked with an asterisk
  [22X(∗)[122X.[133X
  
  [33X[0;0YThe  examples use the [5XGAP[105X Character Table Library, the [5XGAP[105X Library of Tables
  of   Marks,   and   the   [5XGAP[105X  interface [WPN+22]  to  the  [5XAtlas[105X  of  Group
  Representations [WWT+],  so  we  first  load  these  three  packages  in the
  required  versions. The [5XGAP[105X output was adjusted to the versions shown below;
  in  older  versions, features necessary for the computations may be missing,
  and it may happen that with newer versions, the behaviour is different.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCompareVersionNumbers( GAPInfo.Version, "4.5.0" );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XLoadPackage( "ctbllib", "1.2", false );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XLoadPackage( "tomlib", "1.2", false );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XLoadPackage( "atlasrep", "1.5", false );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSome  of  the computations in Section [14X11.5[114X require about [22X800[122X MB of space (on
  [22X32[122X bit machines). Therefore we check whether [5XGAP[105X was started with sufficient
  maximal memory; the command line option for this is [10X-o 800m[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmax:= GAPInfo.CommandLineOptions.o;;[127X[104X
    [4X[25Xgap>[125X [27Xif not ( ( IsSubset( max, "m" ) and[127X[104X
    [4X[25X>[125X [27X              Int( Filtered( max, IsDigitChar ) ) >= 800 ) or[127X[104X
    [4X[25X>[125X [27X            ( IsSubset( max, "g" ) and[127X[104X
    [4X[25X>[125X [27X              Int( Filtered( max, IsDigitChar ) ) >= 1 ) ) then[127X[104X
    [4X[25X>[125X [27X     Print( "the maximal allowed memory might be too small\n" );[127X[104X
    [4X[25X>[125X [27X   fi;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YSeveral  computations  involve  calls to the [5XGAP[105X function [2XRandom[102X ([14XReference:
  Random[114X).  In  order to make the results of individual examples reproducible,
  independent  of  the  rest of the computations, we reset the relevant random
  number  generators  whenever  this  is  appropriate.  For that, we store the
  initial  states  in  the  variable  [10Xstaterandom[110X,  and provide a function for
  resetting  the  random  number  generators.  (The [2XRandom[102X ([14XReference: Random[114X)
  calls   in   the   [5XGAP[105X   library   use  the  two  random  number  generators
  [2XGlobalRandomSource[102X ([14XReference: GlobalRandomSource[114X) and [2XGlobalMersenneTwister[102X
  ([14XReference: GlobalMersenneTwister[114X).)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xstaterandom:= [ State( GlobalRandomSource ),[127X[104X
    [4X[25X>[125X [27X                   State( GlobalMersenneTwister ) ];;[127X[104X
    [4X[25Xgap>[125X [27XResetGlobalRandomNumberGenerators:= function()[127X[104X
    [4X[25X>[125X [27X    Reset( GlobalRandomSource, staterandom[1] );[127X[104X
    [4X[25X>[125X [27X    Reset( GlobalMersenneTwister, staterandom[2] );[127X[104X
    [4X[25X>[125X [27Xend;;[127X[104X
  [4X[32X[104X
  
  
  [1X11.2 [33X[0;0YPrerequisites[133X[101X
  
  
  [1X11.2-1 [33X[0;0YTheoretical Background[133X[101X
  
  [33X[0;0YLet  [22XG[122X  be  a  finite  group, [22XS[122X the socle of [22XG[122X, and denote by [22XG^×[122X the set of
  nonidentity  elements  in [22XG[122X. For [22Xs, g ∈ G^×[122X, let [22XP( g, s ):= |{ h ∈ G; S ⊈ ⟨
  s^h, g ⟩ }| / |G|[122X, the proportion of elements in the class [22Xs^G[122X which fail to
  generate at least [22XS[122X with [22Xg[122X; we set [22XP( G, s ):= max{ P( g, s ); g ∈ G^× }[122X. We
  are interested in finding a class [22Xs^G[122X of elements in [22XS[122X such that [22XP( G, s ) <
  1/3[122X holds.[133X
  
  [33X[0;0YFirst  consider  [22Xg  ∈  S[122X,  and  let  [22XMM(S,s)[122X denote the set of those maximal
  subgroups of [22XS[122X that contain [22Xs[122X. We have[133X
  
  
  [24X[33X[0;6Y|{ h ∈ S; S ⊈ ⟨ s^h, g ⟩ }| = |{ h ∈ S; ⟨ s, h g h^-1 ⟩ ‡ S }| ≤ ∑_M ∈ MM(S,s) |{ h ∈ S; h g h^-1 ∈ M }|[133X[124X
  
  [33X[0;0YSince  [22Xh  g  h^-1 ∈ M[122X holds if and only if the coset [22XM h[122X is fixed by [22Xg[122X under
  the  permutation action of [22XS[122X on the right cosets of [22XM[122X in [22XS[122X, we get that [22X|{ h
  ∈ S; h g h^-1 ∈ M }| = |C_S(g)| ⋅ |g^S ∩ M| = |M| ⋅ 1_M^S(g)[122X, where [22X1_M^S[122X is
  the permutation character of this action, of degree [22X|S|/|M|[122X. Thus[133X
  
  
  [24X[33X[0;6Y|{ h ∈ S; ⟨ s, h g h^-1 ⟩ ‡ S }| / |S| ≤ ∑_M ∈ MM(S,s) 1_M^S(g) / 1_M^S(1) .[133X[124X
  
  [33X[0;0YWe abbreviate the right hand side of this inequality by [22Xσ( g, s )[122X, set [22Xσ( S,
  s  ):= max{ σ( g, s ); g ∈ S^× }[122X, and choose a transversal [22XT[122X of [22XS[122X in [22XG[122X. Then
  [22XP(  g,  s  )  ≤  |T|^-1 ⋅ ∑_t ∈ T σ( g^t, s )[122X and thus [22XP( G, s ) ≤ σ( S, s )[122X
  holds.[133X
  
  [33X[0;0YIf  [22XS  =  G[122X and if [22XMM(G,s)[122X consists of a single maximal subgroup [22XM[122X of [22XG[122X then
  equality holds, i.e., [22XP( g, s ) = σ( g, s ) = 1_M^S(g) / 1_M^S(1)[122X.[133X
  
  [33X[0;0YThe  quantity  [22X1_M^S(g)  / 1_M^S(1) = |g^S ∩ M| / |g^S|[122X is the proportion of
  fixed  points of [22Xg[122X in the permutation action of [22XS[122X on the right cosets of its
  subgroup  [22XM[122X.  This is called the [13Xfixed point ratio[113X of [22Xg[122X w. r. t. [22XS/M[122X, and is
  denoted as [22Xμ(g,S/M)[122X.[133X
  
  [33X[0;0YFor  a  subgroup  [22XM[122X  of [22XS[122X, the number [22Xn[122X of [22XS[122X-conjugates of [22XM[122X containing [22Xs[122X is
  equal  to  [22X|M^S|  ⋅ |s^S ∩ M| / |s^S|[122X. To see this, consider the set [22X{ (s^h,
  M^k);  h, k ∈ S, s^h ∈ M^k }[122X, the cardinality of which can be counted either
  as  [22X|M^S|  ⋅  |s^S  ∩  M|[122X  or  as  [22X|s^S| ⋅ n[122X. So we get [22Xn = |M| ⋅ 1_M^S(s) /
  |N_S(M)|[122X.[133X
  
  [33X[0;0YIf  [22XS[122X is a finite [13Xnonabelian simple[113X group then each maximal subgroup in [22XS[122X is
  self-normalizing,  and  we  have  [22Xn  =  1_M^S(s)[122X  if [22XM[122X is maximal. So we can
  replace  the  summation  over  [22XMM(S,s)[122X  by  one  over  a  set  [22XMM/~(S,s)[122X  of
  representatives of conjugacy classes of maximal subgroups of [22XS[122X, and get that[133X
  
  
  [24X[33X[0;6Yσ( g, s ) = ∑_M ∈ MM/~(S,s) frac1_M^S(s) ⋅ 1_M^S(g)1_M^S(1).[133X[124X
  
  [33X[0;0YFurthermore, we have [22X|MM(S,s)| = ∑_M ∈ MM/~(S,s) 1_M^S(s)[122X.[133X
  
  [33X[0;0YIn the following, we will often deal with the quantities [22Xσ(S):= min{ σ( S, s
  );  s  ∈ S^× }[122X and [22Xtotal(S):= ⌈ 1 / σ(S) - 1 ⌉[122X. These values can be computed
  easily from the primitive permutation characters of [22XS[122X.[133X
  
  [33X[0;0YAnalogously,  we set [22XP(S):= min { P( S, s ); s ∈ S^× }[122X and [22XP(S):= ⌈ 1 / P(S)
  - 1 ⌉[122X. Clearly we have [22XP(S) ≤ σ(S)[122X and [22XP(S) ≥ total(S)[122X.[133X
  
  [33X[0;0YOne  interpretation  of  [22XP(S)[122X  is  that  if this value is at least [22Xk[122X then it
  follows that for any [22Xg_1, g_2, ..., g_k ∈ S^×[122X, there is some [22Xs ∈ S[122X such that
  [22XS  =  ⟨  g_i,  s ⟩[122X, for [22X1 ≤ i ≤ k[122X. In this case, [22XS[122X is said to have [13Xspread[113X at
  least  [22Xk[122X.  (Note that the lower bound [22Xtotal(S)[122X for [22XP(S)[122X can be computed from
  the list of primitive permutation characters of [22XS[122X.)[133X
  
  [33X[0;0YMoreover, [22XP(S) ≥ k[122X implies that the element [22Xs[122X can be chosen uniformly from a
  fixed  conjugacy  class  of  [22XS[122X.  This  is  called  [13Xuniform spread[113X at least [22Xk[122X
  in [BGK08].[133X
  
  [33X[0;0YIt  is proved in [GK00] that all finite simple groups have uniform spread at
  least  [22X1[122X,  that  is,  for  any  element  [22Xx ∈ S^×[122X, there is an element [22Xy[122X in a
  prescribed   class  of  [22XS[122X  such  that  [22XG  =  ⟨  x,  y  ⟩[122X  holds.  In [BGK08,
  Corollary 1.3],  it  is  shown  that  all  finite simple groups have uniform
  spread  at  least  [22X2[122X,  and  the  finite  simple groups with (uniform) spread
  exactly [22X2[122X are listed.[133X
  
  [33X[0;0YConcerning the spread, it should be mentioned that the methods used here and
  in [BGK08]  are  nonconstructive  in  the sense that they cannot be used for
  finding an element [22Xs[122X that generates [22XG[122X together with each of the [22Xk[122X prescribed
  elements [22Xg_1, g_2, ..., g_k[122X.[133X
  
  [33X[0;0YNow  consider  [22Xg  ∈  G  ∖  S[122X. Since [22XP( g^k, s ) ≥ P( g, s )[122X for any positive
  integer  [22Xk[122X, we can assume that [22Xg[122X has prime order [22Xp[122X, say. We set [22XH = ⟨ S, g ⟩
  ≤  G[122X,  with  [22X[H:S]  =  p[122X,  choose a transversal [22XT[122X of [22XH[122X in [22XG[122X, let [22XMM^'(H,s):=
  MM(H,s)  ∖  {  S  }[122X,  and let [22XMM/~^'(H,s)[122X denote a set of representatives of
  [22XH[122X-conjugacy classes of these groups. As above,[133X
  
        [22X|{ h ∈ H; S ⊈ ⟨ s^h, g ⟩ }| / |H|[122X   [22X=[122X   [22X|{ h ∈ H; ⟨ s^h, g ⟩ ‡ H }| / |H|[122X                  
                                            [22X≤[122X   [22X∑_M ∈ MM^'(H,s) |{ h ∈ H; h g h^-1 ∈ M }| / |H|[122X    
                                            [22X=[122X   [22X∑_M ∈ MM^'(H,s) 1_M^H(g) / 1_M^H(1)[122X                
                                            [22X=[122X   [22X∑_M ∈ MM/~^'(H,s) 1_M^H(g) ⋅ 1_M^H(s) / 1_M^H(1)[122X   
  
  [33X[0;0Y(Note  that  no  summand  for  [22XM = S[122X occurs, so each group in [22XMM/~^'(H,s)[122X is
  self-normalizing.)  We  abbreviate  the right hand side by [22Xσ(H,g,s)[122X, and set
  [22Xσ^'( H, s ) = max{ σ(H,g,s); g ∈ H ∖ S, |g| = [H:S] }[122X. Then we get [22XP( g, s )
  ≤ |T|^-1 ⋅ ∑_t ∈ T σ(H^t,g^t,s)[122X and thus[133X
  
  
  [24X[33X[0;6YP( G, s ) ≤ max{ P( S, s ), max{ σ^'( H, s ); S ≤ H ≤ G, [H:S] prime } } .[133X[124X
  
  [33X[0;0YFor convenience, we set [22XP^'(G,s) = max{ P(g,s); g ∈ G ∖ S }[122X.[133X
  
  
  [1X11.2-2 [33X[0;0YComputational Criteria[133X[101X
  
  [33X[0;0YThe  following  criteria  will be used when we have to show the existence or
  nonexistence of [22Xx_1, x_2, ..., x_k[122X, and [22Xs ∈ G[122X with the property [22X⟨ x_i, s ⟩ =
  G[122X  for  [22X1  ≤  i  ≤ k[122X. Note that manipulating lists of integers (representing
  fixed  or  moved points) is much more efficient than testing whether certain
  permutations generate a given group.[133X
  
  [33X[0;0YLemma:[133X
  
  [33X[0;0YLet  [22XG[122X  be a finite group, [22Xs ∈ G^×[122X, and [22XX = ⋃_M ∈ MM(G,s) G/M[122X. For [22Xx_1, x_2,
  ...,  x_k ∈ G[122X, the conjugate [22Xs^'[122X of [22Xs[122X satisfies [22X⟨ x_i, s^' ⟩ = G[122X for [22X1 ≤ i ≤
  k[122X if and only if [22XFix_X(s^') ∩ ⋃_i=1^k Fix_X(x_i) = ∅[122X holds.[133X
  
  [33X[0;0Y[13XProof.[113X  If [22Xs^g ∈ U ≤ G[122X for some [22Xg ∈ G[122X then [22XFix_X(U) = ∅[122X if and only if [22XU = G[122X
  holds;  note  that  [22XFix_X(G) = ∅[122X, and [22XFix_X(U) = ∅[122X implies that [22XU ⊈ h^-1 M h[122X
  holds  for  all [22Xh ∈ G[122X and [22XM ∈ MM(G,s)[122X, thus [22XU = G[122X. Applied to [22XU = ⟨ x_i, s^'
  ⟩[122X, we get [22X⟨ x_i, s^' ⟩ = G[122X if and only if [22XFix_X(s^') ∩ Fix_X(x_i) = Fix_X(U)
  = ∅[122X.[133X
  
  [33X[0;0YCorollary 1:[133X
  
  [33X[0;0YIf  [22XMM(G,s)  =  {  M  }[122X  in the situation of the above Lemma then there is a
  conjugate [22Xs^'[122X of [22Xs[122X that satisfies [22X⟨ x_i, s^' ⟩ = G[122X for [22X1 ≤ i ≤ k[122X if and only
  if [22X⋃_i=1^k Fix_X(x_i) ‡ X[122X.[133X
  
  [33X[0;0YCorollary 2:[133X
  
  [33X[0;0YLet  [22XG[122X  be  a  finite simple group and let [22XX[122X be a [22XG[122X-set such that each [22Xg ∈ G[122X
  fixes  at least one point in [22XX[122X but that [22XFix_X(G) = ∅[122X holds. If [22Xx_1, x_2, ...
  x_k[122X are elements in [22XG[122X such that [22X⋃_i=1^k Fix_X(x_i) = X[122X holds then for each [22Xs
  ∈ G[122X there is at least one [22Xi[122X with [22X⟨ x_i, s ⟩ ‡ G[122X.[133X
  
  
  [1X11.3 [33X[0;0Y[5XGAP[105X[101X[1X Functions for the Computations[133X[101X
  
  [33X[0;0YAfter   the   introduction   of  general  utilities  in  Section [14X11.3-1[114X,  we
  distinguish  two  different  tasks. Section [14X11.3-2[114X introduces functions that
  will  be  used  in  the following to compute [22Xσ(g,s)[122X with character-theoretic
  methods.  Functions  for  computing  [22XP(g,s)[122X or an upper bound for this value
  will be introduced in Section [14X11.3-3[114X.[133X
  
  [33X[0;0YThe  [5XGAP[105X  functions  shown  in  this  section  are  collected  in  the  file
  [11Xtst/probgen.g[111X  that  is  distributed  with  the [5XGAP[105X Character Table Library,
  see [7Xhttp://www.math.rwth-aachen.de/~Thomas.Breuer/ctbllib[107X.[133X
  
  [33X[0;0YThe  functions  have  been  designed for the examples in the later sections,
  they  could  be  generalized and optimized for other examples. It is not our
  aim to provide a package for this functionality.[133X
  
  
  [1X11.3-1 [33X[0;0YGeneral Utilities[133X[101X
  
  [33X[0;0YLet  [10Xlist[110X  be a dense list and [10Xprop[110X be a unary function that returns [9Xtrue[109X or
  [9Xfalse[109X when applied to the entries of [10Xlist[110X. [10XPositionsProperty[110X returns the set
  of positions in [10Xlist[110X for which [9Xtrue[109X is returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xif not IsBound( PositionsProperty ) then[127X[104X
    [4X[25X>[125X [27X     PositionsProperty:= function( list, prop )[127X[104X
    [4X[25X>[125X [27X       return Filtered( [ 1 .. Length( list ) ], i -> prop( list[i] ) );[127X[104X
    [4X[25X>[125X [27X     end;[127X[104X
    [4X[25X>[125X [27X   fi;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  following  two  functions  implement  loops  over  ordered triples (and
  quadruples, respectively) in a Cartesian product. A prescribed function [10Xprop[110X
  is  subsequently  applied  to the triples (quadruples), and if the result of
  this  call  is [9Xtrue[109X then this triple (quadruple) is returned immediately; if
  none of the calls to [10Xprop[110X yields [9Xtrue[109X then [9Xfail[109X is returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XBindGlobal( "TripleWithProperty", function( threelists, prop )[127X[104X
    [4X[25X>[125X [27X    local i, j, k, test;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    for i in threelists[1] do[127X[104X
    [4X[25X>[125X [27X      for j in threelists[2] do[127X[104X
    [4X[25X>[125X [27X        for k in threelists[3] do[127X[104X
    [4X[25X>[125X [27X          test:= [ i, j, k ];[127X[104X
    [4X[25X>[125X [27X          if prop( test ) then[127X[104X
    [4X[25X>[125X [27X              return test;[127X[104X
    [4X[25X>[125X [27X          fi;[127X[104X
    [4X[25X>[125X [27X        od;[127X[104X
    [4X[25X>[125X [27X      od;[127X[104X
    [4X[25X>[125X [27X    od;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    return fail;[127X[104X
    [4X[25X>[125X [27Xend );[127X[104X
    [4X[25Xgap>[125X [27XBindGlobal( "QuadrupleWithProperty", function( fourlists, prop )[127X[104X
    [4X[25X>[125X [27X    local i, j, k, l, test;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    for i in fourlists[1] do[127X[104X
    [4X[25X>[125X [27X      for j in fourlists[2] do[127X[104X
    [4X[25X>[125X [27X        for k in fourlists[3] do[127X[104X
    [4X[25X>[125X [27X          for l in fourlists[4] do[127X[104X
    [4X[25X>[125X [27X            test:= [ i, j, k, l ];[127X[104X
    [4X[25X>[125X [27X            if prop( test ) then[127X[104X
    [4X[25X>[125X [27X              return test;[127X[104X
    [4X[25X>[125X [27X            fi;[127X[104X
    [4X[25X>[125X [27X          od;[127X[104X
    [4X[25X>[125X [27X        od;[127X[104X
    [4X[25X>[125X [27X      od;[127X[104X
    [4X[25X>[125X [27X    od;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    return fail;[127X[104X
    [4X[25X>[125X [27Xend );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YOf course one could do better by considering [13Xun[113Xordered [22Xn[122X-tuples when several
  of  the  argument lists are equal, and in practice, backtrack searches would
  often  allow one to prune parts of the search tree in early stages. However,
  the above loops are not time critical in the examples presented here, so the
  possible improvements are not worth the effort for our purposes.[133X
  
  [33X[0;0YThe  function  [10XPrintFormattedArray[110X  prints  the matrix [10Xarray[110X in a columnwise
  formatted  way.  (The  only diference to the [5XGAP[105X library function [2XPrintArray[102X
  ([14XReference:  PrintArray[114X)  is  that  [10XPrintFormattedArray[110X  chooses each column
  width  according  to  the  entries  only in this column not w.r.t. the whole
  matrix.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XBindGlobal( "PrintFormattedArray", function( array )[127X[104X
    [4X[25X>[125X [27X     local colwidths, n, row;[127X[104X
    [4X[25X>[125X [27X     array:= List( array, row -> List( row, String ) );[127X[104X
    [4X[25X>[125X [27X     colwidths:= List( TransposedMat( array ),[127X[104X
    [4X[25X>[125X [27X                       col -> Maximum( List( col, Length ) ) );[127X[104X
    [4X[25X>[125X [27X     n:= Length( array[1] );[127X[104X
    [4X[25X>[125X [27X     for row in List( array, row -> List( [ 1 .. n ],[127X[104X
    [4X[25X>[125X [27X                  i -> String( row[i], colwidths[i] ) ) ) do[127X[104X
    [4X[25X>[125X [27X       Print( "  ", JoinStringsWithSeparator( row, " " ), "\n" );[127X[104X
    [4X[25X>[125X [27X     od;[127X[104X
    [4X[25X>[125X [27Xend );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YFinally,  [10XCleanWorkspace[110X is a utility for reducing the space needed. This is
  achieved  by unbinding those user variables that are not write protected and
  are  not  mentioned  in  the list [10XNeededVariables[110X of variable names that are
  bound  now,  and  by  flushing  the  caches of tables of marks and character
  tables.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XBindGlobal( "NeededVariables", NamesUserGVars() );[127X[104X
    [4X[25Xgap>[125X [27XBindGlobal( "CleanWorkspace", function()[127X[104X
    [4X[25X>[125X [27X      local name, record;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X      for name in Difference( NamesUserGVars(), NeededVariables ) do[127X[104X
    [4X[25X>[125X [27X       if not IsReadOnlyGlobal( name ) then[127X[104X
    [4X[25X>[125X [27X         UnbindGlobal( name );[127X[104X
    [4X[25X>[125X [27X       fi;[127X[104X
    [4X[25X>[125X [27X     od;[127X[104X
    [4X[25X>[125X [27X     for record in [ LIBTOMKNOWN, LIBTABLE ] do[127X[104X
    [4X[25X>[125X [27X       for name in RecNames( record.LOADSTATUS ) do[127X[104X
    [4X[25X>[125X [27X         Unbind( record.LOADSTATUS.( name ) );[127X[104X
    [4X[25X>[125X [27X         Unbind( record.( name ) );[127X[104X
    [4X[25X>[125X [27X       od;[127X[104X
    [4X[25X>[125X [27X     od;[127X[104X
    [4X[25X>[125X [27Xend );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  function  [10XPossiblePermutationCharacters[110X  takes  two  ordinary character
  tables  [10Xsub[110X  and  [10Xtbl[110X,  computes the possible class fusions from [10Xsub[110X to [10Xtbl[110X,
  then  induces the trivial character of [10Xsub[110X to [10Xtbl[110X, w.r.t. these fusions, and
  returns  the  set  of  these  class  functions.  (So  if [10Xsub[110X and [10Xtbl[110X are the
  character  tables  of groups [22XH[122X and [22XG[122X, respectively, where [22XH[122X is a subgroup of
  [22XG[122X, then the result contains the permutation character [22X1_H^G[122X.)[133X
  
  [33X[0;0YNote  that  the  columns  of the character tables in the [5XGAP[105X Character Table
  Library  are  not explicitly associated with particular conjugacy classes of
  the  corresponding groups, so from the character tables, we can compute only
  [13Xpossible[113X  class  fusions,  i.e., maps between the columns of two tables that
  satisfy  certain  necessary  conditions,  see the section about the function
  [10XPossibleClassFusions[110X  in  the  [5XGAP[105X Reference Manual for details. There is no
  problem if the permutation character is uniquely determined by the character
  tables,  in  all  other  cases  we  give  ad hoc arguments for resolving the
  ambiguities.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xif not IsBound( PossiblePermutationCharacters ) then[127X[104X
    [4X[25X>[125X [27X     BindGlobal( "PossiblePermutationCharacters", function( sub, tbl )[127X[104X
    [4X[25X>[125X [27X       local fus, triv;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X       fus:= PossibleClassFusions( sub, tbl );[127X[104X
    [4X[25X>[125X [27X       if fus = fail then[127X[104X
    [4X[25X>[125X [27X         return fail;[127X[104X
    [4X[25X>[125X [27X       fi;[127X[104X
    [4X[25X>[125X [27X       triv:= [ TrivialCharacter( sub ) ];[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X       return Set([127X[104X
    [4X[25X>[125X [27X           List( fus, map -> Induced( sub, tbl, triv, map )[1] ) );[127X[104X
    [4X[25X>[125X [27X     end );[127X[104X
    [4X[25X>[125X [27X   fi;[127X[104X
  [4X[32X[104X
  
  
  [1X11.3-2 [33X[0;0YCharacter-Theoretic Computations[133X[101X
  
  [33X[0;0YWe want to use the [5XGAP[105X libraries of character tables and of tables of marks,
  and proceed in three steps.[133X
  
  [33X[0;0YFirst  we  extract  the  primitive  permutation  characters from the library
  information   if  this  is  available;  for  that,  we  write  the  function
  [10XPrimitivePermutationCharacters[110X. Then the result can be used as the input for
  the  function  [10XApproxP[110X,  which  computes  the values [22Xσ( g, s )[122X. Finally, the
  functions [10XProbGenInfoSimple[110X and [10XProbGenInfoAlmostSimple[110X compute [22Xtotal( G )[122X.[133X
  
  [33X[0;0YFor  a  group  [22XG[122X  whose  character table [22XT[122X is contained in the [5XGAP[105X character
  table  library,  the complete set of primitive permutation characters can be
  easily  computed  if the character tables of all maximal subgroups and their
  class fusions into [22XT[122X are known (in this case, we check whether the attribute
  [2XMaxes[102X  ([14XCTblLib: Maxes[114X) of [22XT[122X is bound) or if the table of marks of [22XG[122X and the
  class  fusion  from  [22XT[122X  into this table of marks are known (in this case, we
  check  whether  the  attribute  [2XFusionToTom[102X  ([14XCTblLib:  FusionToTom[114X) of [22XT[122X is
  bound).  If  the  attribute  [2XUnderlyingGroup[102X ([14XReference: UnderlyingGroup for
  tables  of  marks[114X)  of [22XT[122X is bound then this group can be used to compute the
  primitive  permutation characters. The latter happens if [22XT[122X was computed from
  the group object in [5XGAP[105X; for tables in the [5XGAP[105X character table library, this
  is not the case by default.[133X
  
  [33X[0;0YThe   [5XGAP[105X  function  [10XPrimitivePermutationCharacters[110X  tries  to  compute  the
  primitive  permutation  characters  of  a  group  using this information; it
  returns  the  required  list of characters if this can be computed this way,
  otherwise  [9Xfail[109X  is  returned. (For convenience, we use the [5XGAP[105X mechanism of
  [13Xattributes[113X  in  order  to  store the permutation characters in the character
  table object once they have been computed.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDeclareAttribute( "PrimitivePermutationCharacters", IsCharacterTable );[127X[104X
    [4X[25Xgap>[125X [27XInstallOtherMethod( PrimitivePermutationCharacters,[127X[104X
    [4X[25X>[125X [27X    [ IsCharacterTable ],[127X[104X
    [4X[25X>[125X [27X    function( tbl )[127X[104X
    [4X[25X>[125X [27X    local maxes, tom, G;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    if HasMaxes( tbl ) then[127X[104X
    [4X[25X>[125X [27X      maxes:= List( Maxes( tbl ), CharacterTable );[127X[104X
    [4X[25X>[125X [27X      if ForAll( maxes, s -> GetFusionMap( s, tbl ) <> fail ) then[127X[104X
    [4X[25X>[125X [27X        return List( maxes, subtbl -> TrivialCharacter( subtbl )^tbl );[127X[104X
    [4X[25X>[125X [27X      fi;[127X[104X
    [4X[25X>[125X [27X    elif HasFusionToTom( tbl ) then[127X[104X
    [4X[25X>[125X [27X      tom:= TableOfMarks( tbl );[127X[104X
    [4X[25X>[125X [27X      maxes:= MaximalSubgroupsTom( tom );[127X[104X
    [4X[25X>[125X [27X      return PermCharsTom( tbl, tom ){ maxes[1] };[127X[104X
    [4X[25X>[125X [27X    elif HasUnderlyingGroup( tbl ) then[127X[104X
    [4X[25X>[125X [27X      G:= UnderlyingGroup( tbl );[127X[104X
    [4X[25X>[125X [27X      return List( MaximalSubgroupClassReps( G ),[127X[104X
    [4X[25X>[125X [27X                   M -> TrivialCharacter( M )^tbl );[127X[104X
    [4X[25X>[125X [27X    fi;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    return fail;[127X[104X
    [4X[25X>[125X [27Xend );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  function  [10XApproxP[110X  takes  a  list  [10Xprimitives[110X  of primitive permutation
  characters  of a group [22XG[122X, say, and the position [10Xspos[110X of the class [22Xs^G[122X in the
  character table of [22XG[122X.[133X
  
  [33X[0;0YAssume  that  the  elements  in [10Xprimitives[110X have the form [22X1_M^G[122X, for suitable
  maximal subgroups [22XM[122X of [22XG[122X, and let [22XMM/~[122X be the set of these groups [22XM[122X. [10XApproxP[110X
  returns the class function [22Xψ[122X of [22XG[122X that is defined by [22Xψ(1) = 0[122X and[133X
  
  
  [24X[33X[0;6Yψ(g) = ∑_M ∈ MM/~ frac1_M^G(s) ⋅ 1_M^G(g)1_M^G(1)[133X[124X
  
  [33X[0;0Yotherwise.[133X
  
  [33X[0;0YIf [10Xprimitives[110X contains all those primitive permutation characters [22X1_M^G[122X of [22XG[122X
  (with  multiplicity  according  to  the number of conjugacy classes of these
  maximal  subgroups)  that  do  not  vanish  at  [22Xs[122X,  and  if  all these [22XM[122X are
  self-normalizing in [22XG[122X –this holds for example if [22XG[122X is a finite simple group–
  then [22Xψ(g) = σ( g, s )[122X holds.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XBindGlobal( "ApproxP", function( primitives, spos )[127X[104X
    [4X[25X>[125X [27X    local sum;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    sum:= ShallowCopy( Sum( List( primitives,[127X[104X
    [4X[25X>[125X [27X                                  pi -> pi[ spos ] * pi / pi[1] ) ) );[127X[104X
    [4X[25X>[125X [27X    sum[1]:= 0;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    return sum;[127X[104X
    [4X[25X>[125X [27Xend );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that  for  computations with permutation characters, it would make the
  functions  more  complicated  (and  not more efficient) if we would consider
  only  elements  [22Xg[122X  of  prime  order,  and  only one representative of Galois
  conjugate classes.[133X
  
  [33X[0;0YThe next functions needed in this context compute [22Xσ(S)[122X and [22Xtotal( S )[122X, for a
  simple  group  [22XS[122X,  and  [22Xσ^'(G,s)[122X  for an almost simple group [22XG[122X with socle [22XS[122X,
  respectively.[133X
  
  [33X[0;0Y[10XProbGenInfoSimple[110X takes the character table [10Xtbl[110X of [22XS[122X as its argument. If the
  full  list  of primitive permutation characters of [22XS[122X cannot be computed with
  [10XPrimitivePermutationCharacters[110X  then  the  function  returns [9Xfail[109X. Otherwise
  [10XProbGenInfoSimple[110X returns a list containing the identifier of the table, the
  value [22Xσ(S)[122X, the integer [22Xtotal( S )[122X, a list of [5XAtlas[105X names of representatives
  of Galois families of those classes of elements [22Xs[122X for which [22Xσ(S) = σ( S, s )[122X
  holds, and the list of the corresponding cardinalities [22X|MM(S,s)|[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XBindGlobal( "ProbGenInfoSimple", function( tbl )[127X[104X
    [4X[25X>[125X [27X    local prim, max, min, bound, s;[127X[104X
    [4X[25X>[125X [27X    prim:= PrimitivePermutationCharacters( tbl );[127X[104X
    [4X[25X>[125X [27X    if prim = fail then[127X[104X
    [4X[25X>[125X [27X      return fail;[127X[104X
    [4X[25X>[125X [27X    fi;[127X[104X
    [4X[25X>[125X [27X    max:= List( [ 1 .. NrConjugacyClasses( tbl ) ],[127X[104X
    [4X[25X>[125X [27X                i -> Maximum( ApproxP( prim, i ) ) );[127X[104X
    [4X[25X>[125X [27X    min:= Minimum( max );[127X[104X
    [4X[25X>[125X [27X    bound:= Inverse( min );[127X[104X
    [4X[25X>[125X [27X    if IsInt( bound ) then[127X[104X
    [4X[25X>[125X [27X      bound:= bound - 1;[127X[104X
    [4X[25X>[125X [27X    else[127X[104X
    [4X[25X>[125X [27X      bound:= Int( bound );[127X[104X
    [4X[25X>[125X [27X    fi;[127X[104X
    [4X[25X>[125X [27X    s:= PositionsProperty( max, x -> x = min );[127X[104X
    [4X[25X>[125X [27X    s:= List( Set( s, i -> ClassOrbit( tbl, i ) ), i -> i[1] );[127X[104X
    [4X[25X>[125X [27X    return [ Identifier( tbl ),[127X[104X
    [4X[25X>[125X [27X             min,[127X[104X
    [4X[25X>[125X [27X             bound,[127X[104X
    [4X[25X>[125X [27X             AtlasClassNames( tbl ){ s },[127X[104X
    [4X[25X>[125X [27X             Sum( List( prim, pi -> pi{ s } ) ) ];[127X[104X
    [4X[25X>[125X [27Xend );[127X[104X
  [4X[32X[104X
  
  [33X[0;0Y[10XProbGenInfoAlmostSimple[110X takes the character tables [10XtblS[110X and [10XtblG[110X of [22XS[122X and [22XG[122X,
  and  a  list  [10XsposS[110X of class positions (w.r.t. [10XtblS[110X) as its arguments. It is
  assumed    that   [22XS[122X   is   simple   and   has   prime   index   in   [22XG[122X.   If
  [10XPrimitivePermutationCharacters[110X  can  compute  the  full  list  of  primitive
  permutation  characters of [22XG[122X then the function returns a list containing the
  identifier  of  [10XtblG[110X,  the  maximum  [22Xm[122X  of [22Xσ^'( G, s )[122X, for [22Xs[122X in the classes
  described  by [10XsposS[110X, a list of [5XAtlas[105X names (in [22XG[122X) of the classes of elements
  [22Xs[122X  for  which  this  maximum  is attained, and the list of the corresponding
  cardinalities [22X|MM^'(G,s)|[122X. When [10XPrimitivePermutationCharacters[110X returns [9Xfail[109X,
  also [10XProbGenInfoAlmostSimple[110X returns [9Xfail[109X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XBindGlobal( "ProbGenInfoAlmostSimple", function( tblS, tblG, sposS )[127X[104X
    [4X[25X>[125X [27X    local p, fus, inv, prim, sposG, outer, approx, l, max, min,[127X[104X
    [4X[25X>[125X [27X          s, cards, i, names;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    p:= Size( tblG ) / Size( tblS );[127X[104X
    [4X[25X>[125X [27X    if not IsPrimeInt( p )[127X[104X
    [4X[25X>[125X [27X       or Length( ClassPositionsOfNormalSubgroups( tblG ) ) <> 3 then[127X[104X
    [4X[25X>[125X [27X      return fail;[127X[104X
    [4X[25X>[125X [27X    fi;[127X[104X
    [4X[25X>[125X [27X    fus:= GetFusionMap( tblS, tblG );[127X[104X
    [4X[25X>[125X [27X    if fus = fail then[127X[104X
    [4X[25X>[125X [27X      return fail;[127X[104X
    [4X[25X>[125X [27X    fi;[127X[104X
    [4X[25X>[125X [27X    inv:= InverseMap( fus );[127X[104X
    [4X[25X>[125X [27X    prim:= PrimitivePermutationCharacters( tblG );[127X[104X
    [4X[25X>[125X [27X    if prim = fail then[127X[104X
    [4X[25X>[125X [27X      return fail;[127X[104X
    [4X[25X>[125X [27X    fi;[127X[104X
    [4X[25X>[125X [27X    sposG:= Set( fus{ sposS } );[127X[104X
    [4X[25X>[125X [27X    outer:= Difference( PositionsProperty([127X[104X
    [4X[25X>[125X [27X                OrdersClassRepresentatives( tblG ), IsPrimeInt ), fus );[127X[104X
    [4X[25X>[125X [27X    approx:= List( sposG, i -> ApproxP( prim, i ){ outer } );[127X[104X
    [4X[25X>[125X [27X    if IsEmpty( outer ) then[127X[104X
    [4X[25X>[125X [27X      max:= List( approx, x -> 0 );[127X[104X
    [4X[25X>[125X [27X    else[127X[104X
    [4X[25X>[125X [27X      max:= List( approx, Maximum );[127X[104X
    [4X[25X>[125X [27X    fi;[127X[104X
    [4X[25X>[125X [27X    min:= Minimum( max);[127X[104X
    [4X[25X>[125X [27X    s:= sposG{ PositionsProperty( max, x -> x = min ) };[127X[104X
    [4X[25X>[125X [27X    cards:= List( prim, pi -> pi{ s } );[127X[104X
    [4X[25X>[125X [27X    for i in [ 1 .. Length( prim ) ] do[127X[104X
    [4X[25X>[125X [27X      # Omit the character that is induced from the simple group.[127X[104X
    [4X[25X>[125X [27X      if ForAll( prim[i], x -> x = 0 or x = prim[i][1] ) then[127X[104X
    [4X[25X>[125X [27X        cards[i]:= 0;[127X[104X
    [4X[25X>[125X [27X      fi;[127X[104X
    [4X[25X>[125X [27X    od;[127X[104X
    [4X[25X>[125X [27X    names:= AtlasClassNames( tblG ){ s };[127X[104X
    [4X[25X>[125X [27X    Perform( names, ConvertToStringRep );[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    return [ Identifier( tblG ),[127X[104X
    [4X[25X>[125X [27X             min,[127X[104X
    [4X[25X>[125X [27X             names,[127X[104X
    [4X[25X>[125X [27X             Sum( cards ) ];[127X[104X
    [4X[25X>[125X [27Xend );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  next  function computes [22Xσ(G,s)[122X from the character table [10Xtbl[110X of a simple
  or  almost  simple  group [22XG[122X, the name [10Xsname[110X of the class of [22Xs[122X in this table,
  the  list [10Xmaxes[110X of the character tables of all subgroups [22XM[122X with [22XM ∈ MM(G,s)[122X,
  and  the list [10Xnumpermchars[110X of the numbers of possible permutation characters
  induced  from  [10Xmaxes[110X. If the string [10X"outer"[110X is given as an optional argument
  then  [22XG[122X  is assumed to be an automorphic extension of a simple group [22XS[122X, with
  [22X[G:S][122X  a  prime, and [22Xσ^'(G,s)[122X is returned. In both situations, the result is
  [9Xfail[109X if the numbers of possible permutation characters induced from [10Xmaxes[110X do
  not coincide with the numbers prescribed in [10Xnumpermchars[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XBindGlobal( "SigmaFromMaxes", function( arg )[127X[104X
    [4X[25X>[125X [27X    local t, sname, maxes, numpermchars, prim, spos, outer;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    t:= arg[1];[127X[104X
    [4X[25X>[125X [27X    sname:= arg[2];[127X[104X
    [4X[25X>[125X [27X    maxes:= arg[3];[127X[104X
    [4X[25X>[125X [27X    numpermchars:= arg[4];[127X[104X
    [4X[25X>[125X [27X    prim:= List( maxes, s -> PossiblePermutationCharacters( s, t ) );[127X[104X
    [4X[25X>[125X [27X    spos:= Position( AtlasClassNames( t ), sname );[127X[104X
    [4X[25X>[125X [27X    if ForAny( [ 1 .. Length( maxes ) ],[127X[104X
    [4X[25X>[125X [27X               i -> Length( prim[i] ) <> numpermchars[i] ) then[127X[104X
    [4X[25X>[125X [27X      return fail;[127X[104X
    [4X[25X>[125X [27X    elif Length( arg ) = 5 and arg[5] = "outer" then[127X[104X
    [4X[25X>[125X [27X      outer:= Difference([127X[104X
    [4X[25X>[125X [27X          PositionsProperty( OrdersClassRepresentatives( t ), IsPrimeInt ),[127X[104X
    [4X[25X>[125X [27X          ClassPositionsOfDerivedSubgroup( t ) );[127X[104X
    [4X[25X>[125X [27X      return Maximum( ApproxP( Concatenation( prim ), spos ){ outer } );[127X[104X
    [4X[25X>[125X [27X    else[127X[104X
    [4X[25X>[125X [27X      return Maximum( ApproxP( Concatenation( prim ), spos ) );[127X[104X
    [4X[25X>[125X [27X    fi;[127X[104X
    [4X[25X>[125X [27Xend );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  following  function allows us to extract information about [22XMM(G,s)[122X from
  the  character  table [10Xtbl[110X of [22XG[122X and a list [10Xsnames[110X of class positions of [22Xs[122X. If
  [10XMaxes(  tbl  )[110X  is  stored  then  the  names  of the character tables of the
  subgroups  in  [22XMM(G,s)[122X  and  the number of conjugates are printed, otherwise
  [9Xfail[109X is printed.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XBindGlobal( "DisplayProbGenMaxesInfo", function( tbl, snames )[127X[104X
    [4X[25X>[125X [27X    local mx, prim, i, spos, nonz, indent, j;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    if not HasMaxes( tbl ) then[127X[104X
    [4X[25X>[125X [27X      Print( Identifier( tbl ), ": fail\n" );[127X[104X
    [4X[25X>[125X [27X      return;[127X[104X
    [4X[25X>[125X [27X    fi;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    mx:= List( Maxes( tbl ), CharacterTable );[127X[104X
    [4X[25X>[125X [27X    prim:= List( mx, s -> TrivialCharacter( s )^tbl );[127X[104X
    [4X[25X>[125X [27X    Assert( 1, SortedList( prim ) =[127X[104X
    [4X[25X>[125X [27X               SortedList( PrimitivePermutationCharacters( tbl ) ) );[127X[104X
    [4X[25X>[125X [27X    for i in [ 1 .. Length( prim ) ] do[127X[104X
    [4X[25X>[125X [27X      # Deal with the case that the subgroup is normal.[127X[104X
    [4X[25X>[125X [27X      if ForAll( prim[i], x -> x = 0 or x = prim[i][1] ) then[127X[104X
    [4X[25X>[125X [27X        prim[i]:= prim[i] / prim[i][1];[127X[104X
    [4X[25X>[125X [27X      fi;[127X[104X
    [4X[25X>[125X [27X    od;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    spos:= List( snames,[127X[104X
    [4X[25X>[125X [27X                 nam -> Position( AtlasClassNames( tbl ), nam ) );[127X[104X
    [4X[25X>[125X [27X    nonz:= List( spos, x -> PositionsProperty( prim, pi -> pi[x] <> 0 ) );[127X[104X
    [4X[25X>[125X [27X    for i in [ 1 .. Length( spos ) ] do[127X[104X
    [4X[25X>[125X [27X      Print( Identifier( tbl ), ", ", snames[i], ": " );[127X[104X
    [4X[25X>[125X [27X      indent:= ListWithIdenticalEntries([127X[104X
    [4X[25X>[125X [27X          Length( Identifier( tbl ) ) + Length( snames[i] ) + 4, ' ' );[127X[104X
    [4X[25X>[125X [27X      if not IsEmpty( nonz[i] ) then[127X[104X
    [4X[25X>[125X [27X        Print( Identifier( mx[ nonz[i][1] ] ), "  (",[127X[104X
    [4X[25X>[125X [27X               prim[ nonz[i][1] ][ spos[i] ], ")\n" );[127X[104X
    [4X[25X>[125X [27X        for j in [ 2 .. Length( nonz[i] ) ] do[127X[104X
    [4X[25X>[125X [27X          Print( indent, Identifier( mx[ nonz[i][j] ] ), "  (",[127X[104X
    [4X[25X>[125X [27X               prim[ nonz[i][j] ][ spos[i] ], ")\n" );[127X[104X
    [4X[25X>[125X [27X        od;[127X[104X
    [4X[25X>[125X [27X      else[127X[104X
    [4X[25X>[125X [27X        Print( "\n" );[127X[104X
    [4X[25X>[125X [27X      fi;[127X[104X
    [4X[25X>[125X [27X    od;[127X[104X
    [4X[25X>[125X [27Xend );[127X[104X
  [4X[32X[104X
  
  
  [1X11.3-3 [33X[0;0YComputations with Groups[133X[101X
  
  [33X[0;0YHere,  the  task  is to compute [22XP(g,s)[122X or [22XP(G,s)[122X using explicit computations
  with [22XG[122X, where the character-theoretic bounds are not sufficient.[133X
  
  [33X[0;0YWe start with small utilities that make the examples shorter.[133X
  
  [33X[0;0YFor  a  finite solvable group [10XG[110X, the function [10XPcConjugacyClassReps[110X returns a
  list  of  representatives  of the conjugacy classes of [10XG[110X, which are computed
  using a polycyclic presentation for [10XG[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XBindGlobal( "PcConjugacyClassReps", function( G )[127X[104X
    [4X[25X>[125X [27X     local iso;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X     iso:= IsomorphismPcGroup( G );[127X[104X
    [4X[25X>[125X [27X     return List( ConjugacyClasses( Image( iso ) ),[127X[104X
    [4X[25X>[125X [27X              c -> PreImagesRepresentative( iso, Representative( c ) ) );[127X[104X
    [4X[25X>[125X [27Xend );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YFor a finite group [10XG[110X, a list [10Xprimes[110X of prime integers, and a normal subgroup
  [10XN[110X  of  [10XG[110X, the function [10XClassesOfPrimeOrder[110X returns a list of those conjugacy
  classes of [10XG[110X that are not contained in [10XN[110X and whose elements' orders occur in
  [10Xprimes[110X.[133X
  
  [33X[0;0YFor  each  prime  [22Xp[122X  in  [10Xprimes[110X, first class representatives of order [22Xp[122X in a
  Sylow  [22Xp[122X  subgroup  of  [10XG[110X  are  computed,  then the representatives in [10XN[110X are
  discarded, and then representatives w. r. t. conjugacy in [10XG[110X are computed.[133X
  
  [33X[0;0Y(Note  that  this  approach  may  be  inappropriate  for  example if a large
  elementary  abelian Sylow [22Xp[122X subgroup occurs, and if the conjugacy tests in [10XG[110X
  are expensive, see Section [14X11.5-14[114X.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XBindGlobal( "ClassesOfPrimeOrder", function( G, primes, N )[127X[104X
    [4X[25X>[125X [27X     local ccl, p, syl, Greps, reps, r, cr;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X     ccl:= [];[127X[104X
    [4X[25X>[125X [27X     for p in primes do[127X[104X
    [4X[25X>[125X [27X       syl:= SylowSubgroup( G, p );[127X[104X
    [4X[25X>[125X [27X       Greps:= [];[127X[104X
    [4X[25X>[125X [27X       reps:= Filtered( PcConjugacyClassReps( syl ),[127X[104X
    [4X[25X>[125X [27X                  r -> Order( r ) = p and not r in N );[127X[104X
    [4X[25X>[125X [27X       for r in reps do[127X[104X
    [4X[25X>[125X [27X         cr:= ConjugacyClass( G, r );[127X[104X
    [4X[25X>[125X [27X         if ForAll( Greps, c -> c <> cr ) then[127X[104X
    [4X[25X>[125X [27X           Add( Greps, cr );[127X[104X
    [4X[25X>[125X [27X         fi;[127X[104X
    [4X[25X>[125X [27X       od;[127X[104X
    [4X[25X>[125X [27X       Append( ccl, Greps );[127X[104X
    [4X[25X>[125X [27X     od;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X     return ccl;[127X[104X
    [4X[25X>[125X [27Xend );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  function  [10XIsGeneratorsOfTransPermGroup[110X  takes  a [13Xtransitive[113X permutation
  group  [10XG[110X  and a list [10Xlist[110X of elements in [10XG[110X, and returns [9Xtrue[109X if the elements
  in  [10Xlist[110X  generate [10XG[110X, and [9Xfalse[109X otherwise. The main point is that the return
  value  [9Xtrue[109X  requires  the group generated by [10Xlist[110X to be transitive, and the
  check  for  transitivity is much cheaper than the test whether this group is
  equal to [10XG[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xif not IsBound( IsGeneratorsOfTransPermGroup) then[127X[104X
    [4X[25X>[125X [27X     BindGlobal( "IsGeneratorsOfTransPermGroup", function( G, list )[127X[104X
    [4X[25X>[125X [27X       local S;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X       if not IsTransitive( G ) then[127X[104X
    [4X[25X>[125X [27X         Error( "<G> must be transitive on its moved points" );[127X[104X
    [4X[25X>[125X [27X       fi;[127X[104X
    [4X[25X>[125X [27X       S:= SubgroupNC( G, list );[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X       return IsTransitive( S, MovedPoints( G ) ) and[127X[104X
    [4X[25X>[125X [27X              Size( S ) = Size( G );[127X[104X
    [4X[25X>[125X [27X     end );[127X[104X
    [4X[25X>[125X [27X   fi;[127X[104X
  [4X[32X[104X
  
  [33X[0;0Y[10XRatioOfNongenerationTransPermGroup[110X  takes  a  [13Xtransitive[113X permutation group [10XG[110X
  and  two  elements  [10Xg[110X  and  [10Xs[110X  of [10XG[110X, and returns the proportion [22XP(g,s)[122X. (The
  function   tests   the   (non)generation   only   for   representatives   of
  [22XC_G(g)[122X-[22XC_G(s)[122X-double cosets. Note that for [22Xc_1 ∈ C_G(g)[122X, [22Xc_2 ∈ C_G(s)[122X, and a
  representative [22Xr ∈ G[122X, we have [22X⟨ g^c_1 r c_2, s ⟩ = ⟨ g^r, s ⟩^c_2[122X.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XBindGlobal( "RatioOfNongenerationTransPermGroup", function( G, g, s )[127X[104X
    [4X[25X>[125X [27X    local nongen, pair;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    if not IsTransitive( G ) then[127X[104X
    [4X[25X>[125X [27X      Error( "<G> must be transitive on its moved points" );[127X[104X
    [4X[25X>[125X [27X    fi;[127X[104X
    [4X[25X>[125X [27X    nongen:= 0;[127X[104X
    [4X[25X>[125X [27X    for pair in DoubleCosetRepsAndSizes( G, Centralizer( G, g ),[127X[104X
    [4X[25X>[125X [27X                    Centralizer( G, s ) ) do[127X[104X
    [4X[25X>[125X [27X      if not IsGeneratorsOfTransPermGroup( G, [ s, g^pair[1] ] ) then[127X[104X
    [4X[25X>[125X [27X        nongen:= nongen + pair[2];[127X[104X
    [4X[25X>[125X [27X      fi;[127X[104X
    [4X[25X>[125X [27X    od;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    return nongen / Size( G );[127X[104X
    [4X[25X>[125X [27Xend );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YLet  [22XG[122X  be  a  group,  and  let  [10Xgroups[110X  be a list [22X[ G_1, G_2, ..., G_n ][122X of
  permutation  groups  such  that  [22XG_i[122X describes the action of [22XG[122X on a set [22XΩ_i[122X,
  say.   Moreover,   we   require  that  for  [22X1  ≤  i,  j  ≤  n[122X,  mapping  the
  [10XGeneratorsOfGroup[110X  list  of  [22XG_i[122X  to  that  of  [22XG_j[122X  defines an isomorphism.
  [10XDiagonalProductOfPermGroups[110X  takes  [10Xgroups[110X  as its argument, and returns the
  action of [22XG[122X on the disjoint union of [22XΩ_1, Ω_2, ..., Ω_n[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XBindGlobal( "DiagonalProductOfPermGroups", function( groups )[127X[104X
    [4X[25X>[125X [27X    local prodgens, deg, i, gens, D, pi;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    prodgens:= GeneratorsOfGroup( groups[1] );[127X[104X
    [4X[25X>[125X [27X    deg:= NrMovedPoints( prodgens );[127X[104X
    [4X[25X>[125X [27X    for i in [ 2 .. Length( groups ) ] do[127X[104X
    [4X[25X>[125X [27X      gens:= GeneratorsOfGroup( groups[i] );[127X[104X
    [4X[25X>[125X [27X      D:= MovedPoints( gens );[127X[104X
    [4X[25X>[125X [27X      pi:= MappingPermListList( D, [ deg+1 .. deg+Length( D ) ] );[127X[104X
    [4X[25X>[125X [27X      deg:= deg + Length( D );[127X[104X
    [4X[25X>[125X [27X      prodgens:= List( [ 1 .. Length( prodgens ) ],[127X[104X
    [4X[25X>[125X [27X                       i -> prodgens[i] * gens[i]^pi );[127X[104X
    [4X[25X>[125X [27X    od;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    return Group( prodgens );[127X[104X
    [4X[25X>[125X [27Xend );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe following two functions are used to reduce checks of generation to class
  representatives of maximal order. Note that if [22X⟨ s, g ⟩[122X is a proper subgroup
  of  [22XG[122X  then  also [22X⟨ s^k, g ⟩[122X is a proper subgroup of [22XG[122X, so we need not check
  powers [22Xs^k[122X different from [22Xs[122X in this situation.[133X
  
  [33X[0;0YFor     an     ordinary     character     table     [10Xtbl[110X,     the    function
  [10XRepresentativesMaximallyCyclicSubgroups[110X  returns  a list of class positions,
  containing  one  class  of  generators  for  each  class of maximally cyclic
  subgroups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XBindGlobal( "RepresentativesMaximallyCyclicSubgroups", function( tbl )[127X[104X
    [4X[25X>[125X [27X    local n, result, orders, p, pmap, i, j;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    # Initialize.[127X[104X
    [4X[25X>[125X [27X    n:= NrConjugacyClasses( tbl );[127X[104X
    [4X[25X>[125X [27X    result:= BlistList( [ 1 .. n ], [ 1 .. n ] );[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    # Omit powers of smaller order.[127X[104X
    [4X[25X>[125X [27X    orders:= OrdersClassRepresentatives( tbl );[127X[104X
    [4X[25X>[125X [27X    for p in PrimeDivisors( Size( tbl ) ) do[127X[104X
    [4X[25X>[125X [27X      pmap:= PowerMap( tbl, p );[127X[104X
    [4X[25X>[125X [27X      for i in [ 1 .. n ] do[127X[104X
    [4X[25X>[125X [27X        if orders[ pmap[i] ] < orders[i] then[127X[104X
    [4X[25X>[125X [27X          result[ pmap[i] ]:= false;[127X[104X
    [4X[25X>[125X [27X        fi;[127X[104X
    [4X[25X>[125X [27X      od;[127X[104X
    [4X[25X>[125X [27X    od;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    # Omit Galois conjugates.[127X[104X
    [4X[25X>[125X [27X    for i in [ 1 .. n ] do[127X[104X
    [4X[25X>[125X [27X      if result[i] then[127X[104X
    [4X[25X>[125X [27X        for j in ClassOrbit( tbl, i ) do[127X[104X
    [4X[25X>[125X [27X          if i <> j then[127X[104X
    [4X[25X>[125X [27X            result[j]:= false;[127X[104X
    [4X[25X>[125X [27X          fi;[127X[104X
    [4X[25X>[125X [27X        od;[127X[104X
    [4X[25X>[125X [27X      fi;[127X[104X
    [4X[25X>[125X [27X    od;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    # Return the result.[127X[104X
    [4X[25X>[125X [27X    return ListBlist( [ 1 .. n ], result );[127X[104X
    [4X[25X>[125X [27Xend );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YLet  [10XG[110X be a finite group, [10Xtbl[110X be the ordinary character table of [10XG[110X, and [10Xcols[110X
  be  a  list  of  class  positions  in  [10Xtbl[110X, for example the list returned by
  [10XRepresentativesMaximallyCyclicSubgroups[110X.             The            function
  [10XClassesPerhapsCorrespondingToTableColumns[110X   returns  the  sublist  of  those
  conjugacy  classes  of  [10XG[110X  for  which the corresponding column in [10Xtbl[110X can be
  contained in [10Xcols[110X, according to element order and class size.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XBindGlobal( "ClassesPerhapsCorrespondingToTableColumns",[127X[104X
    [4X[25X>[125X [27X   function( G, tbl, cols )[127X[104X
    [4X[25X>[125X [27X    local orders, classes, invariants;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    orders:= OrdersClassRepresentatives( tbl );[127X[104X
    [4X[25X>[125X [27X    classes:= SizesConjugacyClasses( tbl );[127X[104X
    [4X[25X>[125X [27X    invariants:= List( cols, i -> [ orders[i], classes[i] ] );[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    return Filtered( ConjugacyClasses( G ),[127X[104X
    [4X[25X>[125X [27X        c -> [ Order( Representative( c ) ), Size(c) ] in invariants );[127X[104X
    [4X[25X>[125X [27Xend );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  next  function  computes,  for a finite group [22XG[122X and subgroups [22XM_1, M_2,
  ...,  M_n[122X of [22XG[122X, an upper bound for [22Xmax { ∑_i=1^n μ(g,G/M_i); g ∈ G ∖ Z(G) }[122X.
  So  if  the  [22XM_i[122X are the groups in [22XMM(G,s)[122X, for some [22Xs ∈ G^×[122X, then we get an
  upper bound for [22Xσ(G,s)[122X.[133X
  
  [33X[0;0YThe idea is that for [22XM ≤ G[122X and [22Xg ∈ G[122X of order [22Xp[122X, we have[133X
  
  
  [24X[33X[0;6Yμ(g,G/M) = |g^G ∩ M| / |g^G| ≤ ∑_h ∈ C |h^M| / |g^G| = ∑_h ∈ C |h^M| ⋅ |C_G(g)| / |G| ,[133X[124X
  
  [33X[0;0Ywhere  [22XC[122X  is  a set of class representatives [22Xh ∈ M[122X of all those classes that
  satisfy  [22X|h|  =  p[122X  and  [22X|C_G(h)|  =  |C_G(g)|[122X,  and in the case that [22XG[122X is a
  permutation  group additionally that [22Xh[122X and [22Xg[122X move the same number of points.
  (Note that it is enough to consider elements of [13Xprime[113X order.)[133X
  
  [33X[0;0YFor  computing the maximum of the rightmost term in this inequality, for [22Xg ∈
  G  ∖ Z(G)[122X, we need not determine the [22XG[122X-conjugacy of class representatives in
  [22XM[122X.  Of  course  we  pay  the  price  that  the result may be larger than the
  leftmost  term.  However,  if  the  maximal sum is in fact taken only over a
  single class representative, we are sure that equality holds. Thus we return
  a  list  of length two, containing the maximum of the right hand side of the
  above inequality and a Boolean value indicating whether this is equal to [22Xmax
  { μ(g,G/M); g ∈ G ∖ Z(G) }[122X or just an upper bound.[133X
  
  [33X[0;0YThe  arguments  for  [10XUpperBoundFixedPointRatios[110X  are  the  group  [10XG[110X,  a list
  [10Xmaxesclasses[110X such that the [22Xi[122X-th entry is a list of conjugacy classes of [22XM_i[122X,
  which  covers  all classes of prime element order in [22XM_i[122X, and either [9Xtrue[109X or
  [9Xfalse[109X, where [9Xtrue[109X means that the [13Xexact[113X value of [22Xσ(G,s)[122X is computed, not just
  an  upper  bound;  this  can be much more expensive because of the conjugacy
  tests  in [22XG[122X that may be necessary. (We try to reduce the number of conjugacy
  tests  in  this  case,  the  second  half  of  the  code  is  not completely
  straightforward. The special treatment of conjugacy checks for elements with
  the  same  sets  of fixed points is essential in the computation of [22Xσ^'(G,s)[122X
  for  [22XG = PGL(6,4)[122X; the critical input line is [10XApproxPForOuterClassesInGL( 6,
  4  )[110X,  see  Section [14X11.5-7[114X. Currently the standard [5XGAP[105X conjugacy test for an
  element  of  order  three  and  its inverse in [22XG ∖ G^'[122X requires hours of CPU
  time,  whereas  the  check  for  existence  of  a conjugating element in the
  stabilizer  of  the common set of fixed points of the two elements is almost
  free of charge.)[133X
  
  [33X[0;0Y[10XUpperBoundFixedPointRatios[110X  can be used to compute [22Xσ^'(G,s)[122X in the case that
  [22XG[122X  is  an automorphic extension of a simple group [22XS[122X, with [22X[G:S] = p[122X a prime;
  if  [22XMM^'(G,s)  =  { M_1, M_2, ..., M_n }[122X then the [22Xi[122X-th entry of [10Xmaxesclasses[110X
  must contain only the classes of element order [22Xp[122X in [22XM_i ∖ (M_i ∩ S)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XBindGlobal( "UpperBoundFixedPointRatios",[127X[104X
    [4X[25X>[125X [27X   function( G, maxesclasses, truetest )[127X[104X
    [4X[25X>[125X [27X    local myIsConjugate, invs, info, c, r, o, inv, pos, sums, max, maxpos,[127X[104X
    [4X[25X>[125X [27X          maxlen, reps, split, i, found, j;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    myIsConjugate:= function( G, x, y )[127X[104X
    [4X[25X>[125X [27X      local movx, movy;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X      movx:= MovedPoints( x );[127X[104X
    [4X[25X>[125X [27X      movy:= MovedPoints( y );[127X[104X
    [4X[25X>[125X [27X      if movx = movy then[127X[104X
    [4X[25X>[125X [27X        G:= Stabilizer( G, movx, OnSets );[127X[104X
    [4X[25X>[125X [27X      fi;[127X[104X
    [4X[25X>[125X [27X      return IsConjugate( G, x, y );[127X[104X
    [4X[25X>[125X [27X    end;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    invs:= [];[127X[104X
    [4X[25X>[125X [27X    info:= [];[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    # First distribute the classes according to invariants.[127X[104X
    [4X[25X>[125X [27X    for c in Concatenation( maxesclasses ) do[127X[104X
    [4X[25X>[125X [27X      r:= Representative( c );[127X[104X
    [4X[25X>[125X [27X      o:= Order( r );[127X[104X
    [4X[25X>[125X [27X      # Take only prime order representatives.[127X[104X
    [4X[25X>[125X [27X      if IsPrimeInt( o ) then[127X[104X
    [4X[25X>[125X [27X        inv:= [ o, Size( Centralizer( G, r ) ) ];[127X[104X
    [4X[25X>[125X [27X        # Omit classes that are central in `G'.[127X[104X
    [4X[25X>[125X [27X        if inv[2] <> Size( G ) then[127X[104X
    [4X[25X>[125X [27X          if IsPerm( r ) then[127X[104X
    [4X[25X>[125X [27X            Add( inv, NrMovedPoints( r ) );[127X[104X
    [4X[25X>[125X [27X          fi;[127X[104X
    [4X[25X>[125X [27X          pos:= First( [ 1 .. Length( invs ) ], i -> inv = invs[i] );[127X[104X
    [4X[25X>[125X [27X          if pos = fail then[127X[104X
    [4X[25X>[125X [27X            # This class is not `G'-conjugate to any of the previous ones.[127X[104X
    [4X[25X>[125X [27X            Add( invs, inv );[127X[104X
    [4X[25X>[125X [27X            Add( info, [ [ r, Size( c ) * inv[2] ] ] );[127X[104X
    [4X[25X>[125X [27X          else[127X[104X
    [4X[25X>[125X [27X            # This class may be conjugate to an earlier one.[127X[104X
    [4X[25X>[125X [27X            Add( info[ pos ], [ r, Size( c ) * inv[2] ] );[127X[104X
    [4X[25X>[125X [27X          fi;[127X[104X
    [4X[25X>[125X [27X        fi;[127X[104X
    [4X[25X>[125X [27X      fi;[127X[104X
    [4X[25X>[125X [27X    od;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    if info = [] then[127X[104X
    [4X[25X>[125X [27X      return [ 0, true ];[127X[104X
    [4X[25X>[125X [27X    fi;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    repeat[127X[104X
    [4X[25X>[125X [27X      # Compute the contributions of the classes with the same invariants.[127X[104X
    [4X[25X>[125X [27X      sums:= List( info, x -> Sum( List( x, y -> y[2] ) ) );[127X[104X
    [4X[25X>[125X [27X      max:= Maximum( sums );[127X[104X
    [4X[25X>[125X [27X      maxpos:= Filtered( [ 1 .. Length( info ) ], i -> sums[i] = max );[127X[104X
    [4X[25X>[125X [27X      maxlen:= List( maxpos, i -> Length( info[i] ) );[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X      # Split the sets with the same invariants if necessary[127X[104X
    [4X[25X>[125X [27X      # and if we want to compute the exact value.[127X[104X
    [4X[25X>[125X [27X      if truetest and not 1 in maxlen then[127X[104X
    [4X[25X>[125X [27X        # Make one conjugacy test.[127X[104X
    [4X[25X>[125X [27X        pos:= Position( maxlen, Minimum( maxlen ) );[127X[104X
    [4X[25X>[125X [27X        reps:= info[ maxpos[ pos ] ];[127X[104X
    [4X[25X>[125X [27X        if myIsConjugate( G, reps[1][1], reps[2][1] ) then[127X[104X
    [4X[25X>[125X [27X          # Fuse the two classes.[127X[104X
    [4X[25X>[125X [27X          reps[1][2]:= reps[1][2] + reps[2][2];[127X[104X
    [4X[25X>[125X [27X          reps[2]:= reps[ Length( reps ) ];[127X[104X
    [4X[25X>[125X [27X          Unbind( reps[ Length( reps ) ] );[127X[104X
    [4X[25X>[125X [27X        else[127X[104X
    [4X[25X>[125X [27X          # Split the list. This may require additional conjugacy tests.[127X[104X
    [4X[25X>[125X [27X          Unbind( info[ maxpos[ pos ] ] );[127X[104X
    [4X[25X>[125X [27X          split:= [ reps[1], reps[2] ];[127X[104X
    [4X[25X>[125X [27X          for i in [ 3 .. Length( reps ) ] do[127X[104X
    [4X[25X>[125X [27X            found:= false;[127X[104X
    [4X[25X>[125X [27X            for j in split do[127X[104X
    [4X[25X>[125X [27X              if myIsConjugate( G, reps[i][1], j[1] ) then[127X[104X
    [4X[25X>[125X [27X                j[2]:= reps[i][2] + j[2];[127X[104X
    [4X[25X>[125X [27X                found:= true;[127X[104X
    [4X[25X>[125X [27X                break;[127X[104X
    [4X[25X>[125X [27X              fi;[127X[104X
    [4X[25X>[125X [27X            od;[127X[104X
    [4X[25X>[125X [27X            if not found then[127X[104X
    [4X[25X>[125X [27X              Add( split, reps[i] );[127X[104X
    [4X[25X>[125X [27X            fi;[127X[104X
    [4X[25X>[125X [27X          od;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X          info:= Compacted( Concatenation( info,[127X[104X
    [4X[25X>[125X [27X                                           List( split, x -> [ x ] ) ) );[127X[104X
    [4X[25X>[125X [27X        fi;[127X[104X
    [4X[25X>[125X [27X      fi;[127X[104X
    [4X[25X>[125X [27X    until 1 in maxlen or not truetest;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    return [ max / Size( G ), 1 in maxlen ];[127X[104X
    [4X[25X>[125X [27Xend );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YSuppose  that  [22XC_1, C_2, C_3[122X are conjugacy classes in [22XG[122X, and that we have to
  prove,  for  each  [22X(x_1,  x_2,  x_3)  ∈ C_1 × C_2 × C_3[122X, the existence of an
  element [22Xs[122X in a prescribed class [22XC[122X of [22XG[122X such that [22X⟨ x_1, s ⟩ = ⟨ x_2, s ⟩ = ⟨
  x_2, s ⟩ = G[122X holds.[133X
  
  [33X[0;0YWe  have  to check only representatives under the conjugation action of [22XG[122X on
  [22XC_1  ×  C_2  ×  C_3[122X.  For each representative, we try a prescribed number of
  random  elements in [22XC[122X. If this is successful then we are done. The following
  two functions implement this idea.[133X
  
  [33X[0;0YFor  a  group  [22XG[122X  and  a  list  [22X[  g_1,  g_2,  ...,  g_n ][122X of elements in [22XG[122X,
  [10XOrbitRepresentativesProductOfClasses[110X returns a list [22XR(G, g_1, g_2, ..., g_n)[122X
  of  representatives of [22XG[122X-orbits on the Cartesian product [22Xg_1^G × g_2^G × ⋯ ×
  g_n^G[122X.[133X
  
  [33X[0;0YThe  idea  behind  this function is to choose [22XR(G, g_1) = { ( g_1 ) }[122X in the
  case [22Xn = 1[122X, and, for [22Xn > 1[122X,[133X
  
  
  [24X[33X[0;6YR(G, g_1, g_2, ..., g_n) = { (h_1, h_2, ..., h_n) ∣ (h_1, h_2, ..., h_n-1) ∈ R(G, g_1, g_2, ..., g_n-1), h_n = g_n^d, for d ∈ D } ,[133X[124X
  
  [33X[0;0Ywhere  [22XD[122X  is  a  set  of  representatives  of  double  cosets [22XC_G(g_n) ∖ G /
  ∩_i=1^n-1 C_G(h_i)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XBindGlobal( "OrbitRepresentativesProductOfClasses",[127X[104X
    [4X[25X>[125X [27X   function( G, classreps )[127X[104X
    [4X[25X>[125X [27X    local cents, n, orbreps;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    cents:= List( classreps, x -> Centralizer( G, x ) );[127X[104X
    [4X[25X>[125X [27X    n:= Length( classreps );[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    orbreps:= function( reps, intersect, pos )[127X[104X
    [4X[25X>[125X [27X      if pos > n then[127X[104X
    [4X[25X>[125X [27X        return [ reps ];[127X[104X
    [4X[25X>[125X [27X      fi;[127X[104X
    [4X[25X>[125X [27X      return Concatenation( List([127X[104X
    [4X[25X>[125X [27X          DoubleCosetRepsAndSizes( G, cents[ pos ], intersect ),[127X[104X
    [4X[25X>[125X [27X            r -> orbreps( Concatenation( reps, [ classreps[ pos ]^r[1] ] ),[127X[104X
    [4X[25X>[125X [27X                 Intersection( intersect, cents[ pos ]^r[1] ), pos+1 ) ) );[127X[104X
    [4X[25X>[125X [27X    end;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    return orbreps( [ classreps[1] ], cents[1], 2 );[127X[104X
    [4X[25X>[125X [27Xend );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  function  [10XRandomCheckUniformSpread[110X takes a transitive permutation group
  [22XG[122X, a list of class representatives [22Xg_i ∈ G[122X, an element [22Xs ∈ G[122X, and a positive
  integer  [22XN[122X.  The return value is [9Xtrue[109X if for each representative of [22XG[122X-orbits
  on  the  product  of the classes [22Xg_i^G[122X, a good conjugate of [22Xs[122X is found in at
  most [22XN[122X random tests.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XBindGlobal( "RandomCheckUniformSpread", function( G, classreps, s, try )[127X[104X
    [4X[25X>[125X [27X    local elms, found, i, conj;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    if not IsTransitive( G, MovedPoints( G ) ) then[127X[104X
    [4X[25X>[125X [27X      Error( "<G> must be transitive on its moved points" );[127X[104X
    [4X[25X>[125X [27X    fi;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    # Compute orbit representatives of G on the direct product,[127X[104X
    [4X[25X>[125X [27X    # and try to find a good conjugate of s for each representative.[127X[104X
    [4X[25X>[125X [27X    for elms in OrbitRepresentativesProductOfClasses( G, classreps ) do[127X[104X
    [4X[25X>[125X [27X      found:= false;[127X[104X
    [4X[25X>[125X [27X      for i in [ 1 .. try ] do[127X[104X
    [4X[25X>[125X [27X        conj:= s^Random( G );[127X[104X
    [4X[25X>[125X [27X        if ForAll( elms,[127X[104X
    [4X[25X>[125X [27X              x -> IsGeneratorsOfTransPermGroup( G, [ x, conj ] ) ) then[127X[104X
    [4X[25X>[125X [27X          found:= true;[127X[104X
    [4X[25X>[125X [27X          break;[127X[104X
    [4X[25X>[125X [27X        fi;[127X[104X
    [4X[25X>[125X [27X      od;[127X[104X
    [4X[25X>[125X [27X      if not found then[127X[104X
    [4X[25X>[125X [27X        return elms;[127X[104X
    [4X[25X>[125X [27X      fi;[127X[104X
    [4X[25X>[125X [27X    od;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    return true;[127X[104X
    [4X[25X>[125X [27Xend );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YOf  course  this approach is not suitable for [13Xdis[113Xproving the existence of [22Xs[122X,
  but it is much cheaper than an exhaustive search in the class [22XC[122X. (Typically,
  [22X|C|[122X is large whereas the [22X|C_i|[122X are small.)[133X
  
  [33X[0;0YThe following function can be used to verify that a given [22Xn[122X-tuple [22X(x_1, x_2,
  ...,  x_n)[122X of elements in a group [22XG[122X has the property that for all elements [22Xg
  ∈  G[122X,  at least one [22Xx_i[122X satisfies [22X⟨ x_i, g ⟩[122X. The arguments are a transitive
  permutation  group  [22XG[122X, a list of class representatives in [22XG[122X, and the [22Xn[122X-tuple
  in  question. The return value is a conjugate [22Xg[122X of the given representatives
  that has the property if such an element exists, and [9Xfail[109X otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XBindGlobal( "CommonGeneratorWithGivenElements",[127X[104X
    [4X[25X>[125X [27X   function( G, classreps, tuple )[127X[104X
    [4X[25X>[125X [27X    local inter, rep, repcen, pair;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    if not IsTransitive( G, MovedPoints( G ) ) then[127X[104X
    [4X[25X>[125X [27X      Error( "<G> must be transitive on its moved points" );[127X[104X
    [4X[25X>[125X [27X    fi;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    inter:= Intersection( List( tuple, x -> Centralizer( G, x ) ) );[127X[104X
    [4X[25X>[125X [27X    for rep in classreps do[127X[104X
    [4X[25X>[125X [27X      repcen:= Centralizer( G, rep );[127X[104X
    [4X[25X>[125X [27X      for pair in DoubleCosetRepsAndSizes( G, repcen, inter ) do[127X[104X
    [4X[25X>[125X [27X        if ForAll( tuple,[127X[104X
    [4X[25X>[125X [27X           x -> IsGeneratorsOfTransPermGroup( G, [ x, rep^pair[1] ] ) ) then[127X[104X
    [4X[25X>[125X [27X          return rep;[127X[104X
    [4X[25X>[125X [27X        fi;[127X[104X
    [4X[25X>[125X [27X      od;[127X[104X
    [4X[25X>[125X [27X    od;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    return fail;[127X[104X
    [4X[25X>[125X [27Xend );[127X[104X
  [4X[32X[104X
  
  
  [1X11.4 [33X[0;0YCharacter-Theoretic Computations[133X[101X
  
  [33X[0;0YIn  this section, we apply the functions introduced in Section [14X11.3-2[114X to the
  character  tables  of  simple groups that are available in the [5XGAP[105X Character
  Table Library.[133X
  
  [33X[0;0YOur  first  examples are the sporadic simple groups, in Section [14X11.4-1[114X, then
  their automorphism groups are considered in Section [14X11.4-2[114X.[133X
  
  [33X[0;0YThen  we  consider  those  other simple groups for which [5XGAP[105X provides enough
  information  for  automatically  computing  an  upper  bound  on [22Xσ(G,s)[122X –see
  Section [14X11.4-3[114X– and their automorphic extensions –see Section [14X11.4-4[114X.[133X
  
  [33X[0;0YAfter that, individual groups are considered.[133X
  
  
  [1X11.4-1 [33X[0;0YSporadic Simple Groups[133X[101X
  
  [33X[0;0YThe  [5XGAP[105X Character Table Library contains the tables of maximal subgroups of
  all  sporadic  simple  groups  except  [22XB[122X and [22XM[122X, so all primitive permutation
  characters  can  be computed via the function [10XPrimitivePermutationCharacters[110X
  for [22X24[122X of the [22X26[122X sporadic simple groups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsporinfo:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xspornames:= AllCharacterTableNames( IsSporadicSimple, true,[127X[104X
    [4X[25X>[125X [27X                                       IsDuplicateTable, false );;[127X[104X
    [4X[25Xgap>[125X [27Xfor tbl in List( spornames, CharacterTable ) do[127X[104X
    [4X[25X>[125X [27X     info:= ProbGenInfoSimple( tbl );[127X[104X
    [4X[25X>[125X [27X     if info <> fail then[127X[104X
    [4X[25X>[125X [27X       Add( sporinfo, info );[127X[104X
    [4X[25X>[125X [27X     fi;[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YWe show the result as a formatted table.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPrintFormattedArray( sporinfo );[127X[104X
    [4X[28X   Co1    421/1545600         3671        [ "35A" ]    [ 4 ][128X[104X
    [4X[28X   Co2          1/270          269        [ "23A" ]    [ 1 ][128X[104X
    [4X[28X   Co3        64/6325           98        [ "21A" ]    [ 4 ][128X[104X
    [4X[28X   F3+ 1/269631216855 269631216854        [ "29A" ]    [ 1 ][128X[104X
    [4X[28X  Fi22         43/585           13        [ "16A" ]    [ 7 ][128X[104X
    [4X[28X  Fi23   2651/2416635          911        [ "23A" ]    [ 2 ][128X[104X
    [4X[28X    HN        4/34375         8593        [ "19A" ]    [ 1 ][128X[104X
    [4X[28X    HS        64/1155           18        [ "15A" ]    [ 2 ][128X[104X
    [4X[28X    He          3/595          198        [ "14C" ]    [ 3 ][128X[104X
    [4X[28X    J1           1/77           76        [ "19A" ]    [ 1 ][128X[104X
    [4X[28X    J2           5/28            5        [ "10C" ]    [ 3 ][128X[104X
    [4X[28X    J3          2/153           76        [ "19A" ]    [ 2 ][128X[104X
    [4X[28X    J4   1/1647124116   1647124115        [ "29A" ]    [ 1 ][128X[104X
    [4X[28X    Ly     1/35049375     35049374        [ "37A" ]    [ 1 ][128X[104X
    [4X[28X   M11            1/3            2        [ "11A" ]    [ 1 ][128X[104X
    [4X[28X   M12            1/3            2        [ "10A" ]    [ 3 ][128X[104X
    [4X[28X   M22           1/21           20        [ "11A" ]    [ 1 ][128X[104X
    [4X[28X   M23         1/8064         8063        [ "23A" ]    [ 1 ][128X[104X
    [4X[28X   M24       108/1265           11        [ "21A" ]    [ 2 ][128X[104X
    [4X[28X   McL      317/22275           70 [ "15A", "30A" ] [ 3, 3 ][128X[104X
    [4X[28X    ON       10/30723         3072        [ "31A" ]    [ 2 ][128X[104X
    [4X[28X    Ru         1/2880         2879        [ "29A" ]    [ 1 ][128X[104X
    [4X[28X   Suz       141/5720           40        [ "14A" ]    [ 3 ][128X[104X
    [4X[28X    Th       2/267995       133997 [ "27A", "27B" ] [ 2, 2 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  see  that  in  all  these cases, [22Xσ(G) < 1/2[122X and thus [22XP( G ) ≥ 2[122X, and all
  sporadic  simple  groups  [22XG[122X except [22XG = M_11[122X and [22XG = M_12[122X satisfy [22Xσ(G) < 1/3[122X.
  See [14X11.5-9[114X  and [14X11.5-10[114X  for a proof that also these two groups have uniform
  spread at least three.[133X
  
  [33X[0;0YThe  structures and multiplicities of the maximal subgroups containing [22Xs[122X are
  as follows.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfor entry in sporinfo do[127X[104X
    [4X[25X>[125X [27X     DisplayProbGenMaxesInfo( CharacterTable( entry[1] ), entry[4] );[127X[104X
    [4X[25X>[125X [27Xod;[127X[104X
    [4X[28XCo1, 35A: (A5xJ2):2  (1)[128X[104X
    [4X[28X          (A6xU3(3)):2  (2)[128X[104X
    [4X[28X          (A7xL2(7)):2  (1)[128X[104X
    [4X[28XCo2, 23A: M23  (1)[128X[104X
    [4X[28XCo3, 21A: U3(5).3.2  (2)[128X[104X
    [4X[28X          L3(4).D12  (1)[128X[104X
    [4X[28X          s3xpsl(2,8).3  (1)[128X[104X
    [4X[28XF3+, 29A: 29:14  (1)[128X[104X
    [4X[28XFi22, 16A: 2^10:m22  (1)[128X[104X
    [4X[28X           (2x2^(1+8)):U4(2):2  (1)[128X[104X
    [4X[28X           2F4(2)'  (4)[128X[104X
    [4X[28X           2^(5+8):(S3xA6)  (1)[128X[104X
    [4X[28XFi23, 23A: 2..11.m23  (1)[128X[104X
    [4X[28X           L2(23)  (1)[128X[104X
    [4X[28XHN, 19A: U3(8).3_1  (1)[128X[104X
    [4X[28XHS, 15A: A8.2  (1)[128X[104X
    [4X[28X         5:4xa5  (1)[128X[104X
    [4X[28XHe, 14C: 2^1+6.psl(3,2)  (1)[128X[104X
    [4X[28X         7^2:2psl(2,7)  (1)[128X[104X
    [4X[28X         7^(1+2):(S3x3)  (1)[128X[104X
    [4X[28XJ1, 19A: 19:6  (1)[128X[104X
    [4X[28XJ2, 10C: 2^1+4b:a5  (1)[128X[104X
    [4X[28X         a5xd10  (1)[128X[104X
    [4X[28X         5^2:D12  (1)[128X[104X
    [4X[28XJ3, 19A: L2(19)  (1)[128X[104X
    [4X[28X         J3M3  (1)[128X[104X
    [4X[28XJ4, 29A: frob  (1)[128X[104X
    [4X[28XLy, 37A: 37:18  (1)[128X[104X
    [4X[28XM11, 11A: L2(11)  (1)[128X[104X
    [4X[28XM12, 10A: A6.2^2  (1)[128X[104X
    [4X[28X          M12M4  (1)[128X[104X
    [4X[28X          2xS5  (1)[128X[104X
    [4X[28XM22, 11A: L2(11)  (1)[128X[104X
    [4X[28XM23, 23A: 23:11  (1)[128X[104X
    [4X[28XM24, 21A: L3(4).3.2_2  (1)[128X[104X
    [4X[28X          2^6:(psl(3,2)xs3)  (1)[128X[104X
    [4X[28XMcL, 15A: 3^(1+4):2S5  (1)[128X[104X
    [4X[28X          2.A8  (1)[128X[104X
    [4X[28X          5^(1+2):3:8  (1)[128X[104X
    [4X[28XMcL, 30A: 3^(1+4):2S5  (1)[128X[104X
    [4X[28X          2.A8  (1)[128X[104X
    [4X[28X          5^(1+2):3:8  (1)[128X[104X
    [4X[28XON, 31A: L2(31)  (1)[128X[104X
    [4X[28X         ONM8  (1)[128X[104X
    [4X[28XRu, 29A: L2(29)  (1)[128X[104X
    [4X[28XSuz, 14A: J2.2  (2)[128X[104X
    [4X[28X          (a4xpsl(3,4)):2  (1)[128X[104X
    [4X[28XTh, 27A: ThN3B  (1)[128X[104X
    [4X[28X         ThM7  (1)[128X[104X
    [4X[28XTh, 27B: ThN3B  (1)[128X[104X
    [4X[28X         ThM7  (1)[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  the  remaining  two sporadic simple groups, [22XB[122X and [22XM[122X, we choose suitable
  elements  [22Xs[122X. If [22XG = B[122X and [22Xs ∈ G[122X is of order [22X47[122X then, by [Wil99], [22XMM(G,s) = {
  47:23 }[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSigmaFromMaxes( CharacterTable( "B" ), "47A",[127X[104X
    [4X[25X>[125X [27X       [ CharacterTable( "47:23" ) ], [ 1 ] );[127X[104X
    [4X[28X1/174702778623598780219392000000[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIf [22XG = M[122X and [22Xs ∈ G[122X is of order [22X59[122X then, by [HW04], [22XMM(G,s) = { L_2(59) }[122X. In
  this  case,  the  permutation  character  is  not uniquely determined by the
  character tables, but all possibilities lead to the same value for [22Xσ(G)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "M" );;[127X[104X
    [4X[25Xgap>[125X [27Xs:= CharacterTable( "L2(59)" );;[127X[104X
    [4X[25Xgap>[125X [27Xpi:= PossiblePermutationCharacters( s, t );;[127X[104X
    [4X[25Xgap>[125X [27XLength( pi );[127X[104X
    [4X[28X5[128X[104X
    [4X[25Xgap>[125X [27Xspos:= Position( OrdersClassRepresentatives( t ), 59 );[127X[104X
    [4X[28X152[128X[104X
    [4X[25Xgap>[125X [27XSet( pi, x -> Maximum( ApproxP( [ x ], spos ) ) );[127X[104X
    [4X[28X[ 1/3385007637938037777290625 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YEssentially  the  same  approach  is  taken  in [GM01].  However, there [22Xs[122X is
  restricted  to  classes  of prime order. Thus the results in the above table
  are  better  for [22XJ_2[122X, [22XHS[122X, [22XM_24[122X, [22XMcL[122X, [22XHe[122X, [22XSuz[122X, [22XCo_3[122X, [22XFi_22[122X, [22XLy[122X, [22XTh[122X, [22XCo_1[122X, and
  [22XJ_4[122X.  Besides that, the value [22X10999[122X claimed in [GM01] for [22Xtotal( HN )[122X is not
  correct.[133X
  
  
  [1X11.4-2 [33X[0;0YAutomorphism Groups of Sporadic Simple Groups[133X[101X
  
  [33X[0;0YNext  we  consider  the  automorphism  groups of the sporadic simple groups.
  There  are  exactly [22X12[122X cases where nontrivial outer automorphisms exist, and
  then the simple group [22XS[122X has index [22X2[122X in its automorphism group [22XG[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsporautnames:= AllCharacterTableNames( IsSporadicSimple, true,[127X[104X
    [4X[25X>[125X [27X                      IsDuplicateTable, false,[127X[104X
    [4X[25X>[125X [27X                      OfThose, AutomorphismGroup );;[127X[104X
    [4X[25Xgap>[125X [27Xsporautnames:= Difference( sporautnames, spornames );[127X[104X
    [4X[28X[ "F3+.2", "Fi22.2", "HN.2", "HS.2", "He.2", "J2.2", "J3.2", "M12.2", [128X[104X
    [4X[28X  "M22.2", "McL.2", "ON.2", "Suz.2" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFirst  we  compute  the values [22Xσ^'(G,s)[122X, for the same [22Xs ∈ S[122X that were chosen
  for the simple group [22XS[122X in Section [14X11.4-1[114X.[133X
  
  [33X[0;0YFor  six  of  the  groups [22XG[122X in question, the character tables of all maximal
  subgroups  are available in the [5XGAP[105X Character Table Library. In these cases,
  the values [22Xσ^'( G, s )[122X can be computed using [10XProbGenInfoAlmostSimple[110X.[133X
  
  [33X[0;0Y[13X(The  above  statement  can  meanwhile be replaced by the statement that the
  character  tables  of  all  maximal  subgroups  are available for all twelve
  groups.  We  show  the  table  results  for  all  these  groups but keep the
  individual computations from the original computations.)[113X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsporautinfo:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xfails:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xfor name in sporautnames do[127X[104X
    [4X[25X>[125X [27X     tbl:= CharacterTable( name{ [ 1 .. Position( name, '.' ) - 1 ] } );[127X[104X
    [4X[25X>[125X [27X     tblG:= CharacterTable( name );[127X[104X
    [4X[25X>[125X [27X     info:= ProbGenInfoSimple( tbl );[127X[104X
    [4X[25X>[125X [27X     info:= ProbGenInfoAlmostSimple( tbl, tblG,[127X[104X
    [4X[25X>[125X [27X         List( info[4], x -> Position( AtlasClassNames( tbl ), x ) ) );[127X[104X
    [4X[25X>[125X [27X     if info = fail then[127X[104X
    [4X[25X>[125X [27X       Add( fails, name );[127X[104X
    [4X[25X>[125X [27X     else[127X[104X
    [4X[25X>[125X [27X       Add( sporautinfo, info );[127X[104X
    [4X[25X>[125X [27X     fi;[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27XPrintFormattedArray( sporautinfo );[127X[104X
    [4X[28X   F3+.2         0         [ "29AB" ]    [ 1 ][128X[104X
    [4X[28X  Fi22.2  251/3861         [ "16AB" ]    [ 7 ][128X[104X
    [4X[28X    HN.2    1/6875         [ "19AB" ]    [ 1 ][128X[104X
    [4X[28X    HS.2    36/275          [ "15A" ]    [ 2 ][128X[104X
    [4X[28X    He.2   37/9520         [ "14CD" ]    [ 3 ][128X[104X
    [4X[28X    J2.2      1/15         [ "10CD" ]    [ 3 ][128X[104X
    [4X[28X    J3.2    1/1080         [ "19AB" ]    [ 1 ][128X[104X
    [4X[28X   M12.2      4/99          [ "10A" ]    [ 1 ][128X[104X
    [4X[28X   M22.2      1/21         [ "11AB" ]    [ 1 ][128X[104X
    [4X[28X   McL.2      1/63 [ "15AB", "30AB" ] [ 3, 3 ][128X[104X
    [4X[28X    ON.2   1/84672         [ "31AB" ]    [ 1 ][128X[104X
    [4X[28X   Suz.2 661/46332          [ "14A" ]    [ 3 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote that for [22XS = McL[122X, the bound [22Xσ^'(G,s)[122X for [22XG = S.2[122X (in the second column)
  is worse than the bound for the simple group [22XS[122X.[133X
  
  [33X[0;0YThe  structures and multiplicities of the maximal subgroups containing [22Xs[122X are
  as follows.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfor entry in sporautinfo do[127X[104X
    [4X[25X>[125X [27X     DisplayProbGenMaxesInfo( CharacterTable( entry[1] ), entry[3] );[127X[104X
    [4X[25X>[125X [27Xod;[127X[104X
    [4X[28XF3+.2, 29AB: F3+  (1)[128X[104X
    [4X[28X             frob  (1)[128X[104X
    [4X[28XFi22.2, 16AB: Fi22  (1)[128X[104X
    [4X[28X              Fi22.2M4  (1)[128X[104X
    [4X[28X              (2x2^(1+8)):(U4(2):2x2)  (1)[128X[104X
    [4X[28X              2F4(2)'.2  (4)[128X[104X
    [4X[28X              2^(5+8):(S3xS6)  (1)[128X[104X
    [4X[28XHN.2, 19AB: HN  (1)[128X[104X
    [4X[28X            U3(8).6  (1)[128X[104X
    [4X[28XHS.2, 15A: HS  (1)[128X[104X
    [4X[28X           S8x2  (1)[128X[104X
    [4X[28X           5:4xS5  (1)[128X[104X
    [4X[28XHe.2, 14CD: He  (1)[128X[104X
    [4X[28X            2^(1+6)_+.L3(2).2  (1)[128X[104X
    [4X[28X            7^2:2.L2(7).2  (1)[128X[104X
    [4X[28X            7^(1+2):(S3x6)  (1)[128X[104X
    [4X[28XJ2.2, 10CD: J2  (1)[128X[104X
    [4X[28X            2^(1+4).S5  (1)[128X[104X
    [4X[28X            (A5xD10).2  (1)[128X[104X
    [4X[28X            5^2:(4xS3)  (1)[128X[104X
    [4X[28XJ3.2, 19AB: J3  (1)[128X[104X
    [4X[28X            19:18  (1)[128X[104X
    [4X[28XM12.2, 10A: M12  (1)[128X[104X
    [4X[28X            (2^2xA5):2  (1)[128X[104X
    [4X[28XM22.2, 11AB: M22  (1)[128X[104X
    [4X[28X             L2(11).2  (1)[128X[104X
    [4X[28XMcL.2, 15AB: McL  (1)[128X[104X
    [4X[28X             3^(1+4):4S5  (1)[128X[104X
    [4X[28X             Isoclinic(2.A8.2)  (1)[128X[104X
    [4X[28X             5^(1+2):(24:2)  (1)[128X[104X
    [4X[28XMcL.2, 30AB: McL  (1)[128X[104X
    [4X[28X             3^(1+4):4S5  (1)[128X[104X
    [4X[28X             Isoclinic(2.A8.2)  (1)[128X[104X
    [4X[28X             5^(1+2):(24:2)  (1)[128X[104X
    [4X[28XON.2, 31AB: ON  (1)[128X[104X
    [4X[28X            31:30  (1)[128X[104X
    [4X[28XSuz.2, 14A: Suz  (1)[128X[104X
    [4X[28X            J2.2x2  (2)[128X[104X
    [4X[28X            (A4xL3(4):2_3):2  (1)[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that  the  maximal subgroups [22XL_2(19)[122X of [22XJ_3[122X do not extend to [22XJ_3.2[122X and
  that  a  class of maximal subgroups of the type [22X19:18[122X appears in [22XJ_3.2[122X whose
  intersection  with  [22XJ_3[122X  is  not  maximal  in  [22XJ_3[122X.  Similarly,  the maximal
  subgroups [22XA_6.2^2[122X of [22XM_12[122X do not extend to [22XM_12.2[122X.[133X
  
  [33X[0;0YFor the other six groups, we use individual computations.[133X
  
  [33X[0;0YIn  the case [22XS = Fi_24^'[122X, the unique maximal subgroup [22X29:14[122X that contains an
  element  [22Xs[122X  of order [22X29[122X extends to a group of the type [22X29:28[122X in [22XFi_24[122X, which
  is a nonsplit extension of [22X29:14[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSigmaFromMaxes( CharacterTable( "Fi24'.2" ), "29AB",[127X[104X
    [4X[25X>[125X [27X       [ CharacterTable( "29:28" ) ], [ 1 ], "outer" );[127X[104X
    [4X[28X0[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  the  case  [22XS  =  Fi_22[122X, there are four classes of maximal subgroups that
  contain [22Xs[122X of order [22X16[122X. They extend to [22XG = Fi_22.2[122X, and none of the [13Xnovelties[113X
  in  [22XG[122X  (i. e.,  subgroups of [22XG[122X that are maximal in [22XG[122X but whose intersections
  with [22XS[122X are not maximal in [22XS[122X) contains [22Xs[122X, cf. [CCN+85, p. 163].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27X16 in OrdersClassRepresentatives( CharacterTable( "U4(2).2" ) );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27X16 in OrdersClassRepresentatives( CharacterTable( "G2(3).2" ) );[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  character  tables  of three of the four extensions are available in the
  [5XGAP[105X  Character Table Library. The permutation character on the cosets of the
  fourth  extension  can  be  obtained  as  the  extension  of the permutation
  character  of [22XS[122X on the cosets of its maximal subgroup of the type [22X2^5+8:(S_3
  × A_6)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt2:= CharacterTable( "Fi22.2" );;[127X[104X
    [4X[25Xgap>[125X [27Xprim:= List( [ "Fi22.2M4", "(2x2^(1+8)):(U4(2):2x2)", "2F4(2)" ],[127X[104X
    [4X[25X>[125X [27X       n -> PossiblePermutationCharacters( CharacterTable( n ), t2 ) );;[127X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "Fi22" );;[127X[104X
    [4X[25Xgap>[125X [27Xpi:= PossiblePermutationCharacters([127X[104X
    [4X[25X>[125X [27X            CharacterTable( "2^(5+8):(S3xA6)" ), t );[127X[104X
    [4X[28X[ Character( CharacterTable( "Fi22" ),[128X[104X
    [4X[28X  [ 3648645, 56133, 10629, 2245, 567, 729, 405, 81, 549, 165, 133, [128X[104X
    [4X[28X      37, 69, 20, 27, 81, 9, 39, 81, 19, 1, 13, 33, 13, 1, 0, 13, 13, [128X[104X
    [4X[28X      5, 1, 0, 0, 0, 8, 4, 0, 0, 9, 3, 15, 3, 1, 1, 1, 1, 3, 3, 1, 0, [128X[104X
    [4X[28X      0, 0, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2 ] ) ][128X[104X
    [4X[25Xgap>[125X [27Xtorso:= CompositionMaps( pi[1], InverseMap( GetFusionMap( t, t2 ) ) );[127X[104X
    [4X[28X[ 3648645, 56133, 10629, 2245, 567, 729, 405, 81, 549, 165, 133, 37, [128X[104X
    [4X[28X  69, 20, 27, 81, 9, 39, 81, 19, 1, 13, 33, 13, 1, 0, 13, 13, 5, 1, [128X[104X
    [4X[28X  0, 0, 0, 8, 4, 0, 9, 3, 15, 3, 1, 1, 1, 3, 3, 1, 0, 0, 2, 1, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 1, 1, 2 ][128X[104X
    [4X[25Xgap>[125X [27Xext:= PermChars( t2, rec( torso:= torso ) );;[127X[104X
    [4X[25Xgap>[125X [27XAdd( prim, ext );[127X[104X
    [4X[25Xgap>[125X [27Xprim:= Concatenation( prim );;  Length( prim );[127X[104X
    [4X[28X4[128X[104X
    [4X[25Xgap>[125X [27Xspos:= Position( OrdersClassRepresentatives( t2 ), 16 );;[127X[104X
    [4X[25Xgap>[125X [27XList( prim, x -> x[ spos ] );[127X[104X
    [4X[28X[ 1, 1, 4, 1 ][128X[104X
    [4X[25Xgap>[125X [27Xsigma:= ApproxP( prim, spos );;[127X[104X
    [4X[25Xgap>[125X [27XMaximum( sigma{ Difference( PositionsProperty([127X[104X
    [4X[25X>[125X [27X                       OrdersClassRepresentatives( t2 ), IsPrimeInt ),[127X[104X
    [4X[25X>[125X [27X                       ClassPositionsOfDerivedSubgroup( t2 ) ) } );[127X[104X
    [4X[28X251/3861[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  the  case [22XS = HN[122X, the unique maximal subgroup [22XU_3(8).3[122X that contains the
  fixed element [22Xs[122X of order [22X19[122X extends to a group of the type [22XU_3(8).6[122X in [22XHN.2[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSigmaFromMaxes( CharacterTable( "HN.2" ), "19AB",[127X[104X
    [4X[25X>[125X [27X       [ CharacterTable( "U3(8).6" ) ], [ 1 ], "outer" );[127X[104X
    [4X[28X1/6875[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  the case [22XS = HS[122X, there are two classes of maximal subgroups that contain
  [22Xs[122X  of  order  [22X15[122X.  They  extend  to [22XG = HS.2[122X, and none of the novelties in [22XG[122X
  contains [22Xs[122X (cf. [CCN+85, p. 80]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSigmaFromMaxes( CharacterTable( "HS.2" ), "15A",[127X[104X
    [4X[25X>[125X [27X     [ CharacterTable( "S8x2" ),[127X[104X
    [4X[25X>[125X [27X       CharacterTable( "5:4" ) * CharacterTable( "A5.2" ) ], [ 1, 1 ],[127X[104X
    [4X[25X>[125X [27X     "outer" );[127X[104X
    [4X[28X36/275[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  the  case  [22XS  =  He[122X,  there  are three classes of maximal subgroups that
  contain  [22Xs[122X  in  the  class  [10X14C[110X.  They  extend  to [22XG = He.2[122X, and none of the
  novelties  in [22XG[122X contains [22Xs[122X (cf. [CCN+85, p. 104]). We compute the extensions
  of the corresponding primitive permutation characters of [22XS[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "He" );;[127X[104X
    [4X[25Xgap>[125X [27Xt2:= CharacterTable( "He.2" );;[127X[104X
    [4X[25Xgap>[125X [27Xprim:= PrimitivePermutationCharacters( t );;[127X[104X
    [4X[25Xgap>[125X [27Xspos:= Position( AtlasClassNames( t ), "14C" );;[127X[104X
    [4X[25Xgap>[125X [27Xprim:= Filtered( prim, x -> x[ spos ] <> 0 );;[127X[104X
    [4X[25Xgap>[125X [27Xmap:= InverseMap( GetFusionMap( t, t2 ) );;[127X[104X
    [4X[25Xgap>[125X [27Xtorso:= List( prim, pi -> CompositionMaps( pi, map ) );[127X[104X
    [4X[28X[ [ 187425, 945, 449, 0, 21, 21, 25, 25, 0, 0, 5, 0, 0, 7, 1, 0, 0, [128X[104X
    [4X[28X      1, 0, 1, 0, 0, 0, 0, 0, 0 ], [128X[104X
    [4X[28X  [ 244800, 0, 64, 0, 84, 0, 0, 16, 0, 0, 4, 24, 45, 3, 4, 0, 0, 0, [128X[104X
    [4X[28X      0, 1, 0, 0, 0, 0, 0, 0 ], [128X[104X
    [4X[28X  [ 652800, 0, 512, 120, 72, 0, 0, 0, 0, 0, 8, 8, 22, 1, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 1, 0, 0, 1, 1, 2, 0 ] ][128X[104X
    [4X[25Xgap>[125X [27Xext:= List( torso, x -> PermChars( t2, rec( torso:= x ) ) );[127X[104X
    [4X[28X[ [ Character( CharacterTable( "He.2" ),[128X[104X
    [4X[28X      [ 187425, 945, 449, 0, 21, 21, 25, 25, 0, 0, 5, 0, 0, 7, 1, 0, [128X[104X
    [4X[28X          0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 315, 15, 0, 0, 3, 7, 7, 3, 0, [128X[104X
    [4X[28X          0, 0, 1, 1, 0, 1, 1, 0, 0, 0 ] ) ], [128X[104X
    [4X[28X  [ Character( CharacterTable( "He.2" ),[128X[104X
    [4X[28X      [ 244800, 0, 64, 0, 84, 0, 0, 16, 0, 0, 4, 24, 45, 3, 4, 0, 0, [128X[104X
    [4X[28X          0, 0, 1, 0, 0, 0, 0, 0, 0, 360, 0, 0, 0, 6, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X          3, 2, 2, 0, 0, 0, 0, 0, 0 ] ) ], [128X[104X
    [4X[28X  [ Character( CharacterTable( "He.2" ),[128X[104X
    [4X[28X      [ 652800, 0, 512, 120, 72, 0, 0, 0, 0, 0, 8, 8, 22, 1, 0, 0, 0, [128X[104X
    [4X[28X          0, 0, 1, 0, 0, 1, 1, 2, 0, 480, 0, 120, 0, 12, 0, 0, 0, 0, [128X[104X
    [4X[28X          0, 4, 0, 0, 0, 0, 0, 0, 1, 1 ] ) ] ][128X[104X
    [4X[25Xgap>[125X [27Xspos:= Position( AtlasClassNames( t2 ), "14CD" );;[127X[104X
    [4X[25Xgap>[125X [27Xsigma:= ApproxP( Concatenation( ext ), spos );;[127X[104X
    [4X[25Xgap>[125X [27XMaximum( sigma{ Difference( PositionsProperty([127X[104X
    [4X[25X>[125X [27X                       OrdersClassRepresentatives( t2 ), IsPrimeInt ),[127X[104X
    [4X[25X>[125X [27X                       ClassPositionsOfDerivedSubgroup( t2 ) ) } );[127X[104X
    [4X[28X37/9520[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  the  case  [22XS  =  O'N[122X,  the  two classes of maximal subgroups of the type
  [22XL_2(31)[122X  do  not  extend  to  [22XG  =  O'N.2[122X,  and  a class of novelties of the
  structure [22X31:30[122X appears (see [CCN+85, p. 132]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSigmaFromMaxes( CharacterTable( "ON.2" ), "31AB",[127X[104X
    [4X[25X>[125X [27X       [ CharacterTable( "P:Q", [ 31, 30 ] ) ], [ 1 ], "outer" );[127X[104X
    [4X[28X1/84672[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow we consider also [22Xσ(G,hats)[122X, for suitable [22Xhats ∈ G ∖ S[122X; this yields lower
  bounds for the spread of the nonsimple groups [22XG[122X. (These results are shown in
  the last two columns of [BGK08, Table 9].)[133X
  
  [33X[0;0YAs  above,  we  use  the  known character tables of the maximal subgroups in
  order  to  compute  the  optimal  choice  for  [22Xhats ∈ G ∖ S[122X. (We may use the
  function  [10XProbGenInfoSimple[110X  although the groups are not simple; all we need
  is that the relevant maximal subgroups are self-normalizing.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsporautinfo2:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xfor name in List( sporautinfo, x -> x[1] ) do[127X[104X
    [4X[25X>[125X [27X     Add( sporautinfo2, ProbGenInfoSimple( CharacterTable( name ) ) );[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27XPrintFormattedArray( sporautinfo2 );[127X[104X
    [4X[28X   F3+.2    19/5684  299        [ "42E" ]   [ 10 ][128X[104X
    [4X[28X  Fi22.2 1165/20592   17        [ "24G" ]    [ 3 ][128X[104X
    [4X[28X    HN.2     1/1425 1424        [ "24B" ]    [ 4 ][128X[104X
    [4X[28X    HS.2     21/550   26        [ "20C" ]    [ 4 ][128X[104X
    [4X[28X    He.2    33/4165  126        [ "24A" ]    [ 2 ][128X[104X
    [4X[28X    J2.2       1/15   14        [ "14A" ]    [ 1 ][128X[104X
    [4X[28X    J3.2   77/10260  133        [ "34A" ]    [ 1 ][128X[104X
    [4X[28X   M12.2    113/495    4        [ "12B" ]    [ 3 ][128X[104X
    [4X[28X   M22.2       8/33    4        [ "10A" ]    [ 4 ][128X[104X
    [4X[28X   McL.2      1/135  134        [ "22A" ]    [ 1 ][128X[104X
    [4X[28X    ON.2  61/109368 1792 [ "22A", "38A" ] [ 1, 1 ][128X[104X
    [4X[28X   Suz.2      1/351  350        [ "28A" ]    [ 1 ][128X[104X
    [4X[25Xgap>[125X [27Xfor entry in sporautinfo2 do[127X[104X
    [4X[25X>[125X [27X     DisplayProbGenMaxesInfo( CharacterTable( entry[1] ), entry[4] );[127X[104X
    [4X[25X>[125X [27Xod;[127X[104X
    [4X[28XF3+.2, 42E: 2^12.M24  (2)[128X[104X
    [4X[28X            2^2.U6(2):S3x2  (1)[128X[104X
    [4X[28X            2^(3+12).(L3(2)xS6)  (2)[128X[104X
    [4X[28X            (S3xS3xG2(3)):2  (1)[128X[104X
    [4X[28X            S6xL2(8):3  (1)[128X[104X
    [4X[28X            7:6xS7  (1)[128X[104X
    [4X[28X            7^(1+2)_+:(6xS3).2  (2)[128X[104X
    [4X[28XFi22.2, 24G: Fi22.2M4  (1)[128X[104X
    [4X[28X             2^(5+8):(S3xS6)  (1)[128X[104X
    [4X[28X             3^5:(2xU4(2).2)  (1)[128X[104X
    [4X[28XHN.2, 24B: 2^(1+8)_+.(A5xA5).2^2  (1)[128X[104X
    [4X[28X           5^2.5.5^2.4S5  (2)[128X[104X
    [4X[28X           HN.2M13  (1)[128X[104X
    [4X[28XHS.2, 20C: (2xA6.2^2).2  (1)[128X[104X
    [4X[28X           HS.2N5  (2)[128X[104X
    [4X[28X           5:4xS5  (1)[128X[104X
    [4X[28XHe.2, 24A: 2^(1+6)_+.L3(2).2  (1)[128X[104X
    [4X[28X           S4xL3(2).2  (1)[128X[104X
    [4X[28XJ2.2, 14A: L3(2).2x2  (1)[128X[104X
    [4X[28XJ3.2, 34A: L2(17)x2  (1)[128X[104X
    [4X[28XM12.2, 12B: L2(11).2  (1)[128X[104X
    [4X[28X            2^3.(S4x2)  (1)[128X[104X
    [4X[28X            3^(1+2):D8  (1)[128X[104X
    [4X[28XM22.2, 10A: M22.2M4  (1)[128X[104X
    [4X[28X            A6.2^2  (1)[128X[104X
    [4X[28X            L2(11).2  (2)[128X[104X
    [4X[28XMcL.2, 22A: 2xM11  (1)[128X[104X
    [4X[28XON.2, 22A: J1x2  (1)[128X[104X
    [4X[28XON.2, 38A: J1x2  (1)[128X[104X
    [4X[28XSuz.2, 28A: (A4xL3(4):2_3):2  (1)[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  the  other  six  cases,  we  do not have the complete lists of primitive
  permutation characters, so we choose a suitable element [22Xhats[122X for each group.
  It is sufficient to prescribe [22X|hats|[122X, as follows.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsporautchoices:= [[127X[104X
    [4X[25X>[125X [27X       [ "Fi22",  "Fi22.2",  42 ],[127X[104X
    [4X[25X>[125X [27X       [ "Fi24'", "Fi24'.2", 46 ],[127X[104X
    [4X[25X>[125X [27X       [ "He",    "He.2",    42 ],[127X[104X
    [4X[25X>[125X [27X       [ "HN",    "HN.2",    44 ],[127X[104X
    [4X[25X>[125X [27X       [ "HS",    "HS.2",    30 ],[127X[104X
    [4X[25X>[125X [27X       [ "ON",    "ON.2",    38 ], ];;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YFirst  we list the maximal subgroups of the corresponding simple groups that
  contain the square of [22Xhats[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfor triple in sporautchoices do[127X[104X
    [4X[25X>[125X [27X     tbl:= CharacterTable( triple[1] );[127X[104X
    [4X[25X>[125X [27X     tbl2:= CharacterTable( triple[2] );[127X[104X
    [4X[25X>[125X [27X     spos2:= PowerMap( tbl2, 2,[127X[104X
    [4X[25X>[125X [27X         Position( OrdersClassRepresentatives( tbl2 ), triple[3] ) );[127X[104X
    [4X[25X>[125X [27X     spos:= Position( GetFusionMap( tbl, tbl2 ), spos2 );[127X[104X
    [4X[25X>[125X [27X     DisplayProbGenMaxesInfo( tbl, AtlasClassNames( tbl ){ [ spos ] } );[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[28XFi22, 21A: O8+(2).3.2  (1)[128X[104X
    [4X[28X           S3xU4(3).2_2  (1)[128X[104X
    [4X[28X           A10.2  (1)[128X[104X
    [4X[28X           A10.2  (1)[128X[104X
    [4X[28XF3+, 23A: Fi23  (1)[128X[104X
    [4X[28X          F3+M7  (1)[128X[104X
    [4X[28XHe, 21B: 3.A7.2  (1)[128X[104X
    [4X[28X         7^(1+2):(S3x3)  (1)[128X[104X
    [4X[28X         7:3xpsl(3,2)  (2)[128X[104X
    [4X[28XHN, 22A: 2.HS.2  (1)[128X[104X
    [4X[28XHS, 15A: A8.2  (1)[128X[104X
    [4X[28X         5:4xa5  (1)[128X[104X
    [4X[28XON, 19B: L3(7).2  (1)[128X[104X
    [4X[28X         ONM2  (1)[128X[104X
    [4X[28X         J1  (1)[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAccording to [CCN+85], exactly the following maximal subgroups of the simple
  group  [22XS[122X  in  the  above  list  do  [13Xnot[113X  extend to [22XAut(S)[122X: The two [22XS_10[122X type
  subgroups of [22XFi_22[122X and the two [22XL_3(7).2[122X type subgroups of [22XO'N[122X.[133X
  
  [33X[0;0YFurthermore,  the  following  maximal  subgroups of [22XAut(S)[122X with the property
  that  the  intersection  with  [22XS[122X  is  not maximal in [22XS[122X have to be considered
  whether  they contain [22Xs^'[122X: [22XG_2(3).2[122X and [22X3^5:(2 × U_4(2).2)[122X in [22XFi_22.2[122X. (Note
  that the order of the [22X7^1+2_+:(3 × D_16)[122X type subgroup in [22XO'N.2[122X is obviously
  not divisible by [22X19[122X.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27X42 in OrdersClassRepresentatives( CharacterTable( "G2(3).2" ) );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XSize( CharacterTable( "U4(2)" ) ) mod 7 = 0;[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo  we  take  the  extensions  of  the above maximal subgroups, as described
  in [CCN+85].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSigmaFromMaxes( CharacterTable( "Fi22.2" ), "42A",[127X[104X
    [4X[25X>[125X [27X    [ CharacterTable( "O8+(2).3.2" ) * CharacterTable( "Cyclic", 2 ),[127X[104X
    [4X[25X>[125X [27X      CharacterTable( "S3" ) * CharacterTable( "U4(3).(2^2)_{122}" ) ],[127X[104X
    [4X[25X>[125X [27X    [ 1, 1 ] );[127X[104X
    [4X[28X163/1170[128X[104X
    [4X[25Xgap>[125X [27XSigmaFromMaxes( CharacterTable( "Fi24'.2" ), "46A",[127X[104X
    [4X[25X>[125X [27X     [ CharacterTable( "Fi23" ) * CharacterTable( "Cyclic", 2 ),[127X[104X
    [4X[25X>[125X [27X       CharacterTable( "2^12.M24" ) ],[127X[104X
    [4X[25X>[125X [27X     [ 1, 1 ] );[127X[104X
    [4X[28X566/5481[128X[104X
    [4X[25Xgap>[125X [27XSigmaFromMaxes( CharacterTable( "He.2" ), "42A",[127X[104X
    [4X[25X>[125X [27X     [ CharacterTable( "3.A7.2" ) * CharacterTable( "Cyclic", 2 ),[127X[104X
    [4X[25X>[125X [27X       CharacterTable( "7^(1+2):(S3x6)" ),[127X[104X
    [4X[25X>[125X [27X       CharacterTable( "7:6" ) * CharacterTable( "L3(2)" ) ],[127X[104X
    [4X[25X>[125X [27X     [ 1, 1, 1 ] );[127X[104X
    [4X[28X1/119[128X[104X
    [4X[25Xgap>[125X [27XSigmaFromMaxes( CharacterTable( "HN.2" ), "44A",[127X[104X
    [4X[25X>[125X [27X     [ CharacterTable( "4.HS.2" ) ],[127X[104X
    [4X[25X>[125X [27X     [ 1 ] );[127X[104X
    [4X[28X997/192375[128X[104X
    [4X[25Xgap>[125X [27XSigmaFromMaxes( CharacterTable( "HS.2" ), "30A",[127X[104X
    [4X[25X>[125X [27X     [ CharacterTable( "S8" ) * CharacterTable( "C2" ),[127X[104X
    [4X[25X>[125X [27X       CharacterTable( "5:4" ) * CharacterTable( "S5" ) ],[127X[104X
    [4X[25X>[125X [27X     [ 1, 1 ] );[127X[104X
    [4X[28X36/275[128X[104X
    [4X[25Xgap>[125X [27XSigmaFromMaxes( CharacterTable( "ON.2" ), "38A",[127X[104X
    [4X[25X>[125X [27X     [ CharacterTable( "J1" ) * CharacterTable( "C2" ) ],[127X[104X
    [4X[25X>[125X [27X     [ 1 ] );[127X[104X
    [4X[28X61/109368[128X[104X
  [4X[32X[104X
  
  
  [1X11.4-3 [33X[0;0YOther Simple Groups – Easy Cases[133X[101X
  
  [33X[0;0YWe  are  interested  in simple groups [22XG[122X for which [10XProbGenInfoSimple[110X does not
  guarantee  [22Xtotal(G) ≥ 3[122X. So we examine the remaining tables of simple groups
  in  the  [5XGAP[105X  Character  Table  Library, and distinguish the following three
  cases:  Either [10XProbGenInfoSimple[110X yields the lower bound at least three, or a
  smaller  bound, or the computation of a lower bound fails because not enough
  information is available to compute the primitive permutation characters.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xnames:= AllCharacterTableNames( IsSimple, true, IsAbelian, false,[127X[104X
    [4X[25X>[125X [27X                                   IsDuplicateTable, false );;[127X[104X
    [4X[25Xgap>[125X [27Xnames:= Difference( names, spornames );;[127X[104X
    [4X[25Xgap>[125X [27Xfails:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xlessthan3:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xatleast3:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xfor name in names do[127X[104X
    [4X[25X>[125X [27X     tbl:= CharacterTable( name );[127X[104X
    [4X[25X>[125X [27X     info:= ProbGenInfoSimple( tbl );[127X[104X
    [4X[25X>[125X [27X     if info = fail then[127X[104X
    [4X[25X>[125X [27X       Add( fails, name );[127X[104X
    [4X[25X>[125X [27X     elif info[3] < 3 then[127X[104X
    [4X[25X>[125X [27X       Add( lessthan3, info );[127X[104X
    [4X[25X>[125X [27X     else[127X[104X
    [4X[25X>[125X [27X       Add( atleast3, info );[127X[104X
    [4X[25X>[125X [27X     fi;[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  the  following  simple  groups,  (currently)  not enough information is
  available  in  the  [5XGAP[105X  Character  Table  Library and in the [5XGAP[105X Library of
  Tables  of Marks, for computing a lower bound for [22Xσ(G)[122X. Some of these groups
  will  be  dealt with in later sections, and for the other groups, the bounds
  derived  with theoretical arguments in [BGK08] are sufficient, so we need no
  [5XGAP[105X computations for them.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfails;[127X[104X
    [4X[28X[ "2E6(2)", "2F4(8)", "3D4(3)", "3D4(4)", "A14", "A15", "A16", "A17", [128X[104X
    [4X[28X  "A18", "A19", "E6(2)", "L4(4)", "L4(5)", "L4(9)", "L5(3)", "L8(2)", [128X[104X
    [4X[28X  "O10+(2)", "O10+(3)", "O10-(2)", "O10-(3)", "O12+(2)", "O12+(3)", [128X[104X
    [4X[28X  "O12-(2)", "O12-(3)", "O7(5)", "O8+(7)", "O8-(3)", "O9(3)", [128X[104X
    [4X[28X  "R(27)", "S10(2)", "S12(2)", "S4(7)", "S4(8)", "S4(9)", "S6(4)", [128X[104X
    [4X[28X  "S6(5)", "S8(3)", "U4(4)", "U4(5)", "U5(3)", "U5(4)", "U6(4)", [128X[104X
    [4X[28X  "U7(2)" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  following  simple  groups  appear  in [BGK08, Table 1–6]. More detailed
  computations  can  be found in the sections [14X11.5-2[114X, [14X11.5-3[114X, [14X11.5-4[114X, [14X11.5-12[114X,
  [14X11.5-13[114X, [14X11.5-20[114X, [14X11.5-23[114X, [14X11.5-24[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPrintFormattedArray( lessthan3 );[127X[104X
    [4X[28X      A5      1/3 2                [ "5A" ]       [ 1 ][128X[104X
    [4X[28X      A6      2/3 1                [ "5A" ]       [ 2 ][128X[104X
    [4X[28X      A7      2/5 2                [ "7A" ]       [ 2 ][128X[104X
    [4X[28X   O7(3)  199/351 1               [ "14A" ]       [ 3 ][128X[104X
    [4X[28X  O8+(2)  334/315 0 [ "15A", "15B", "15C" ] [ 7, 7, 7 ][128X[104X
    [4X[28X  O8+(3) 863/1820 2 [ "20A", "20B", "20C" ] [ 8, 8, 8 ][128X[104X
    [4X[28X   S6(2)      4/7 1                [ "9A" ]       [ 4 ][128X[104X
    [4X[28X   S8(2)     8/15 1               [ "17A" ]       [ 3 ][128X[104X
    [4X[28X   U4(2)    21/40 1               [ "12A" ]       [ 2 ][128X[104X
    [4X[28X   U4(3)   53/135 2                [ "7A" ]       [ 7 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  the  following  simple groups [22XG[122X, the inequality [22Xσ(G) < 1/3[122X follows from
  the  loop  above.  The  columns  show  the  name  of  [22XG[122X, the values [22Xσ(G)[122X and
  [22Xtotal(G)[122X,  the  class  names  of  [22Xs[122X for which these values are attained, and
  [22X|MM(G,s)|[122X.[133X
  
  [33X[0;0Y(We increase the line length for this table. Even with this width, the entry
  for the group [22XL_7(2)[122X would not fit on one screen line, we show it separately
  below.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xoldsize:= SizeScreen();;[127X[104X
    [4X[25Xgap>[125X [27XSizeScreen( [ 80 ] );;[127X[104X
    [4X[25Xgap>[125X [27XPrintFormattedArray( Filtered( atleast3, l -> l[1] <> "L7(2)" ) );[127X[104X
    [4X[28X  2F4(2)'  118/1755   14                           [ "16A" ]             [ 2 ][128X[104X
    [4X[28X   3D4(2)    1/5292 5291                           [ "13A" ]             [ 1 ][128X[104X
    [4X[28X      A10      3/10    3                           [ "21A" ]             [ 1 ][128X[104X
    [4X[28X      A11     2/105   52                           [ "11A" ]             [ 2 ][128X[104X
    [4X[28X      A12       2/9    4                           [ "35A" ]             [ 1 ][128X[104X
    [4X[28X      A13    4/1155  288                           [ "13A" ]             [ 5 ][128X[104X
    [4X[28X       A8      3/14    4                           [ "15A" ]             [ 1 ][128X[104X
    [4X[28X       A9      9/35    3                      [ "9A", "9B" ]          [ 4, 4 ][128X[104X
    [4X[28X    F4(2)     9/595   66                           [ "13A" ]             [ 5 ][128X[104X
    [4X[28X    G2(3)       1/7    6                           [ "13A" ]             [ 3 ][128X[104X
    [4X[28X    G2(4)      1/21   20                           [ "13A" ]             [ 2 ][128X[104X
    [4X[28X    G2(5)      1/31   30                     [ "7A", "21A" ]         [ 10, 1 ][128X[104X
    [4X[28X  L2(101)     1/101  100                    [ "51A", "17A" ]          [ 1, 1 ][128X[104X
    [4X[28X  L2(103)   53/5253   99             [ "52A", "26A", "13A" ]       [ 1, 1, 1 ][128X[104X
    [4X[28X  L2(107)   55/5671  103 [ "54A", "27A", "18A", "9A", "6A" ] [ 1, 1, 1, 1, 1 ][128X[104X
    [4X[28X  L2(109)     1/109  108                    [ "55A", "11A" ]          [ 1, 1 ][128X[104X
    [4X[28X   L2(11)      7/55    7                            [ "6A" ]             [ 1 ][128X[104X
    [4X[28X  L2(113)     1/113  112                    [ "57A", "19A" ]          [ 1, 1 ][128X[104X
    [4X[28X  L2(121)     1/121  120                           [ "61A" ]             [ 1 ][128X[104X
    [4X[28X  L2(125)     1/125  124        [ "63A", "21A", "9A", "7A" ]    [ 1, 1, 1, 1 ][128X[104X
    [4X[28X   L2(13)      1/13   12                            [ "7A" ]             [ 1 ][128X[104X
    [4X[28X   L2(16)      1/15   14                           [ "17A" ]             [ 1 ][128X[104X
    [4X[28X   L2(17)      1/17   16                            [ "9A" ]             [ 1 ][128X[104X
    [4X[28X   L2(19)    11/171   15                           [ "10A" ]             [ 1 ][128X[104X
    [4X[28X   L2(23)    13/253   19                     [ "6A", "12A" ]          [ 1, 1 ][128X[104X
    [4X[28X   L2(25)      1/25   24                           [ "13A" ]             [ 1 ][128X[104X
    [4X[28X   L2(27)     5/117   23                     [ "7A", "14A" ]          [ 1, 1 ][128X[104X
    [4X[28X   L2(29)      1/29   28                           [ "15A" ]             [ 1 ][128X[104X
    [4X[28X   L2(31)    17/465   27                     [ "8A", "16A" ]          [ 1, 1 ][128X[104X
    [4X[28X   L2(32)      1/31   30              [ "3A", "11A", "33A" ]       [ 1, 1, 1 ][128X[104X
    [4X[28X   L2(37)      1/37   36                           [ "19A" ]             [ 1 ][128X[104X
    [4X[28X   L2(41)      1/41   40                     [ "21A", "7A" ]          [ 1, 1 ][128X[104X
    [4X[28X   L2(43)    23/903   39                    [ "22A", "11A" ]          [ 1, 1 ][128X[104X
    [4X[28X   L2(47)   25/1081   43        [ "24A", "12A", "8A", "6A" ]    [ 1, 1, 1, 1 ][128X[104X
    [4X[28X   L2(49)      1/49   48                           [ "25A" ]             [ 1 ][128X[104X
    [4X[28X   L2(53)      1/53   52                     [ "27A", "9A" ]          [ 1, 1 ][128X[104X
    [4X[28X   L2(59)   31/1711   55       [ "30A", "15A", "10A", "6A" ]    [ 1, 1, 1, 1 ][128X[104X
    [4X[28X   L2(61)      1/61   60                           [ "31A" ]             [ 1 ][128X[104X
    [4X[28X   L2(64)      1/63   62                    [ "65A", "13A" ]          [ 1, 1 ][128X[104X
    [4X[28X   L2(67)   35/2211   63                    [ "34A", "17A" ]          [ 1, 1 ][128X[104X
    [4X[28X   L2(71)   37/2485   67 [ "36A", "18A", "12A", "9A", "6A" ] [ 1, 1, 1, 1, 1 ][128X[104X
    [4X[28X   L2(73)      1/73   72                           [ "37A" ]             [ 1 ][128X[104X
    [4X[28X   L2(79)   41/3081   75       [ "40A", "20A", "10A", "8A" ]    [ 1, 1, 1, 1 ][128X[104X
    [4X[28X    L2(8)       1/7    6                      [ "3A", "9A" ]          [ 1, 1 ][128X[104X
    [4X[28X   L2(81)      1/81   80                           [ "41A" ]             [ 1 ][128X[104X
    [4X[28X   L2(83)   43/3403   79 [ "42A", "21A", "14A", "7A", "6A" ] [ 1, 1, 1, 1, 1 ][128X[104X
    [4X[28X   L2(89)      1/89   88              [ "45A", "15A", "9A" ]       [ 1, 1, 1 ][128X[104X
    [4X[28X   L2(97)      1/97   96                     [ "49A", "7A" ]          [ 1, 1 ][128X[104X
    [4X[28X   L3(11)    1/6655 6654                   [ "19A", "133A" ]          [ 1, 1 ][128X[104X
    [4X[28X    L3(2)       1/4    3                            [ "7A" ]             [ 1 ][128X[104X
    [4X[28X    L3(3)      1/24   23                           [ "13A" ]             [ 1 ][128X[104X
    [4X[28X    L3(4)       1/5    4                            [ "7A" ]             [ 3 ][128X[104X
    [4X[28X    L3(5)     1/250  249                           [ "31A" ]             [ 1 ][128X[104X
    [4X[28X    L3(7)    1/1372 1371                           [ "19A" ]             [ 1 ][128X[104X
    [4X[28X    L3(8)    1/1792 1791                           [ "73A" ]             [ 1 ][128X[104X
    [4X[28X    L3(9)    1/2880 2879                           [ "91A" ]             [ 1 ][128X[104X
    [4X[28X    L4(3)   53/1053   19                           [ "20A" ]             [ 1 ][128X[104X
    [4X[28X    L5(2)    1/5376 5375                           [ "31A" ]             [ 1 ][128X[104X
    [4X[28X    L6(2) 365/55552  152                    [ "21A", "63A" ]          [ 2, 2 ][128X[104X
    [4X[28X   O8-(2)      1/63   62                           [ "17A" ]             [ 1 ][128X[104X
    [4X[28X    S4(4)      4/15    3                           [ "17A" ]             [ 2 ][128X[104X
    [4X[28X    S4(5)       1/5    4                           [ "13A" ]             [ 1 ][128X[104X
    [4X[28X    S6(3)     1/117  116                           [ "14A" ]             [ 2 ][128X[104X
    [4X[28X   Sz(32)    1/1271 1270                     [ "5A", "25A" ]          [ 1, 1 ][128X[104X
    [4X[28X    Sz(8)      1/91   90                            [ "5A" ]             [ 1 ][128X[104X
    [4X[28X   U3(11)    1/6655 6654                           [ "37A" ]             [ 1 ][128X[104X
    [4X[28X    U3(3)     16/63    3                     [ "6A", "12A" ]          [ 2, 2 ][128X[104X
    [4X[28X    U3(4)     1/160  159                           [ "13A" ]             [ 1 ][128X[104X
    [4X[28X    U3(5)    46/525   11                           [ "10A" ]             [ 2 ][128X[104X
    [4X[28X    U3(7)    1/1372 1371                           [ "43A" ]             [ 1 ][128X[104X
    [4X[28X    U3(8)    1/1792 1791                           [ "19A" ]             [ 1 ][128X[104X
    [4X[28X    U3(9)    1/3600 3599                           [ "73A" ]             [ 1 ][128X[104X
    [4X[28X    U5(2)      1/54   53                           [ "11A" ]             [ 1 ][128X[104X
    [4X[28X    U6(2)      5/21    4                           [ "11A" ]             [ 4 ][128X[104X
    [4X[25Xgap>[125X [27XSizeScreen( oldsize );;[127X[104X
    [4X[25Xgap>[125X [27XFirst( atleast3, l -> l[1] = "L7(2)" );[127X[104X
    [4X[28X[ "L7(2)", 1/4388290560, 4388290559, [ "127A" ], [ 1 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YIt should be mentioned that [BW75] states the following lower bounds for the
  uniform spread of the groups [22XL_2(q)[122X.[133X
  
      │ [22Xq-2[122X │ if [22X4 ≤ q[122X is even,     │ 
      │ [22Xq-1[122X │ if [22X11 ≤ q ≡ 1 mod 4[122X,  │ 
      │ [22Xq-4[122X │ if [22X11 ≤ q ≡ -1 mod 4[122X. │ 
  
  [33X[0;0YThese  bounds  appear  in  the third column of the above table. Furthermore,
  [BW75] states that the (uniform) spread of alternating groups of even degree
  at least [22X8[122X is exactly [22X4[122X.[133X
  
  [33X[0;0YFor the sake of completeness, Table II gives an overview of the sets [22XMM(G,s)[122X
  for those cases in the above list that are needed in [BGK08] but that do not
  require  a  further  discussion here. The structure of the maximal subgroups
  and the order of [22Xs[122X in the table refer to the matrix groups not to the simple
  groups.  The  number  of  the  subgroups has been shown above, the structure
  follows from [CCN+85].[133X
  
      ┌─────────────────────┬────────────────────────────────────────────────┬─────┬──────────────┐
      │ [22XG[122X                   │ [22XMM(G,s)[122X                                        │ [22X|s|[122X │ see [CCN+85] │ 
      ├─────────────────────┼────────────────────────────────────────────────┼─────┼──────────────┤
      ├─────────────────────┼────────────────────────────────────────────────┼─────┼──────────────┤
      │ [22XSL(3,4) = 3.L_3(4)[122X  │ [22X3 × L_3(2), 3 × L_3(2), 3 × L_3(2)[122X             │  [22X21[122X │        p. 23 │ 
      ├─────────────────────┼────────────────────────────────────────────────┼─────┼──────────────┤
      │ [22XΩ^-(8,2) = O^-_8(2)[122X │ [22XΩ^-(4,4).2 = L_2(16).2[122X                         │  [22X17[122X │        p. 89 │ 
      ├─────────────────────┼────────────────────────────────────────────────┼─────┼──────────────┤
      │ [22XSp(4,4) = S_4(4)[122X    │ [22XΩ^-(4,4).2 = L_2(16).2, Sp(2,16).2 = L_2(16).2[122X │  [22X17[122X │        p. 44 │ 
      │ [22XSp(6,3) = 2.S_6(3)[122X  │ [22X(4 × U_3(3)).2, Sp(2,17).3 = 2.L_2(27).3[122X       │  [22X28[122X │       p. 113 │ 
      ├─────────────────────┼────────────────────────────────────────────────┼─────┼──────────────┤
      │ [22XSU(3,3) = U_3(3)[122X    │ [22X3^1+2_+:8, GU(2,3) = 4.S_4[122X                     │   [22X6[122X │        p. 14 │ 
      │ [22XSU(3,5) = 3.U_3(5)[122X  │ [22X3 × 5^1+2_+:8, GU(2,5) = 3 × 2S_5[122X              │  [22X30[122X │        p. 34 │ 
      │ [22XSU(5,2) = U_5(2)[122X    │ [22XL_2(11)[122X                                        │  [22X11[122X │        p. 73 │ 
      └─────────────────────┴────────────────────────────────────────────────┴─────┴──────────────┘
  
       [1XTable:[101X Table II: Maximal subgroups/>
  
  
  
  [1X11.4-4 [33X[0;0YAutomorphism Groups of other Simple Groups – Easy Cases[133X[101X
  
  [33X[0;0YWe  deal  with automorphic extensions of those simple groups that are listed
  in Table I and that have been treated successfully in Section [14X11.4-3[114X.[133X
  
  [33X[0;0YFor  the  following  groups, [10XProbGenInfoAlmostSimple[110X can be used because [5XGAP[105X
  can compute their primitive permutation characters.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xlist:= [[127X[104X
    [4X[25X>[125X [27X  [ "A5", "A5.2" ],[127X[104X
    [4X[25X>[125X [27X  [ "A6", "A6.2_1" ],[127X[104X
    [4X[25X>[125X [27X  [ "A6", "A6.2_2" ],[127X[104X
    [4X[25X>[125X [27X  [ "A6", "A6.2_3" ],[127X[104X
    [4X[25X>[125X [27X  [ "A7", "A7.2" ],[127X[104X
    [4X[25X>[125X [27X  [ "A8", "A8.2" ],[127X[104X
    [4X[25X>[125X [27X  [ "A9", "A9.2" ],[127X[104X
    [4X[25X>[125X [27X  [ "A11", "A11.2" ],[127X[104X
    [4X[25X>[125X [27X  [ "L3(2)", "L3(2).2" ],[127X[104X
    [4X[25X>[125X [27X  [ "L3(3)", "L3(3).2" ],[127X[104X
    [4X[25X>[125X [27X  [ "L3(4)", "L3(4).2_1" ],[127X[104X
    [4X[25X>[125X [27X  [ "L3(4)", "L3(4).2_2" ],[127X[104X
    [4X[25X>[125X [27X  [ "L3(4)", "L3(4).2_3" ],[127X[104X
    [4X[25X>[125X [27X  [ "L3(4)", "L3(4).3" ],[127X[104X
    [4X[25X>[125X [27X  [ "S4(4)", "S4(4).2" ],[127X[104X
    [4X[25X>[125X [27X  [ "U3(3)", "U3(3).2" ],[127X[104X
    [4X[25X>[125X [27X  [ "U3(5)", "U3(5).2" ],[127X[104X
    [4X[25X>[125X [27X  [ "U3(5)", "U3(5).3" ],[127X[104X
    [4X[25X>[125X [27X  [ "U4(2)", "U4(2).2" ],[127X[104X
    [4X[25X>[125X [27X  [ "U4(3)", "U4(3).2_1" ],[127X[104X
    [4X[25X>[125X [27X  [ "U4(3)", "U4(3).2_3" ],[127X[104X
    [4X[25X>[125X [27X];;[127X[104X
    [4X[25Xgap>[125X [27Xautinfo:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xfails:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xfor pair in list do[127X[104X
    [4X[25X>[125X [27X     tbl:= CharacterTable( pair[1] );[127X[104X
    [4X[25X>[125X [27X     tblG:= CharacterTable( pair[2] );[127X[104X
    [4X[25X>[125X [27X     info:= ProbGenInfoSimple( tbl );[127X[104X
    [4X[25X>[125X [27X     spos:= List( info[4], x -> Position( AtlasClassNames( tbl ), x ) );[127X[104X
    [4X[25X>[125X [27X     Add( autinfo, ProbGenInfoAlmostSimple( tbl, tblG, spos ) );[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27XPrintFormattedArray( autinfo );[127X[104X
    [4X[28X       A5.2      0        [ "5AB" ]    [ 1 ][128X[104X
    [4X[28X     A6.2_1    2/3        [ "5AB" ]    [ 2 ][128X[104X
    [4X[28X     A6.2_2    1/6         [ "5A" ]    [ 1 ][128X[104X
    [4X[28X     A6.2_3      0        [ "5AB" ]    [ 1 ][128X[104X
    [4X[28X       A7.2   1/15        [ "7AB" ]    [ 1 ][128X[104X
    [4X[28X       A8.2  13/28       [ "15AB" ]    [ 1 ][128X[104X
    [4X[28X       A9.2    1/4        [ "9AB" ]    [ 1 ][128X[104X
    [4X[28X      A11.2  1/945       [ "11AB" ]    [ 1 ][128X[104X
    [4X[28X    L3(2).2    1/4        [ "7AB" ]    [ 1 ][128X[104X
    [4X[28X    L3(3).2   1/18       [ "13AB" ]    [ 1 ][128X[104X
    [4X[28X  L3(4).2_1   3/10        [ "7AB" ]    [ 3 ][128X[104X
    [4X[28X  L3(4).2_2  11/60         [ "7A" ]    [ 1 ][128X[104X
    [4X[28X  L3(4).2_3   1/12        [ "7AB" ]    [ 1 ][128X[104X
    [4X[28X    L3(4).3   1/64         [ "7A" ]    [ 1 ][128X[104X
    [4X[28X    S4(4).2      0       [ "17AB" ]    [ 2 ][128X[104X
    [4X[28X    U3(3).2    2/7 [ "6A", "12AB" ] [ 2, 2 ][128X[104X
    [4X[28X    U3(5).2   2/21        [ "10A" ]    [ 2 ][128X[104X
    [4X[28X    U3(5).3 46/525        [ "10A" ]    [ 2 ][128X[104X
    [4X[28X    U4(2).2  16/45       [ "12AB" ]    [ 2 ][128X[104X
    [4X[28X  U4(3).2_1 76/135         [ "7A" ]    [ 3 ][128X[104X
    [4X[28X  U4(3).2_3 31/162        [ "7AB" ]    [ 3 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  see  that  from  this  list, the two groups [22XA_6.2_1 = S_6[122X and [22XU_4(3).2_1[122X
  require further computations (see Sections [14X11.5-3[114X and [14X11.5-24[114X, respectively)
  because the bound in the second column is larger than [22X1/2[122X.[133X
  
  [33X[0;0YAlso  [22XU_4(2)[122X  is  not  done  by  the  above, because in [BGK08, Table 4], an
  element [22Xs[122X of order [22X9[122X is chosen for the simple group, see Section [14X11.5-23[114X.[133X
  
  [33X[0;0YFinally, we deal with automorphic extensions of the groups [22XL_4(3)[122X, [22XO_8^-(2)[122X,
  [22XS_6(3)[122X, and [22XU_5(2)[122X.[133X
  
  [33X[0;0YFor [22XS = L_4(3)[122X and [22Xs ∈ S[122X of order [22X20[122X, we have [22XMM(S,s) = { (4 × A_6):2 }[122X, the
  subgroup has index [22X2106[122X, see [CCN+85, p. 69].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "L4(3)" );;[127X[104X
    [4X[25Xgap>[125X [27Xprim:= PrimitivePermutationCharacters( t );;[127X[104X
    [4X[25Xgap>[125X [27Xspos:= Position( AtlasClassNames( t ), "20A" );;[127X[104X
    [4X[25Xgap>[125X [27Xprim:= Filtered( prim, x -> x[ spos ] <> 0 );[127X[104X
    [4X[28X[ Character( CharacterTable( "L4(3)" ),[128X[104X
    [4X[28X  [ 2106, 106, 42, 0, 27, 27, 0, 46, 6, 6, 1, 7, 7, 0, 3, 3, 0, 0, 0, [128X[104X
    [4X[28X      1, 1, 1, 0, 0, 0, 0, 0, 1, 1 ] ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  the  three  automorphic extensions of the structure [22XG = S.2[122X, we compute
  the extensions of the permutation character, and the bounds [22Xσ^'(G,s)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfor name in [ "L4(3).2_1", "L4(3).2_2", "L4(3).2_3" ] do[127X[104X
    [4X[25X>[125X [27X     t2:= CharacterTable( name );[127X[104X
    [4X[25X>[125X [27X     map:= InverseMap( GetFusionMap( t, t2 ) );[127X[104X
    [4X[25X>[125X [27X     torso:= List( prim, pi -> CompositionMaps( pi, map ) );[127X[104X
    [4X[25X>[125X [27X     ext:= Concatenation( List( torso,[127X[104X
    [4X[25X>[125X [27X                             x -> PermChars( t2, rec( torso:= x ) ) ) );[127X[104X
    [4X[25X>[125X [27X     sigma:= ApproxP( ext, Position( OrdersClassRepresentatives( t2 ), 20 ) );[127X[104X
    [4X[25X>[125X [27X     max:= Maximum( sigma{ Difference( PositionsProperty([127X[104X
    [4X[25X>[125X [27X                          OrdersClassRepresentatives( t2 ), IsPrimeInt ),[127X[104X
    [4X[25X>[125X [27X                          ClassPositionsOfDerivedSubgroup( t2 ) ) } );[127X[104X
    [4X[25X>[125X [27X     Print( name, ":\n", ext, "\n", max, "\n" );[127X[104X
    [4X[25X>[125X [27Xod;[127X[104X
    [4X[28XL4(3).2_1:[128X[104X
    [4X[28X[ Character( CharacterTable( "L4(3).2_1" ), [128X[104X
    [4X[28X    [ 2106, 106, 42, 0, 27, 0, 46, 6, 6, 1, 7, 0, 3, 0, 0, 1, 1, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 1, 1, 0, 4, 0, 0, 6, 6, 6, 6, 2, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      1, 1, 1, 1 ] ) ][128X[104X
    [4X[28X0[128X[104X
    [4X[28XL4(3).2_2:[128X[104X
    [4X[28X[ Character( CharacterTable( "L4(3).2_2" ), [128X[104X
    [4X[28X    [ 2106, 106, 42, 0, 27, 27, 0, 46, 6, 6, 1, 7, 7, 0, 3, 3, 0, 0, [128X[104X
    [4X[28X      0, 1, 1, 1, 0, 0, 0, 1, 306, 306, 42, 6, 10, 10, 0, 0, 15, 15, [128X[104X
    [4X[28X      3, 3, 3, 3, 0, 0, 1, 1, 0, 1, 1, 0, 0 ] ) ][128X[104X
    [4X[28X17/117[128X[104X
    [4X[28XL4(3).2_3:[128X[104X
    [4X[28X[ Character( CharacterTable( "L4(3).2_3" ), [128X[104X
    [4X[28X    [ 2106, 106, 42, 0, 27, 0, 46, 6, 6, 1, 7, 0, 3, 0, 0, 1, 1, 0, [128X[104X
    [4X[28X      0, 0, 1, 36, 0, 0, 6, 6, 2, 2, 2, 1, 1, 0, 0, 0 ] ) ][128X[104X
    [4X[28X2/117[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor [22XS = O_8^-(2)[122X and [22Xs ∈ S[122X of order [22X17[122X, we have [22XMM(S,s) = { L_2(16).2 }[122X, the
  subgroup  extends  to  [22XL_2(16).4[122X  in  [22XS.2[122X,  see [CCN+85,  p. 89].  This is a
  non-split extension, so [22Xσ^'(S.2,s) = 0[122X holds.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSigmaFromMaxes( CharacterTable( "O8-(2).2" ), "17AB",[127X[104X
    [4X[25X>[125X [27X       [ CharacterTable( "L2(16).4" ) ], [ 1 ], "outer" );[127X[104X
    [4X[28X0[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  [22XS  = S_6(3)[122X and [22Xs ∈ S[122X irreducible of order [22X14[122X, we have [22XMM(S,s) = { (2 ×
  U_3(3)).2,  L_2(27).3  }[122X. In [22XG = S.2[122X, the subgroups extend to [22X(4 × U_3(3)).2[122X
  and  [22XL_2(27).6[122X,  respectively,  see [CCN+85,  p. 113]. In order to show that
  [22Xσ^'(G,s)  = 7/3240[122X holds, we compute the primitive permutation characters of
  [22XS[122X  (cf. Section [14X11.4-3[114X)  and  the  unique extensions to [22XG[122X of those which are
  nonzero on [22Xs[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "S6(3)" );;[127X[104X
    [4X[25Xgap>[125X [27Xt2:= CharacterTable( "S6(3).2" );;[127X[104X
    [4X[25Xgap>[125X [27Xprim:= PrimitivePermutationCharacters( t );;[127X[104X
    [4X[25Xgap>[125X [27Xspos:= Position( AtlasClassNames( t ), "14A" );;[127X[104X
    [4X[25Xgap>[125X [27Xprim:= Filtered( prim, x -> x[ spos ] <> 0 );;[127X[104X
    [4X[25Xgap>[125X [27Xmap:= InverseMap( GetFusionMap( t, t2 ) );;[127X[104X
    [4X[25Xgap>[125X [27Xtorso:= List( prim, pi -> CompositionMaps( pi, map ) );;[127X[104X
    [4X[25Xgap>[125X [27Xext:= List( torso, pi -> PermChars( t2, rec( torso:= pi ) ) );[127X[104X
    [4X[28X[ [ Character( CharacterTable( "S6(3).2" ),[128X[104X
    [4X[28X      [ 155520, 0, 288, 0, 0, 0, 216, 54, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X          0, 0, 0, 0, 6, 1, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X          0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 144, 288, 0, 0, 0, [128X[104X
    [4X[28X          6, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, [128X[104X
    [4X[28X          0 ] ) ], [128X[104X
    [4X[28X  [ Character( CharacterTable( "S6(3).2" ),[128X[104X
    [4X[28X      [ 189540, 1620, 568, 0, 486, 0, 0, 27, 540, 84, 24, 0, 0, 0, 0, [128X[104X
    [4X[28X          0, 54, 0, 0, 10, 0, 7, 1, 6, 6, 0, 0, 0, 0, 0, 0, 18, 0, 0, [128X[104X
    [4X[28X          0, 6, 12, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 234, 64, [128X[104X
    [4X[28X          30, 8, 0, 3, 90, 6, 0, 4, 10, 6, 0, 2, 1, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X          0, 1, 1, 0, 0 ] ) ] ][128X[104X
    [4X[25Xgap>[125X [27Xspos:= Position( AtlasClassNames( t2 ), "14A" );;[127X[104X
    [4X[25Xgap>[125X [27Xsigma:= ApproxP( Concatenation( ext ), spos );;[127X[104X
    [4X[25Xgap>[125X [27XMaximum( sigma{ Difference([127X[104X
    [4X[25X>[125X [27X     PositionsProperty( OrdersClassRepresentatives( t2 ), IsPrimeInt ),[127X[104X
    [4X[25X>[125X [27X     ClassPositionsOfDerivedSubgroup( t2 ) ) } );[127X[104X
    [4X[28X7/3240[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  [22XS  =  U_5(2)[122X  and [22Xs ∈ S[122X of order [22X11[122X, we have [22XMM(S,s) = { L_2(11) }[122X, the
  subgroup extends to [22XL_2(11).2[122X in [22XS.2[122X, see [CCN+85, p. 73].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSigmaFromMaxes( CharacterTable( "U5(2).2" ), "11AB",[127X[104X
    [4X[25X>[125X [27X       [ CharacterTable( "L2(11).2" ) ], [ 1 ], "outer" );[127X[104X
    [4X[28X1/288[128X[104X
  [4X[32X[104X
  
  [33X[0;0YHere  we clean the workspace for the first time. This may save more than [22X100[122X
  megabytes, due to the fact that the caches for tables of marks and character
  tables are flushed.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCleanWorkspace();[127X[104X
  [4X[32X[104X
  
  
  [1X11.4-5 [33X[0;0Y[22XO_8^-(3)[122X[101X[1X[133X[101X
  
  [33X[0;0YWe show that [22XS = O_8^-(3) = Ω^-(8,3)[122X satisfies the following.[133X
  
  [8X(a)[108X
        [33X[0;6YFor  [22Xs  ∈  S[122X  of  order  [22X41[122X, [22XMM(S,s)[122X consists of one group of the type
        [22XL_2(81).2_1 = Ω^-(4,9).2[122X.[133X
  
  [8X(b)[108X
        [33X[0;6Y[22Xσ(S,s) = 1/567[122X.[133X
  
  [33X[0;0YThe  only  maximal  subgroups  of [22XS[122X containing elements of order [22X41[122X have the
  type  [22XL_2(81).2_1[122X,  and  there is one class of these subgroups, see [CCN+85,
  p. 141].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSigmaFromMaxes( CharacterTable( "O8-(3)" ), "41A",[127X[104X
    [4X[25X>[125X [27X   [ CharacterTable( "L2(81).2_1" ) ], [ 1 ] );[127X[104X
    [4X[28X1/567[128X[104X
  [4X[32X[104X
  
  
  [1X11.4-6 [33X[0;0Y[22XO_10^+(2)[122X[101X[1X[133X[101X
  
  [33X[0;0YWe show that [22XS = O_10^+(2) = Ω^+(10,2)[122X satisfies the following.[133X
  
  [8X(a)[108X
        [33X[0;6YFord[22Xs  ∈ S[122X of order [22X45[122X, [22XMM(S,s)[122X consists of one group of the type [22X(A_5
        × U_4(2)).2 = (Ω^-(4,2) × Ω^-(6,2)).2[122X.[133X
  
  [8X(b)[108X
        [33X[0;6Y[22Xσ(S,s) = 43/4216[122X.[133X
  
  [8X(c)[108X
        [33X[0;6YFor [22Xs[122X as in (a), the maximal subgroup in (a) extends to [22XS_5 × U_4(2).2[122X
        in [22XG = Aut(S) = S.2[122X, and [22Xσ^'(G,s) = 23/248[122X.[133X
  
  [33X[0;0YThe  only  maximal  subgroups  of  [22XS[122X containing elements of order [22X45[122X are one
  class  of groups [22XH = (A_5 × U_4(2)):2[122X, see [CCN+85, p. 146]. (Note that none
  of  the groups [22XS_8(2)[122X, [22XO_8^+(2)[122X, [22XL_5(2)[122X, [22XO_8^-(2)[122X, and [22XA_8[122X contains elements
  of  order [22X45[122X.) [22XH[122X extends to subgroups of the type [22XH.2 = S_5 × U_4(2):2[122X in [22XG[122X,
  so we can compute [22X1_H^S = (1_H.2^G)_S[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XForAny( [ "S8(2)", "O8+(2)", "L5(2)", "O8-(2)", "A8" ],[127X[104X
    [4X[25X>[125X [27X           x -> 45 in OrdersClassRepresentatives( CharacterTable( x ) ) );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "O10+(2)" );;[127X[104X
    [4X[25Xgap>[125X [27Xt2:= CharacterTable( "O10+(2).2" );;[127X[104X
    [4X[25Xgap>[125X [27Xs2:= CharacterTable( "A5.2" ) * CharacterTable( "U4(2).2" );[127X[104X
    [4X[28XCharacterTable( "A5.2xU4(2).2" )[128X[104X
    [4X[25Xgap>[125X [27Xpi:= PossiblePermutationCharacters( s2, t2 );;[127X[104X
    [4X[25Xgap>[125X [27Xspos:= Position( OrdersClassRepresentatives( t2 ), 45 );;[127X[104X
    [4X[25Xgap>[125X [27Xapprox:= ApproxP( pi, spos );;[127X[104X
    [4X[25Xgap>[125X [27XMaximum( approx{ ClassPositionsOfDerivedSubgroup( t2 ) } );[127X[104X
    [4X[28X43/4216[128X[104X
  [4X[32X[104X
  
  [33X[0;0YStatement (c)  follows  from  considering the outer classes of prime element
  order.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMaximum( approx{ Difference([127X[104X
    [4X[25X>[125X [27X     PositionsProperty( OrdersClassRepresentatives( t2 ), IsPrimeInt ),[127X[104X
    [4X[25X>[125X [27X     ClassPositionsOfDerivedSubgroup( t2 ) ) } );[127X[104X
    [4X[28X23/248[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAlternatively, we can use [10XSigmaFromMaxes[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSigmaFromMaxes( t2, "45AB", [ s2 ], [ 1 ], "outer" );[127X[104X
    [4X[28X23/248[128X[104X
  [4X[32X[104X
  
  
  [1X11.4-7 [33X[0;0Y[22XO_10^-(2)[122X[101X[1X[133X[101X
  
  [33X[0;0YWe show that [22XS = O_10^-(2) = Ω^-(10,2)[122X satisfies the following.[133X
  
  [8X(a)[108X
        [33X[0;6YFor  [22Xs  ∈ S[122X of order [22X33[122X, [22XMM(S,s)[122X consists of one group of the type [22X3 ×
        U_5(2) = GU(5,2)[122X.[133X
  
  [8X(b)[108X
        [33X[0;6Y[22Xσ(S,s) = 1/119[122X.[133X
  
  [8X(c)[108X
        [33X[0;6YFor [22Xs[122X as in (a), the maximal subgroup in (a) extends to [22X(3 × U_5(2)).2[122X
        in [22XG[122X, and [22Xσ^'(G,s) = 1/595[122X.[133X
  
  [33X[0;0YThe  only  maximal  subgroups  of [22XS[122X containing elements of order [22X11[122X have the
  types [22XA_12[122X and [22X3 × U_5(2)[122X, see [CCN+85, p. 147]. So [22X3 × U_5(2)[122X is the unique
  class   of   subgroups   containing   elements   of  order  [22X33[122X.  This  shows
  statement (a), and statement (b) follows using [10XSigmaFromMaxes[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSigmaFromMaxes( CharacterTable( "O10-(2)" ), "33A",[127X[104X
    [4X[25X>[125X [27X   [ CharacterTable( "Cyclic", 3 ) * CharacterTable( "U5(2)" ) ], [ 1 ] );[127X[104X
    [4X[28X1/119[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe structure of the maximal subgroup of [22XG[122X follows from [CCN+85, p. 147]. We
  create  its character table with a generic construction that is based on the
  fact  that  the  outer  automorphism  acts  nontrivially  on  the two direct
  factors;  this  determines  the  character  table  uniquely. (See [Brec] for
  details.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XtblG:= CharacterTable( "U5(2)" );;[127X[104X
    [4X[25Xgap>[125X [27XtblMG:= CharacterTable( "Cyclic", 3 ) * tblG;;[127X[104X
    [4X[25Xgap>[125X [27XtblGA:= CharacterTable( "U5(2).2" );;[127X[104X
    [4X[25Xgap>[125X [27Xacts:= PossibleActionsForTypeMGA( tblMG, tblG, tblGA );;[127X[104X
    [4X[25Xgap>[125X [27Xposs:= Concatenation( List( acts, pi ->[127X[104X
    [4X[25X>[125X [27X           PossibleCharacterTablesOfTypeMGA( tblMG, tblG, tblGA, pi,[127X[104X
    [4X[25X>[125X [27X               "(3xU5(2)).2" ) ) );[127X[104X
    [4X[28X[ rec( [128X[104X
    [4X[28X      MGfusMGA := [ 1, 2, 3, 4, 4, 5, 5, 6, 7, 8, 9, 10, 11, 12, 12, [128X[104X
    [4X[28X          13, 13, 14, 14, 15, 15, 16, 17, 17, 18, 18, 19, 20, 21, 21, [128X[104X
    [4X[28X          22, 22, 23, 23, 24, 24, 25, 25, 26, 27, 27, 28, 28, 29, 29, [128X[104X
    [4X[28X          30, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, [128X[104X
    [4X[28X          44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, [128X[104X
    [4X[28X          59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, [128X[104X
    [4X[28X          74, 75, 76, 77, 31, 32, 33, 35, 34, 37, 36, 38, 39, 40, 41, [128X[104X
    [4X[28X          42, 43, 45, 44, 47, 46, 49, 48, 51, 50, 52, 54, 53, 56, 55, [128X[104X
    [4X[28X          57, 58, 60, 59, 62, 61, 64, 63, 66, 65, 68, 67, 69, 71, 70, [128X[104X
    [4X[28X          73, 72, 75, 74, 77, 76 ], [128X[104X
    [4X[28X      table := CharacterTable( "(3xU5(2)).2" ) ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow statement (c) follows using [10XSigmaFromMaxes[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSigmaFromMaxes( CharacterTable( "O10-(2).2" ), "33AB",[127X[104X
    [4X[25X>[125X [27X       [ poss[1].table ], [ 1 ], "outer" );[127X[104X
    [4X[28X1/595[128X[104X
  [4X[32X[104X
  
  
  [1X11.4-8 [33X[0;0Y[22XO_12^+(2)[122X[101X[1X[133X[101X
  
  [33X[0;0YWe show that [22XS = O_12^+(2) = Ω^+(12,2)[122X satisfies the following.[133X
  
  [8X(a)[108X
        [33X[0;6YFor  [22Xs  ∈  S[122X of the type [22X4^- perp 8^-[122X (i. e., [22Xs[122X decomposes the natural
        [22X12[122X-dimensional  module  for [22XGO^+_12(2) = S.2[122X into an orthogonal sum of
        two  irreducible  modules of the dimensions [22X4[122X and [22X8[122X, respectively) and
        of order [22X85[122X, [22XMM(S,s)[122X consists of one group of the type [22XG_8 = (Ω^-(4,2)
        ×  Ω^-(8,2)).2[122X  and  two  groups of the type [22XL_4(4).2^2 = Ω^+(6,4).2^2[122X
        that  are conjugate in [22XG = Aut(S) = S.2 = SO^+(12,2)[122X but [13Xnot[113X conjugate
        in [22XS[122X.[133X
  
  [8X(b)[108X
        [33X[0;6Y[22Xσ(S,s) = 7675/1031184[122X.[133X
  
  [8X(c)[108X
        [33X[0;6Y[22Xσ^'(G,s) = 73/1008[122X.[133X
  
  [33X[0;0YThe  element  [22Xs[122X  is  a  ppd[22X(12,2;8)[122X-element in the sense of [GPPS99], so the
  maximal  subgroups  of [22XS[122X that contain [22Xs[122X are among the nine cases (2.1)–(2.9)
  listed  in this paper; in the notation of this paper, we have [22Xq = 2[122X, [22Xd = 12[122X,
  [22Xe  =  8[122X, and [22Xr = 17[122X. Case (2.1) does not occur for orthogonal groups and [22Xq =
  2[122X,  according  to [KL90];  case (2.2) contributes a unique maximal subgroup,
  the  stabilizer  [22XG_8[122X  of  the  orthogonal  decomposition;  the  cases (2.3),
  (2.4) (a),  (2.5),  and  (2.6) (a)  do  not  occur  because  [22Xr  ‡ e+1[122X in our
  situation;  case (2.4) (b) describes extension field type subgroups that are
  contained  in  [22XΓL(6,4)[122X, which yields the candidates [22XGU(6,2).2 ≅ 3.U_6(2).S_3[122X
  –but  [22X3.U_6(2).3[122X  does  not contain elements of order [22X85[122X– and [22XΩ^+(6,4).2^2 ≅
  L_4(4).2^2[122X  (two  classes  by [KL90, Prop. 4.3.14]); the cases (2.6) (b)–(c)
  and (2.8) do not occur because they require [22Xd ≤ 8[122X; case (2.7) does not occur
  because [GPPS99,  Table 5]  contains  no  entry  for [22Xr = 2e+1 = 17[122X; finally,
  case (2.9) does not occur because it requires [22Xe ∈ { d-1, d }[122X in the case [22Xr =
  2e+1[122X.[133X
  
  [33X[0;0YSo  we  need  the  permutation  characters  of  the actions on the cosets of
  [22XL_4(4).2^2[122X  (two classes) and [22XG_8[122X. According to [KL90, Prop. 4.1.6], [22XG_8[122X has
  the structure [22X(Ω^-(4,2) × Ω^-(8,2)).2[122X.[133X
  
  [33X[0;0YNewer  versions  of  the  [5XGAP[105X  Character Table Library contain the character
  table  of  [22XS[122X,  but it is still easier to work with the table of [22XG[122X, which was
  already  available at the times when the first version of these examples was
  created.[133X
  
  [33X[0;0YThe  two  classes of [22XL_4(4).2^2[122X type subgroups in [22XS[122X are fused in [22XG[122X. This can
  be  seen from the fact that inducing the trivial character of a subgroup [22XH_1
  =  L_4(4).2^2[122X  of [22XS[122X to [22XG[122X yields a character [22Xψ[122X whose values are not all even;
  note  that  if  [22XH_1[122X would extend in [22XG[122X to a subgroup of twice the size of [22XH_1[122X
  then  [22Xψ[122X  would be induced from a degree two character of this subgroup whose
  values are all even, and induction preserves this property.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "O12+(2).2" );;[127X[104X
    [4X[25Xgap>[125X [27Xh1:= CharacterTable( "L4(4).2^2" );;[127X[104X
    [4X[25Xgap>[125X [27Xpsi:= PossiblePermutationCharacters( h1, t );;[127X[104X
    [4X[25Xgap>[125X [27XLength( psi );[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27XForAny( psi[1], IsOddInt );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  fixed element [22Xs[122X of order [22X85[122X is contained in a member of each of the two
  conjugacy  classes  of  the  type [22XL_4(4).2^2[122X in [22XS[122X, since [22XS[122X contains only one
  class  of  subgroups  of  the  order [22X85[122X; note that the order of the Sylow [22X17[122X
  centralizer (in both [22XS[122X and [22XG[122X) is not divisible by [22X25[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSizesCentralizers( t ){ PositionsProperty([127X[104X
    [4X[25X>[125X [27X       OrdersClassRepresentatives( t ), x -> x = 17 ) } / 25;[127X[104X
    [4X[28X[ 408/5, 408/5 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThis implies that the restriction of [22Xψ[122X to [22XS[122X is the sum [22Xψ_S = π_1 + π_2[122X, say,
  of the first two interesting permutation characters of [22XS[122X.[133X
  
  [33X[0;0YThe subgroup [22XG_8[122X of [22XS[122X extends to a group of the structure [22XH_2 = Ω^-(4,2).2 ×
  Ω^-(8,2).2[122X  in  [22XG[122X,  inducing  the  trivial  characters  of [22XH_2[122X to [22XG[122X yields a
  permutation character [22Xφ[122X of [22XG[122X whose restriction to [22XS[122X is the third permutation
  character [22Xφ_S = π_3[122X, say.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xh2:= CharacterTable( "S5" ) * CharacterTable( "O8-(2).2" );;[127X[104X
    [4X[25Xgap>[125X [27Xphi:= PossiblePermutationCharacters( h2, t );;[127X[104X
    [4X[25Xgap>[125X [27XLength( phi );[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  have  [22Xπ_1(1)  =  π_2(1)[122X  and [22Xπ_1(s) = π_2(s)[122X, the latter again because [22XS[122X
  contains only one class of subgroups of order [22X85[122X.[133X
  
  [33X[0;0YNow statement (a) follows from the fact that [22Xπ_i(s) = 1[122X holds for [22X1 ≤ i ≤ 3[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xprim:= Concatenation( psi, phi );;[127X[104X
    [4X[25Xgap>[125X [27Xspos:= Position( OrdersClassRepresentatives( t ), 85 );[127X[104X
    [4X[28X213[128X[104X
    [4X[25Xgap>[125X [27XList( prim, x -> x[ spos ] );[127X[104X
    [4X[28X[ 2, 1 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  statement (b),  we  compute  [22Xσ(S,s)[122X. Note that we have to consider only
  classes inside [22XS = G^'[122X, and that[133X
  
  
  [24X[33X[0;6Yσ( g, s ) = ∑_i=1^3 fracπ_i(s) ⋅ π_i(g)π_i(1) = fracψ(s) ⋅ ψ(g)ψ(1) + fracφ(s) ⋅ φ(g)φ(1)[133X[124X
  
  [33X[0;0Yholds for [22Xg ∈ S^×[122X, so the characters [22Xψ[122X and [22Xφ[122X are sufficient.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xapprox:= ApproxP( prim, spos );;[127X[104X
    [4X[25Xgap>[125X [27XMaximum( approx{ ClassPositionsOfDerivedSubgroup( t ) } );[127X[104X
    [4X[28X7675/1031184[128X[104X
  [4X[32X[104X
  
  [33X[0;0YStatement (c)  follows  from  considering the outer involution classes. Note
  that  by [BGK08,  Remark after Proposition 5.14], only the subgroup [22XH_2[122X need
  to be considered, no novelties appear.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMaximum( approx{ Difference([127X[104X
    [4X[25X>[125X [27X     PositionsProperty( OrdersClassRepresentatives( t ), IsPrimeInt ),[127X[104X
    [4X[25X>[125X [27X     ClassPositionsOfDerivedSubgroup( t ) ) } );[127X[104X
    [4X[28X73/1008[128X[104X
  [4X[32X[104X
  
  
  [1X11.4-9 [33X[0;0Y[22XO_12^-(2)[122X[101X[1X[133X[101X
  
  [33X[0;0YWe show that [22XS = O_12^-(2) = Ω^-(12,2)[122X satisfies the following.[133X
  
  [8X(a)[108X
        [33X[0;6YFor  [22Xs  ∈  S[122X  irreducible of order [22X2^6+1 = 65[122X, [22XMM(S,s)[122X consists of two
        groups  of the types [22XU_4(4).2 = Ω^-(6,4).2[122X and [22XL_2(64).3 = Ω^-(4,8).3[122X,
        respectively.[133X
  
  [8X(b)[108X
        [33X[0;6Y[22Xσ(S,s) = 1/1023[122X.[133X
  
  [8X(c)[108X
        [33X[0;6Y[22Xσ^'(Aut(S),s) = 1/347820[122X.[133X
  
  [33X[0;0YBy [Ber00],  [22XMM(S,s)[122X  consists  of extension field subgroups, which have the
  structures    [22XU_4(4).2[122X    and   [22XL_2(64).3[122X,   respectively,   and   by [KL90,
  Prop. 4.3.16], there is just one class of each of these types.[133X
  
  [33X[0;0YNewer  versions  of  the  [5XGAP[105X  Character Table Library contain the character
  table  of  [22XS[122X,  but  using this table for the computations is not easier than
  using  the table of [22XG = Aut(S) = O_12^-(2).2[122X, which was already available at
  the  times  when  the  first  version  of  these examples was created. So we
  compute  the permutation characters [22Xπ_1, π_2[122X of the extensions of the groups
  in  [22XMM(S,s)[122X  to  [22XG[122X –these maximal subgroups have the structures [22XU_4(4).4[122X and
  [22XL_2(64).6[122X,   respectively–  and  compute  the  fixed  point  ratios  of  the
  restrictions to [22XS[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "O12-(2).2" );;[127X[104X
    [4X[25Xgap>[125X [27Xs1:= CharacterTable( "U4(4).4" );;[127X[104X
    [4X[25Xgap>[125X [27Xpi1:= PossiblePermutationCharacters( s1, t );;[127X[104X
    [4X[25Xgap>[125X [27Xs2:= CharacterTable( "L2(64).6" );;[127X[104X
    [4X[25Xgap>[125X [27Xpi2:= PossiblePermutationCharacters( s2, t );;[127X[104X
    [4X[25Xgap>[125X [27Xprim:= Concatenation( pi1, pi2 );;  Length( prim );[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow statement (a) follows from the fact that [22Xπ_1(s) = π_2(s) = 1[122X holds.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xspos:= Position( OrdersClassRepresentatives( t ), 65 );;[127X[104X
    [4X[25Xgap>[125X [27XList( prim, x -> x[ spos ] );[127X[104X
    [4X[28X[ 1, 1 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  statement (b),  we  compute  [22Xσ(S,s)[122X; note that we have to consider only
  classes inside [22XS = G^'[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xapprox:= ApproxP( prim, spos );;[127X[104X
    [4X[25Xgap>[125X [27XMaximum( approx{ ClassPositionsOfDerivedSubgroup( t ) } );[127X[104X
    [4X[28X1/1023[128X[104X
  [4X[32X[104X
  
  [33X[0;0YStatement (c) follows from the values on the outer involution classes.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMaximum( approx{ Difference([127X[104X
    [4X[25X>[125X [27X     PositionsProperty( OrdersClassRepresentatives( t ), IsPrimeInt ),[127X[104X
    [4X[25X>[125X [27X     ClassPositionsOfDerivedSubgroup( t ) ) } );[127X[104X
    [4X[28X1/347820[128X[104X
  [4X[32X[104X
  
  
  [1X11.4-10 [33X[0;0Y[22XS_6(4)[122X[101X[1X[133X[101X
  
  [33X[0;0YWe show that [22XS = S_6(4) = Sp(6,4)[122X satisfies the following.[133X
  
  [8X(a)[108X
        [33X[0;6YFor  [22Xs  ∈ S[122X irreducible of order [22X65[122X, [22XMM(S,s)[122X consists of two groups of
        the   types   [22XU_4(4).2   =  Ω^-(6,4).2[122X  and  [22XL_2(64).3  =  Sp(2,64).3[122X,
        respectively.[133X
  
  [8X(b)[108X
        [33X[0;6Y[22Xσ(S,s) = 16/63[122X.[133X
  
  [8X(c)[108X
        [33X[0;6Y[22Xσ^'(Aut(S),s) = 0[122X.[133X
  
  [33X[0;0YBy [Ber00],  the  element  [22Xs[122X  is contained in maximal subgroups of the given
  types,  and  by [KL90,  Prop. 4.3.10,  4.8.6], there is exactly one class of
  these subgroups.[133X
  
  [33X[0;0YThe  character  tables of these two subgroups are currently not contained in
  the  [5XGAP[105X  Character  Table  Library.  We  compute  the permutation character
  induced  from the first subgroup as the unique character of the right degree
  that is combinatorially possible (cf. [BP98]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "S6(4)" );;[127X[104X
    [4X[25Xgap>[125X [27Xdegree:= Size( t ) / ( 2 * Size( CharacterTable( "U4(4)" ) ) );;[127X[104X
    [4X[25Xgap>[125X [27Xpi1:= PermChars( t, rec( torso:= [ degree ] ) );;[127X[104X
    [4X[25Xgap>[125X [27XLength( pi1 );[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  index  of  the  second  subgroup  is  too  large  for this simpleminded
  approach;  therefore,  we  first  restrict  the  set of possible irreducible
  constituents  of the permutation character to those of [22X1_H^G[122X, where [22XH[122X is the
  derived subgroup of [22XL_2(64).3[122X, for which the character table is available.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCharacterTable( "L2(64).3" );  CharacterTable( "U4(4).2" );[127X[104X
    [4X[28Xfail[128X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27Xs:= CharacterTable( "L2(64)" );;[127X[104X
    [4X[25Xgap>[125X [27Xsubpi:= PossiblePermutationCharacters( s, t );;[127X[104X
    [4X[25Xgap>[125X [27XLength( subpi );[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27Xscp:= MatScalarProducts( t, Irr( t ), subpi );;[127X[104X
    [4X[25Xgap>[125X [27Xnonzero:= PositionsProperty( scp[1], x -> x <> 0 );[127X[104X
    [4X[28X[ 1, 11, 13, 14, 17, 18, 32, 33, 56, 58, 59, 73, 74, 77, 78, 79, 80, [128X[104X
    [4X[28X  93, 95, 96, 103, 116, 117, 119, 120 ][128X[104X
    [4X[25Xgap>[125X [27Xconst:= RationalizedMat( Irr( t ){ nonzero } );;[127X[104X
    [4X[25Xgap>[125X [27Xdegree:= Size( t ) / ( 3 * Size( s ) );[127X[104X
    [4X[28X5222400[128X[104X
    [4X[25Xgap>[125X [27Xpi2:= PermChars( t, rec( torso:= [ degree ], chars:= const ) );;[127X[104X
    [4X[25Xgap>[125X [27XLength( pi2 );[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27Xprim:= Concatenation( pi1, pi2 );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YNow statement (a) follows from the fact that [22Xπ_1(s) = π_2(s) = 1[122X holds.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xspos:= Position( OrdersClassRepresentatives( t ), 65 );;[127X[104X
    [4X[25Xgap>[125X [27XList( prim, x -> x[ spos ] );[127X[104X
    [4X[28X[ 1, 1 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor statement (b), we compute [22Xσ(G,s)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMaximum( ApproxP( prim, spos ) );[127X[104X
    [4X[28X16/63[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  order to prove statement (c), we have to consider only the extensions of
  the  above  permutation  characters  of  [22XS[122X  to  [22XAut(S)  ≅  S.2[122X  (cf. [BGK08,
  Section 2.2]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt2:= CharacterTable( "S6(4).2" );;[127X[104X
    [4X[25Xgap>[125X [27Xtfust2:= GetFusionMap( t, t2 );;[127X[104X
    [4X[25Xgap>[125X [27Xcand:= List( prim, x -> CompositionMaps( x, InverseMap( tfust2 ) ) );;[127X[104X
    [4X[25Xgap>[125X [27Xext:= List( cand, pi -> PermChars( t2, rec( torso:= pi ) ) );[127X[104X
    [4X[28X[ [ Character( CharacterTable( "S6(4).2" ),[128X[104X
    [4X[28X      [ 2016, 512, 96, 128, 32, 120, 0, 6, 16, 40, 24, 0, 8, 136, 1, [128X[104X
    [4X[28X          6, 6, 1, 32, 0, 8, 6, 2, 0, 2, 0, 0, 4, 0, 16, 32, 1, 8, 2, [128X[104X
    [4X[28X          6, 2, 1, 2, 4, 0, 0, 1, 6, 0, 1, 10, 0, 1, 1, 0, 10, 10, 4, [128X[104X
    [4X[28X          0, 1, 0, 2, 0, 2, 1, 2, 2, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, [128X[104X
    [4X[28X          0, 0, 0, 32, 0, 0, 8, 0, 0, 0, 0, 8, 8, 0, 0, 0, 0, 8, 0, [128X[104X
    [4X[28X          0, 0, 2, 2, 0, 2, 2, 0, 2, 2, 2, 0, 0 ] ) ], [128X[104X
    [4X[28X  [ Character( CharacterTable( "S6(4).2" ),[128X[104X
    [4X[28X      [ 5222400, 0, 0, 0, 1280, 0, 960, 120, 0, 0, 0, 0, 0, 0, 1600, [128X[104X
    [4X[28X          0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 1, 0, 0, 15, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X          0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, [128X[104X
    [4X[28X          0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, [128X[104X
    [4X[28X          0, 0, 960, 0, 0, 0, 16, 0, 24, 12, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X          0, 0, 0, 0, 4, 1, 0, 0, 3, 0, 0, 0, 0, 0 ] ) ] ][128X[104X
    [4X[25Xgap>[125X [27Xspos2:= Position( OrdersClassRepresentatives( t2 ), 65 );;[127X[104X
    [4X[25Xgap>[125X [27Xsigma:= ApproxP( Concatenation( ext ), spos2 );;[127X[104X
    [4X[25Xgap>[125X [27XMaximum( approx{ Difference([127X[104X
    [4X[25X>[125X [27X     PositionsProperty( OrdersClassRepresentatives( t2 ), IsPrimeInt ),[127X[104X
    [4X[25X>[125X [27X     ClassPositionsOfDerivedSubgroup( t2 ) ) } );[127X[104X
    [4X[28X0[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor the simple group, we can [13Xalternatively[113X consider a reducible element [22Xs: 2
  perp  4[122X  of order [22X85[122X, which is a multiple of the primitive prime divisor [22Xr =
  17[122X  of  [22X4^4-1[122X.  So  we  have  [22Xe  =  4[122X,  [22Xd = 6[122X, and [22Xq = 4[122X, in the terminology
  of [GPPS99].  Then [22XMM(S,s)[122X consists of two groups, of the types [22XΩ^+(6,4).2 ≅
  L_4(4).2_2[122X  and  [22XSp(2,4)  ×  Sp(4,4)[122X. This can be shown by checking [GPPS99,
  Ex. 2.1–2.9].  Ex. 2.1 yields the candidates [22XΩ^±(6,4).2[122X, but only [22XΩ^+(6,4).2[122X
  contains   elements  of  order  [22X85[122X.  Ex. 2.2  yields  the  stabilizer  of  a
  two-dimensional  subspace,  which  has  the  structure  [22XSp(2,4)  ×  Sp(4,4)[122X,
  by [KL90].  All other cases except Ex. 2.4 (b) are excluded by the fact that
  [22Xr = 4e+1[122X, and Ex. 2.4 (b) does not apply because [22Xd/gcd(d,e)[122X is odd.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSigmaFromMaxes( CharacterTable( "S6(4)" ), "85A",[127X[104X
    [4X[25X>[125X [27X   [ CharacterTable( "L4(4).2_2" ),[127X[104X
    [4X[25X>[125X [27X     CharacterTable( "A5" ) * CharacterTable( "S4(4)" ) ], [ 1, 1 ] );[127X[104X
    [4X[28X142/455[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThis  bound  is not as good as the one obtained from the irreducible element
  of order [22X65[122X used above.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27X16/63 < 142/455;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X11.4-11 [33X[0;0Y[22X∗[122X[101X[1X [22XS_6(5)[122X[101X[1X[133X[101X
  
  [33X[0;0YWe show that [22XS = S_6(5) = PSp(6,5)[122X satisfies the following.[133X
  
  [8X(a)[108X
        [33X[0;6YFor  [22Xs ∈ S[122X of the type [22X2 perp 4[122X (i. e., the preimage of [22Xs[122X in [22XSp(6,5) =
        2.G[122X  decomposes  the  natural [22X6[122X-dimensional module for [22XSp(6,5)[122X into an
        orthogonal  sum  of two irreducible modules of the dimensions [22X2[122X and [22X4[122X,
        respectively)  and  of  order [22X78[122X, [22XMM(S,s)[122X consists of one group of the
        type [22XG_2 = 2.(PSp(2,5) × PSp(4,5))[122X.[133X
  
  [8X(b)[108X
        [33X[0;6Y[22Xσ(S,s) = 9/217[122X.[133X
  
  [33X[0;0YThe order of [22Xs[122X is a multiple of the primitive prime divisor [22Xr = 13[122X of [22X5^4-1[122X,
  so  we  have  [22Xe  =  4[122X,  [22Xd = 6[122X, and [22Xq = 5[122X, in the terminology of [GPPS99]. We
  check [GPPS99,  Ex. 2.1–2.9]. Ex. 2.1 does not apply because the classes [22XC_5[122X
  and  [22XC_8[122X  are  empty  by [KL90,  Table 3.5.C],  Ex. 2.2  yields  exactly the
  stabilizer  [22XG_2[122X  of  a  [22X2[122X-dimensional  subspace,  Ex. 2.4 (b) does not apply
  because [22Xd/gcd(d,e)[122X is odd, and all other cases are excluded by the fact that
  [22Xr = 3e+1[122X.[133X
  
  [33X[0;0YThe  group [22XG_2[122X has the structure [22X2.(PSp(2,5) × PSp(4,5))[122X, which is a central
  product of [22XSp(2,5) ≅ 2.A_5[122X and [22XSp(4,5) = 2.S_4(5)[122X (see [KL90, Prop. 4.1.3]).
  The character table of [22XG_2[122X can be derived from that of the direct product of
  [22X2.A_5[122X  and [22X2.S_4(5)[122X, by factoring out the diagonal central subgroup of order
  two.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "S6(5)" );;[127X[104X
    [4X[25Xgap>[125X [27Xs1:= CharacterTable( "2.A5" );;[127X[104X
    [4X[25Xgap>[125X [27Xs2:= CharacterTable( "2.S4(5)" );;[127X[104X
    [4X[25Xgap>[125X [27Xdp:= s1 * s2;[127X[104X
    [4X[28XCharacterTable( "2.A5x2.S4(5)" )[128X[104X
    [4X[25Xgap>[125X [27Xc:= Difference( ClassPositionsOfCentre( dp ), Union([127X[104X
    [4X[25X>[125X [27X                       GetFusionMap( s1, dp ), GetFusionMap( s2, dp ) ) );[127X[104X
    [4X[28X[ 62 ][128X[104X
    [4X[25Xgap>[125X [27Xs:= dp / c;[127X[104X
    [4X[28XCharacterTable( "2.A5x2.S4(5)/[ 1, 62 ]" )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow we compute [22Xσ(S,s)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSigmaFromMaxes( t, "78A", [ s ], [ 1 ] );[127X[104X
    [4X[28X9/217[128X[104X
  [4X[32X[104X
  
  
  [1X11.4-12 [33X[0;0Y[22XS_8(3)[122X[101X[1X[133X[101X
  
  [33X[0;0YWe show that [22XS = S_8(3) = PSp(8,3)[122X satisfies the following.[133X
  
  [8X(a)[108X
        [33X[0;6YFor  [22Xs ∈ S[122X irreducible of order [22X41[122X, [22XMM(S,s)[122X consists of one group [22XM[122X of
        the type [22XS_4(9).2 = PSp(4,9).2[122X.[133X
  
  [8X(b)[108X
        [33X[0;6Y[22Xσ(S,s) = 1/546[122X.[133X
  
  [8X(c)[108X
        [33X[0;6YThe preimage of [22Xs[122X in the matrix group [22X2.S_8(3) = Sp(8,3)[122X can be chosen
        of order [22X82[122X, and the preimage of [22XM[122X is [22X2.S_4(9).2 = Sp(4,9).2[122X.[133X
  
  [33X[0;0YBy [Ber00],  the  only  maximal  subgroups  of  [22XS[122X  that  contain irreducible
  elements  of order [22X(3^4+1)/2 = 41[122X are of extension field type, and by [KL90,
  Prop. 4.3.10], these groups have the structure [22XS_4(9).2[122X and there is exactly
  one class of these groups.[133X
  
  [33X[0;0YThe group [22XU = S_4(9)[122X has three nontrivial outer automorphisms, the character
  table  of the subgroup [22XU.2[122X in question has the identifier [10X"S4(9).2_1"[110X, which
  follows  from  the  fact  that  the  extensions  of [22XU[122X by the other two outer
  automorphisms do not admit a class fusion into [22XS[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "S8(3)" );;[127X[104X
    [4X[25Xgap>[125X [27Xpi:= List( [ "S4(9).2_1", "S4(9).2_2", "S4(9).2_3" ],[127X[104X
    [4X[25X>[125X [27X              name -> PossiblePermutationCharacters([127X[104X
    [4X[25X>[125X [27X                          CharacterTable( name ), t ) );;[127X[104X
    [4X[25Xgap>[125X [27XList( pi, Length );[127X[104X
    [4X[28X[ 1, 0, 0 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow statement (a) follows from the fact that [22X(1_U.2)^S(s) = 1[122X holds.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xspos:= Position( OrdersClassRepresentatives( t ), 41 );;[127X[104X
    [4X[25Xgap>[125X [27Xpi[1][1][ spos ];[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow we compute [22Xσ(S,s)[122X in order to show statement (b).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMaximum( ApproxP( pi[1], spos ) );[127X[104X
    [4X[28X1/546[128X[104X
  [4X[32X[104X
  
  [33X[0;0YStatement (c)  is  clear  from  the  description  of  extension  field  type
  subgroups in [KL90].[133X
  
  
  [1X11.4-13 [33X[0;0Y[22XU_4(4)[122X[101X[1X[133X[101X
  
  [33X[0;0YWe show that [22XS = U_4(4) = SU(4,4)[122X satisfies the following.[133X
  
  [8X(a)[108X
        [33X[0;6YFor  [22Xs  ∈  S[122X  of  the  type  [22X1 perp 3[122X (i. e., [22Xs[122X decomposes the natural
        [22X4[122X-dimensional  module  for  [22XSU(4,4)[122X  into  an  orthogonal  sum  of two
        irreducible  modules  of  the dimensions [22X1[122X and [22X3[122X, respectively) and of
        order  [22X4^3+1 = 65[122X, [22XMM(S,s)[122X consists of one group of the type [22XG_1 = 5 ×
        U_3(4) = GU(3,4)[122X.[133X
  
  [8X(b)[108X
        [33X[0;6Y[22Xσ(S,s) = 209/3264[122X.[133X
  
  [33X[0;0YBy [MSW94],  the only maximal subgroups of [22XS[122X that contain [22Xs[122X are one class of
  stabilizers  [22XH ≅ 5 × U_3(4)[122X of this decomposition, and clearly there is only
  one such group containing [22Xs[122X.[133X
  
  [33X[0;0YNote that [22XH[122X has index [22X3264[122X in [22XS[122X, since [22XS[122X has two orbits on the [22X1[122X-dimensional
  subspaces,  of lengths [22X1105[122X and [22X3264[122X, respectively, and elements of order [22X13
  = 65/5[122X lie in the stabilizers of points in the latter orbit.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:= SU(4,4);;[127X[104X
    [4X[25Xgap>[125X [27Xorbs:= OrbitsDomain( g, NormedRowVectors( GF(16)^4 ), OnLines );;[127X[104X
    [4X[25Xgap>[125X [27Xorblen:= List( orbs, Length );[127X[104X
    [4X[28X[ 1105, 3264 ][128X[104X
    [4X[25Xgap>[125X [27XList( orblen, x -> x mod 13 );[127X[104X
    [4X[28X[ 0, 1 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe   compute  the  permutation  character  [22X1_G_1^S[122X;  there  is  exactly  one
  combinatorially possible permutation character of degree [22X3264[122X (cf. [BP98]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "U4(4)" );;[127X[104X
    [4X[25Xgap>[125X [27Xpi:= PermChars( t, rec( torso:= [ orblen[2] ] ) );;[127X[104X
    [4X[25Xgap>[125X [27XLength( pi );[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow we compute [22Xσ(S,s)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xspos:= Position( OrdersClassRepresentatives( t ), 65 );;[127X[104X
    [4X[25Xgap>[125X [27XMaximum( ApproxP( pi, spos ) );[127X[104X
    [4X[28X209/3264[128X[104X
  [4X[32X[104X
  
  
  [1X11.4-14 [33X[0;0Y[22XU_6(2)[122X[101X[1X[133X[101X
  
  [33X[0;0YWe show that [22XS = U_6(2) = PSU(6,2)[122X satisfies the following.[133X
  
  [8X(a)[108X
        [33X[0;6YFor  [22Xs  ∈  S[122X  of  order  [22X11[122X, [22XMM(S,s)[122X consists of one group of the type
        [22XU_5(2) = SU(5,2)[122X and three groups of the type [22XM_22[122X.[133X
  
  [8X(b)[108X
        [33X[0;6Y[22Xσ(S,s) = 5/21[122X.[133X
  
  [8X(c)[108X
        [33X[0;6YThe preimage of [22Xs[122X in the matrix group [22XSU(6,2) = 3.U_6(2)[122X can be chosen
        of  order  [22X33[122X,  and  the  preimages  of the groups in [22XMM(S,s)[122X have the
        structures [22X3 × U_5(2) ≅ GU(5,2)[122X and [22X3.M_22[122X, respectively.[133X
  
  [8X(d)[108X
        [33X[0;6YWith  [22Xs[122X  as  in (a),  the automorphic extensions [22XS.2[122X, [22XS.3[122X of [22XS[122X satisfy
        [22Xσ^'(S.2,s) = 5/96[122X and [22Xσ^'(S.3,s) = 59/224[122X.[133X
  
  [33X[0;0YAccording  to  the  list of maximal subgroups of [22XS[122X in [CCN+85, p. 115], [22Xs[122X is
  contained  exactly  in maximal subgroups of the types [22XU_5(2)[122X (one class) and
  [22XM_22[122X (three classes).[133X
  
  [33X[0;0YThe  permutation  character  of  the  action  on  the  cosets of [22XU_5(2)[122X type
  subgroups  is  uniquely  determined  by  the  character tables. We get three
  possibilities  for  the  permutation  character  on  the cosets of [22XM_22[122X type
  subgroups;  they  correspond to the three classes of such subgroups, because
  each  of  these  classes  contains  elements in exactly one of the conjugacy
  classes  [10X4C[110X,  [10X4D[110X, and [10X4E[110X of elements in [22XS[122X, and these classes are fused under
  the outer automorphism of [22XS[122X of order three.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "U6(2)" );;[127X[104X
    [4X[25Xgap>[125X [27Xs1:= CharacterTable( "U5(2)" );;[127X[104X
    [4X[25Xgap>[125X [27Xpi1:= PossiblePermutationCharacters( s1, t );;[127X[104X
    [4X[25Xgap>[125X [27XLength( pi1 );[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27Xs2:= CharacterTable( "M22" );;[127X[104X
    [4X[25Xgap>[125X [27Xpi2:= PossiblePermutationCharacters( s2, t );[127X[104X
    [4X[28X[ Character( CharacterTable( "U6(2)" ),[128X[104X
    [4X[28X  [ 20736, 0, 384, 0, 0, 0, 54, 0, 0, 0, 0, 48, 0, 16, 6, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 6, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( "U6(2)" ),[128X[104X
    [4X[28X  [ 20736, 0, 384, 0, 0, 0, 54, 0, 0, 0, 48, 0, 0, 16, 6, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 6, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( "U6(2)" ),[128X[104X
    [4X[28X  [ 20736, 0, 384, 0, 0, 0, 54, 0, 0, 48, 0, 0, 0, 16, 6, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 6, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27Ximgs:= Set( pi2, x -> Position( x, 48 ) );[127X[104X
    [4X[28X[ 10, 11, 12 ][128X[104X
    [4X[25Xgap>[125X [27XAtlasClassNames( t ){ imgs };[127X[104X
    [4X[28X[ "4C", "4D", "4E" ][128X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( t, CharacterTable( "U6(2).3" ) ){ imgs };[127X[104X
    [4X[28X[ 10, 10, 10 ][128X[104X
    [4X[25Xgap>[125X [27Xprim:= Concatenation( pi1, pi2 );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YNow statement (a) follows from the fact that the permutation characters have
  the value [22X1[122X on [22Xs[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xspos:= Position( OrdersClassRepresentatives( t ), 11 );;[127X[104X
    [4X[25Xgap>[125X [27XList( prim, x -> x[ spos ] );[127X[104X
    [4X[28X[ 1, 1, 1, 1 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor statement (b), we compute [22Xσ(S,s)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMaximum( ApproxP( prim, spos ) );[127X[104X
    [4X[28X5/21[128X[104X
  [4X[32X[104X
  
  [33X[0;0YStatement (c) follows from [CCN+85], plus the information that [22X3.U_6(2)[122X does
  not contain groups of the structure [22X3 × M_22[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPossibleClassFusions([127X[104X
    [4X[25X>[125X [27X       CharacterTable( "Cyclic", 3 ) * CharacterTable( "M22" ),[127X[104X
    [4X[25X>[125X [27X       CharacterTable( "3.U6(2)" ) );[127X[104X
    [4X[28X[  ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  statement (d),  we  need that the relevant maximal subgroups of [22XS.2[122X are
  [22XU_5(2).2[122X  and one subgroup [22XM_22.2[122X, and that the relevant maximal subgroup of
  [22XS.3[122X is [22XU_5(2) × 3[122X, see [CCN+85, p. 115].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSigmaFromMaxes( CharacterTable( "U6(2).2" ), "11AB",[127X[104X
    [4X[25X>[125X [27X       [ CharacterTable( "U5(2).2" ), CharacterTable( "M22.2" ) ],[127X[104X
    [4X[25X>[125X [27X       [ 1, 1 ], "outer" );[127X[104X
    [4X[28X5/96[128X[104X
    [4X[25Xgap>[125X [27XSigmaFromMaxes( CharacterTable( "U6(2).3" ), "11A",[127X[104X
    [4X[25X>[125X [27X       [ CharacterTable( "U5(2)" ) * CharacterTable( "Cyclic", 3 ) ],[127X[104X
    [4X[25X>[125X [27X       [ 1 ], "outer" );[127X[104X
    [4X[28X59/224[128X[104X
  [4X[32X[104X
  
  
  [1X11.5 [33X[0;0YComputations using Groups[133X[101X
  
  [33X[0;0YBefore we start the computations using groups, we clean the workspace.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCleanWorkspace();[127X[104X
  [4X[32X[104X
  
  
  [1X11.5-1 [33X[0;0Y[22XA_2m+1[122X[101X[1X, [22X2 ≤ m ≤ 11[122X[101X[1X[133X[101X
  
  [33X[0;0YFor alternating groups of odd degree [22Xn = 2m+1[122X, we choose [22Xs[122X to be an [22Xn[122X-cycle.
  The interesting cases in [BGK08, Proposition 6.7] are [22X5 ≤ n ≤ 23[122X.[133X
  
  [33X[0;0YIn  each  case,  we compute representatives of the maximal subgroups of [22XA_n[122X,
  consider   only  those  that  contain  an  [22Xn[122X-cycle,  and  then  compute  the
  permutation  characters.  Additionally, we show also the names that are used
  for  the  subgroups in the [5XGAP[105X Library of Transitive Groups, see [Hul05] and
  the documentation of this library in the [5XGAP[105X Reference Manual.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPrimitivesInfoForOddDegreeAlternatingGroup:= function( n )[127X[104X
    [4X[25X>[125X [27X    local G, max, cycle, spos, prim, nonz;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    G:= AlternatingGroup( n );[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    # Compute representatives of the classes of maximal subgroups.[127X[104X
    [4X[25X>[125X [27X    max:= MaximalSubgroupClassReps( G );[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    # Omit subgroups that cannot contain an `n'-cycle.[127X[104X
    [4X[25X>[125X [27X    max:= Filtered( max, m -> IsTransitive( m, [ 1 .. n ] ) );[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    # Compute the permutation characters.[127X[104X
    [4X[25X>[125X [27X    cycle:= [];[127X[104X
    [4X[25X>[125X [27X    cycle[ n-1 ]:= 1;[127X[104X
    [4X[25X>[125X [27X    spos:= PositionProperty( ConjugacyClasses( CharacterTable( G ) ),[127X[104X
    [4X[25X>[125X [27X               c -> CycleStructurePerm( Representative( c ) ) = cycle );[127X[104X
    [4X[25X>[125X [27X    prim:= List( max, m -> TrivialCharacter( m )^G );[127X[104X
    [4X[25X>[125X [27X    nonz:= PositionsProperty( prim, x -> x[ spos ] <> 0 );[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    # Compute the subgroup names and the multiplicities.[127X[104X
    [4X[25X>[125X [27X    return rec( spos := spos,[127X[104X
    [4X[25X>[125X [27X                prim := prim{ nonz },[127X[104X
    [4X[25X>[125X [27X                grps := List( max{ nonz },[127X[104X
    [4X[25X>[125X [27X                              m -> TransitiveGroup( n,[127X[104X
    [4X[25X>[125X [27X                                       TransitiveIdentification( m ) ) ),[127X[104X
    [4X[25X>[125X [27X                mult := List( prim{ nonz }, x -> x[ spos ] ) );[127X[104X
    [4X[25X>[125X [27Xend;;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  sets [22XMM/~(s)[122X and the values [22Xσ(A_n,s)[122X are as follows. For each degree in
  question,  the  first  list shows names for representatives of the conjugacy
  classes of maximal subgroups containing a fixed [22Xn[122X-cycle, and the second list
  shows the number of conjugates in each class.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfor n in [ 5, 7 .. 23 ] do[127X[104X
    [4X[25X>[125X [27X     prim:= PrimitivesInfoForOddDegreeAlternatingGroup( n );[127X[104X
    [4X[25X>[125X [27X     bound:= Maximum( ApproxP( prim.prim, prim.spos ) );[127X[104X
    [4X[25X>[125X [27X     Print( n, ": ", prim.grps, ", ", prim.mult, ", ", bound, "\n" );[127X[104X
    [4X[25X>[125X [27Xod;[127X[104X
    [4X[28X5: [ D(5) = 5:2 ], [ 1 ], 1/3[128X[104X
    [4X[28X7: [ L(7) = L(3,2), L(7) = L(3,2) ], [ 1, 1 ], 2/5[128X[104X
    [4X[28X9: [ 1/2[S(3)^3]S(3), L(9):3=P|L(2,8) ], [ 1, 3 ], 9/35[128X[104X
    [4X[28X11: [ M(11), M(11) ], [ 1, 1 ], 2/105[128X[104X
    [4X[28X13: [ F_78(13)=13:6, L(13)=PSL(3,3), L(13)=PSL(3,3) ], [ 1, 2, 2 ], 4/[128X[104X
    [4X[28X1155[128X[104X
    [4X[28X15: [ 1/2[S(3)^5]S(5), 1/2[S(5)^3]S(3), L(15)=A_8(15)=PSL(4,2), [128X[104X
    [4X[28X  L(15)=A_8(15)=PSL(4,2) ], [ 1, 1, 1, 1 ], 29/273[128X[104X
    [4X[28X17: [ L(17):4=PYL(2,16), L(17):4=PYL(2,16) ], [ 1, 1 ], 2/135135[128X[104X
    [4X[28X19: [ F_171(19)=19:9 ], [ 1 ], 1/6098892800[128X[104X
    [4X[28X21: [ t21n150, t21n161, t21n91 ], [ 1, 1, 2 ], 29/285[128X[104X
    [4X[28X23: [ M(23), M(23) ], [ 1, 1 ], 2/130945815[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn   the   above   output,  a  subgroup  printed  as  [10X1/2[S([110X[22Xn_1[122X[10X)^[110X[22Xn_2[122X[10X]S([110X[22Xn_2[122X[10X)[110X,
  [10X1/2[S([110X[22Xn_1[122X[10X)^[110X[22Xn_2[122X[10X]S([110X[22Xn_2[122X[10X)[110X,  where [22Xn = n_1 n_2[122X holds, denotes the intersection of
  [22XA_n[122X  with  the  wreath  product  [22XS_n_1  ≀  S_n_2 ≤ S_n[122X. (Note that the [5XAtlas[105X
  denotes  the subgroup [10X1/2[S(3)^3]S(3)[110X of [22XA_9[122X as [22X3^3:S_4[122X.) The groups printed
  as  [10XP|L(2,8)[110X  and [10XPYL(2,16)[110X denote [22XPΓL(2,8)[122X and [22XPΓL(2,16)[122X, respectively. And
  the  three  subgroups of [22XA_21[122X have the structures [22X(S_3 ≀ S_7) ∩ A_21[122X, [22X(S_7 ≀
  S_3) ∩ A_21[122X, and [22XPGL(3,4)[122X, respectively.[133X
  
  [33X[0;0YNote  that  [22XA_9[122X  contains  two conjugacy classes of maximal subgroups of the
  type  [22XPΓL(2,8)  ≅  L_2(8):3[122X,  and  that  each [22X9[122X-cycle in [22XA_9[122X is contained in
  exactly  three [13Xconjugate[113X subgroups of this type. For [22Xn ∈ { 13, 15, 17 }[122X, [22XA_n[122X
  contains  two  conjugacy  classes  of isomorphic maximal subgroups of linear
  type,  and  each [22Xn[122X-cycle is contained in subgroups from each class. Finally,
  [22XA_21[122X contains only one class of maximal subgroups of linear type.[133X
  
  [33X[0;0YFor  the  two  groups  [22XA_5[122X  and  [22XA_7[122X,  the  values  computed  above  are not
  sufficient. See Section [14X11.5-2[114X and [14X11.5-4[114X for a further treatment.[133X
  
  [33X[0;0YThe  above  computations look like a brute-force approach, but note that the
  computation  of the maximal subgroups of alternating and symmetric groups in
  [5XGAP[105X  uses  the  classification  of  these  subgroups, and also the conjugacy
  classes  of  elements  in  alternating  and symmetric groups can be computed
  cheaply.[133X
  
  [33X[0;0YAlternative  (character-theoretic)  computations for [22Xn ∈ { 5, 7, 9, 11, 13 }[122X
  were shown in Section [14X11.4-3[114X. (A hand calculation for the case [22Xn = 19[122X can be
  found in [BW75].)[133X
  
  
  [1X11.5-2 [33X[0;0Y[22XA_5[122X[101X[1X[133X[101X
  
  [33X[0;0YWe show that [22XS = A_5[122X satisfies the following.[133X
  
  [8X(a)[108X
        [33X[0;6Y[22Xσ(S)  =  1/3[122X,  and this value is attained exactly for [22Xσ(S,s)[122X with [22Xs[122X of
        order [22X5[122X.[133X
  
  [8X(b)[108X
        [33X[0;6YFor [22Xs ∈ S[122X of order [22X5[122X, [22XMM(S,s)[122X consists of one group of the type [22XD_10[122X.[133X
  
  [8X(c)[108X
        [33X[0;6Y[22XP(S)  =  1/3[122X,  and this value is attained exactly for [22XP(S,s)[122X with [22Xs[122X of
        order [22X5[122X.[133X
  
  [8X(d)[108X
        [33X[0;6YEach  element  in  [22XS[122X  together  with  one  of  [22X(1,2)(3,4)[122X, [22X(1,3)(2,4)[122X,
        [22X(1,4)(2,3)[122X generates a proper subgroup of [22XS[122X.[133X
  
  [8X(e)[108X
        [33X[0;6YBoth   the  spread  and  the  uniform  spread  of  [22XS[122X  is  exactly  two
        (see [BW75]), with [22Xs[122X of order [22X5[122X.[133X
  
  [33X[0;0YStatement (a)   follows   from   inspection  of  the  primitive  permutation
  characters, cf. Section [14X11.4-3[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "A5" );;[127X[104X
    [4X[25Xgap>[125X [27XProbGenInfoSimple( t );[127X[104X
    [4X[28X[ "A5", 1/3, 2, [ "5A" ], [ 1 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YStatement (b) can be read off from the primitive permutation characters, and
  the fact that the unique class of maximal subgroups that contain elements of
  order [22X5[122X consists of groups of the structure [22XD_10[122X, see [CCN+85, p. 2].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XOrdersClassRepresentatives( t );[127X[104X
    [4X[28X[ 1, 2, 3, 5, 5 ][128X[104X
    [4X[25Xgap>[125X [27XPrimitivePermutationCharacters( t );[127X[104X
    [4X[28X[ Character( CharacterTable( "A5" ), [ 5, 1, 2, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 6, 2, 0, 1, 1 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 10, 2, 1, 0, 0 ] ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  statement (c),  we  compute that for all nonidentity elements [22Xs ∈ S[122X and
  involutions  [22Xg  ∈  S[122X, [22XP(g,s) ≥ 1/3[122X holds, with equality if and only if [22Xs[122X has
  order [22X5[122X. We actually compute, for class representatives [22Xs[122X, the proportion of
  involutions [22Xg[122X such that [22X⟨ g, s ⟩ ‡ S[122X holds.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:= AlternatingGroup( 5 );;[127X[104X
    [4X[25Xgap>[125X [27Xinv:= g.1^2 * g.2;[127X[104X
    [4X[28X(1,4)(2,5)[128X[104X
    [4X[25Xgap>[125X [27Xcclreps:= List( ConjugacyClasses( g ), Representative );;[127X[104X
    [4X[25Xgap>[125X [27XSortParallel( List( cclreps, Order ), cclreps );[127X[104X
    [4X[25Xgap>[125X [27XList( cclreps, Order );[127X[104X
    [4X[28X[ 1, 2, 3, 5, 5 ][128X[104X
    [4X[25Xgap>[125X [27XSize( ConjugacyClass( g, inv ) );[127X[104X
    [4X[28X15[128X[104X
    [4X[25Xgap>[125X [27Xprop:= List( cclreps,[127X[104X
    [4X[25X>[125X [27X                r -> RatioOfNongenerationTransPermGroup( g, inv, r ) );[127X[104X
    [4X[28X[ 1, 1, 3/5, 1/3, 1/3 ][128X[104X
    [4X[25Xgap>[125X [27XMinimum( prop );[127X[104X
    [4X[28X1/3[128X[104X
  [4X[32X[104X
  
  [33X[0;0YStatement (d) follows by explicit computations.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtriple:= [ (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) ];;[127X[104X
    [4X[25Xgap>[125X [27XCommonGeneratorWithGivenElements( g, cclreps, triple );[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAs  for  statement (e),  we know from (a) that the uniform spread of [22XS[122X is at
  least two, and from (d) that the spread is less than three.[133X
  
  
  [1X11.5-3 [33X[0;0Y[22XA_6[122X[101X[1X[133X[101X
  
  [33X[0;0YWe show that [22XS = A_6[122X satisfies the following.[133X
  
  [8X(a)[108X
        [33X[0;6Y[22Xσ(S)  =  2/3[122X,  and this value is attained exactly for [22Xσ(S,s)[122X with [22Xs[122X of
        order [22X5[122X.[133X
  
  [8X(b)[108X
        [33X[0;6YFor  [22Xs[122X  of order [22X5[122X, [22XMM(S,s)[122X consists of two nonconjugate groups of the
        type [22XA_5[122X.[133X
  
  [8X(c)[108X
        [33X[0;6Y[22XP(S)  =  5/9[122X,  and this value is attained exactly for [22XP(S,s)[122X with [22Xs[122X of
        order [22X5[122X.[133X
  
  [8X(d)[108X
        [33X[0;6YEach  element  in  [22XS[122X  together  with  one  of  [22X(1,2)(3,4)[122X, [22X(1,3)(2,4)[122X,
        [22X(1,4)(2,3)[122X generates a proper subgroup of [22XS[122X.[133X
  
  [8X(e)[108X
        [33X[0;6YBoth   the  spread  and  the  uniform  spread  of  [22XS[122X  is  exactly  two
        (see [BW75]), with [22Xs[122X of order [22X4[122X.[133X
  
  [8X(f)[108X
        [33X[0;6YFor  [22Xx[122X, [22Xy ∈ S_6^×[122X, there is [22Xs ∈ S_6[122X such that [22XS ⊆ ⟨ x, s ⟩ ∩ ⟨ y, s ⟩[122X.
        It  is  [13Xnot[113X  possible  to  find  [22Xs  ∈  S[122X with this property, or [22Xs[122X in a
        prescribed conjugacy class of [22XS_6[122X.[133X
  
  [8X(g)[108X
        [33X[0;6Y[22Xσ(  PGL(2,9)  )  =  1/6[122X and [22Xσ( M_10 ) = 1/9[122X, with [22Xs[122X of order [22X10[122X and [22X8[122X,
        respectively.[133X
  
  [33X[0;0Y(Note  that in this example, the optimal choice of [22Xs[122X for [22XP(S)[122X cannot be used
  to obtain the result on the exact spread.)[133X
  
  [33X[0;0YStatement (a)   follows   from   inspection  of  the  primitive  permutation
  characters, cf. Section [14X11.4-3[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "A6" );;[127X[104X
    [4X[25Xgap>[125X [27XProbGenInfoSimple( t );[127X[104X
    [4X[28X[ "A6", 2/3, 1, [ "5A" ], [ 2 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YStatement (b)  can be read off from the permutation characters, and the fact
  that  the  two classes of maximal subgroups that contain elements of order [22X5[122X
  consist of groups of the structure [22XA_5[122X, see [CCN+85, p. 4].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XOrdersClassRepresentatives( t );[127X[104X
    [4X[28X[ 1, 2, 3, 3, 4, 5, 5 ][128X[104X
    [4X[25Xgap>[125X [27Xprim:= PrimitivePermutationCharacters( t );[127X[104X
    [4X[28X[ Character( CharacterTable( "A6" ), [ 6, 2, 3, 0, 0, 1, 1 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A6" ), [ 6, 2, 0, 3, 0, 1, 1 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A6" ), [ 10, 2, 1, 1, 2, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A6" ), [ 15, 3, 3, 0, 1, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A6" ), [ 15, 3, 0, 3, 1, 0, 0 ] ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  statement (c), we first compute that for all nonidentity elements [22Xs ∈ S[122X
  and  involutions  [22Xg  ∈ S[122X, [22XP(g,s) ≥ 5/9[122X holds, with equality if and only if [22Xs[122X
  has  order  [22X5[122X.  We  actually  compute,  for  class  representatives  [22Xs[122X,  the
  proportion of involutions [22Xg[122X such that [22X⟨ g, s ⟩ ‡ S[122X holds.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS:= AlternatingGroup( 6 );;[127X[104X
    [4X[25Xgap>[125X [27Xinv:= (S.1*S.2)^2;[127X[104X
    [4X[28X(1,3)(2,5)[128X[104X
    [4X[25Xgap>[125X [27Xcclreps:= List( ConjugacyClasses( S ), Representative );;[127X[104X
    [4X[25Xgap>[125X [27XSortParallel( List( cclreps, Order ), cclreps );[127X[104X
    [4X[25Xgap>[125X [27XList( cclreps, Order );[127X[104X
    [4X[28X[ 1, 2, 3, 3, 4, 5, 5 ][128X[104X
    [4X[25Xgap>[125X [27XC:= ConjugacyClass( S, inv );;[127X[104X
    [4X[25Xgap>[125X [27XSize( C );[127X[104X
    [4X[28X45[128X[104X
    [4X[25Xgap>[125X [27Xprop:= List( cclreps,[127X[104X
    [4X[25X>[125X [27X                r -> RatioOfNongenerationTransPermGroup( S, inv, r ) );[127X[104X
    [4X[28X[ 1, 1, 1, 1, 29/45, 5/9, 5/9 ][128X[104X
    [4X[25Xgap>[125X [27XMinimum( prop );[127X[104X
    [4X[28X5/9[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  statement (c) follows from the fact that for [22Xg ∈ S[122X of order larger than
  two, [22Xσ(S,g) ≤ 1/2 < 5/9[122X holds.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XApproxP( prim, 6 );[127X[104X
    [4X[28X[ 0, 2/3, 1/2, 1/2, 0, 1/3, 1/3 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YStatement (d) follows by explicit computations.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtriple:= [ (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) ];;[127X[104X
    [4X[25Xgap>[125X [27XCommonGeneratorWithGivenElements( S, cclreps, triple );[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAn alternative triple to that in statement (d) is the one given in [BW75].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtriple:= [ (1,3)(2,4), (1,5)(2,6), (3,6)(4,5) ];;[127X[104X
    [4X[25Xgap>[125X [27XCommonGeneratorWithGivenElements( S, cclreps, triple );[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  [33X[0;0YOf course we can also construct such a triple, as follows.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XTripleWithProperty( [ [ inv ], C, C ],[127X[104X
    [4X[25X>[125X [27X       l -> ForAll( S, elm ->[127X[104X
    [4X[25X>[125X [27X  ForAny( l, x -> not IsGeneratorsOfTransPermGroup( S, [ elm, x ] ) ) ) );[127X[104X
    [4X[28X[ (1,3)(2,5), (1,3)(2,6), (1,3)(2,4) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor statement (e), we use the random approach described in Section [14X11.3-3[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:= (1,2,3,4)(5,6);;[127X[104X
    [4X[25Xgap>[125X [27Xreps:= Filtered( cclreps, x -> Order( x ) > 1 );;[127X[104X
    [4X[25Xgap>[125X [27XResetGlobalRandomNumberGenerators();[127X[104X
    [4X[25Xgap>[125X [27Xfor pair in UnorderedTuples( reps, 2 ) do[127X[104X
    [4X[25X>[125X [27X     if RandomCheckUniformSpread( S, pair, s, 40 ) <> true then[127X[104X
    [4X[25X>[125X [27X       Print( "#E  nongeneration!\n" );[127X[104X
    [4X[25X>[125X [27X     fi;[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  get no output, so a suitable element of order [22X4[122X works in all cases. Note
  that  we  cannot  use an element of order [22X5[122X, because it fixes a point in the
  natural  permutation representation, and we may take [22Xx_1 = (1,2,3)[122X and [22Xx_2 =
  (4,5,6)[122X.  With  this  argument, only elements of order [22X4[122X and double [22X3[122X-cycles
  are  possible choices for [22Xs[122X, and the latter are excluded by the fact that an
  outer  automorphism maps the class of double [22Xs[122X-cycles in [22XA_6[122X to the class of
  [22X3[122X-cycles. So no element in [22XA_6[122X of order different from [22X4[122X works.[133X
  
  [33X[0;0YNext  we show statement (f). Already in [22XA_6.2_1 = S_6[122X, elements [22Xs[122X of order [22X4[122X
  do in general not work because they do not generate with transpositions.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:= SymmetricGroup( 6 );;[127X[104X
    [4X[25Xgap>[125X [27XRatioOfNongenerationTransPermGroup( G, s, (1,2) );[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAlso,  choosing  [22Xs[122X from a prescribed conjugacy class of [22XS_6[122X (that is, also [22Xs[122X
  outside  [22XA_6[122X is allowed) with the property that [22XA_6 ⊆ ⟨ x, s ⟩ ∩ ⟨ y, s ⟩[122X is
  not  possible.  Note  that  only  [22X6[122X-cycles are possible for [22Xs[122X if [22Xx[122X and [22Xy[122X are
  commuting  transpositions,  and –applying the outer automorphism– no [22X6[122X-cycle
  works  for  two  commuting fixed-point free involutions. (The group is small
  enough for a brute force test.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgoods:= Filtered( Elements( G ),[127X[104X
    [4X[25X>[125X [27X     s -> IsGeneratorsOfTransPermGroup( G, [ s, (1,2) ] ) and[127X[104X
    [4X[25X>[125X [27X          IsGeneratorsOfTransPermGroup( G, [ s, (3,4) ] ) );;[127X[104X
    [4X[25Xgap>[125X [27XCollected( List( goods, CycleStructurePerm ) );[127X[104X
    [4X[28X[ [ [ ,,,, 1 ], 24 ] ][128X[104X
    [4X[25Xgap>[125X [27Xgoods:= Filtered( Elements( G ),[127X[104X
    [4X[25X>[125X [27X     s -> IsGeneratorsOfTransPermGroup( G, [ s, (1,2)(3,4)(5,6) ] ) and[127X[104X
    [4X[25X>[125X [27X          IsGeneratorsOfTransPermGroup( G, [ s, (1,3)(2,4)(5,6) ] ) );;[127X[104X
    [4X[25Xgap>[125X [27XCollected( List( goods, CycleStructurePerm ) );[127X[104X
    [4X[28X[ [ [ 1, 1 ], 24 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YHowever,  for  each pair of nonidentity element [22Xx[122X, [22Xy ∈ S_6[122X, there is [22Xs ∈ S_6[122X
  such  that  [22X⟨ x, s ⟩[122X and [22X⟨ y, s ⟩[122X both contain [22XA_6[122X. (If [22Xs[122X works for the pair
  [22X(x,y)[122X  then  [22Xs^g[122X  works  for [22X(x^g,y^g)[122X, so it is sufficient to consider only
  orbit representatives [22X(x,y)[122X under the conjugation action of [22XG[122X on pairs. Thus
  we check conjugacy class representatives [22Xx[122X and, for fixed [22Xx[122X, representatives
  of   orbits  of  [22XC_G(x)[122X  on  the  classes  [22Xy^G[122X,  i. e.,  representatives  of
  [22XC_G(y)[122X-[22XC_G(x)[122X-double  cosets  in  [22XG[122X.  Moreover,  clearly we can restrict the
  checks to elements [22Xx[122X, [22Xy[122X of prime order.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSgens:= GeneratorsOfGroup( S );;[127X[104X
    [4X[25Xgap>[125X [27Xprimord:= Filtered( List( ConjugacyClasses( G ), Representative ),[127X[104X
    [4X[25X>[125X [27X                       x -> IsPrimeInt( Order( x ) ) );;[127X[104X
    [4X[25Xgap>[125X [27Xfor x in primord do[127X[104X
    [4X[25X>[125X [27X     for y in primord do[127X[104X
    [4X[25X>[125X [27X       for pair in DoubleCosetRepsAndSizes( G, Centralizer( G, y ),[127X[104X
    [4X[25X>[125X [27X                       Centralizer( G, x ) ) do[127X[104X
    [4X[25X>[125X [27X         if not ForAny( G, s -> IsSubset( Group( x,s ), S ) and [127X[104X
    [4X[25X>[125X [27X                                IsSubset( Group( y^pair[1], s ), S ) ) then[127X[104X
    [4X[25X>[125X [27X           Error( [ x, y ] );[127X[104X
    [4X[25X>[125X [27X         fi;[127X[104X
    [4X[25X>[125X [27X       od;[127X[104X
    [4X[25X>[125X [27X     od;[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  other words, the spread of [22XS_6[122X is [22X2[122X but the uniform spread of [22XS_6[122X is not
  [22X2[122X but only [22X1[122X.[133X
  
  [33X[0;0YWe  cannot  always  find  [22Xs  ∈  A_6[122X  with  the  required property: If [22Xx[122X is a
  transposition then any [22Xs[122X with [22XS ⊆⟨ x, s ⟩[122X must be a [22X5[122X-cycle.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfilt:= Filtered( S, s -> IsSubset( Group( (1,2), s ), S ) );;[127X[104X
    [4X[25Xgap>[125X [27XCollected( List( filt, Order ) );[127X[104X
    [4X[28X[ [ 5, 48 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YMoreover,  clearly  such  [22Xs[122X  fixes  one  of the moved points of [22Xx[122X, so we may
  prescribe a transposition [22Xy ‡ x[122X that commutes with [22Xx[122X, it satisfies [22XS ⊈⟨ y, s
  ⟩[122X.[133X
  
  [33X[0;0YFor  the  other  two automorphic extensions [22XA_6.2_2 = PGL(2,9)[122X and [22XA_6.2_3 =
  M_10[122X,  we  compute  the  character-theoretic  bounds  [22Xσ(A_6.2_2)  =  1/6[122X and
  [22Xσ(A_6.2_3) = 1/9[122X, which shows statement (g).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XProbGenInfoSimple( CharacterTable( "A6.2_2" ) );[127X[104X
    [4X[28X[ "A6.2_2", 1/6, 5, [ "10A" ], [ 1 ] ][128X[104X
    [4X[25Xgap>[125X [27XProbGenInfoSimple( CharacterTable( "A6.2_3" ) );[127X[104X
    [4X[28X[ "A6.2_3", 1/9, 8, [ "8C" ], [ 1 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that [22Xσ^'( PGL(2,9), s ) = 1/6[122X, with [22Xs[122X of order [22X5[122X, and [22Xσ^'( M_10, s ) =
  0[122X for any [22Xs ∈ A_6[122X since [22XM_10[122X is a non-split extension of [22XA_6[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "A6" );;[127X[104X
    [4X[25Xgap>[125X [27Xt2:= CharacterTable( "A6.2_2" );;[127X[104X
    [4X[25Xgap>[125X [27Xspos:= PositionsProperty( OrdersClassRepresentatives( t ), x -> x = 5 );;[127X[104X
    [4X[25Xgap>[125X [27XProbGenInfoAlmostSimple( t, t2, spos );[127X[104X
    [4X[28X[ "A6.2_2", 1/6, [ "5A", "5B" ], [ 1, 1 ] ][128X[104X
  [4X[32X[104X
  
  
  [1X11.5-4 [33X[0;0Y[22XA_7[122X[101X[1X[133X[101X
  
  [33X[0;0YWe show that [22XS = A_7[122X satisfies the following.[133X
  
  [8X(a)[108X
        [33X[0;6Y[22Xσ(S)  =  2/5[122X,  and this value is attained exactly for [22Xσ(S,s)[122X with [22Xs[122X of
        order [22X7[122X.[133X
  
  [8X(b)[108X
        [33X[0;6YFor  [22Xs[122X  of  order [22X7[122X, [22XMM(S,s)[122X consists of two nonconjugate subgroups of
        the type [22XL_2(7)[122X.[133X
  
  [8X(c)[108X
        [33X[0;6Y[22XP(S)  =  2/5[122X,  and this value is attained exactly for [22XP(S,s)[122X with [22Xs[122X of
        order [22X7[122X.[133X
  
  [8X(d)[108X
        [33X[0;6YThe uniform spread of [22XS[122X is exactly three, with [22Xs[122X of order [22X7[122X.[133X
  
  [33X[0;0YStatement (a)   follows   from   inspection  of  the  primitive  permutation
  characters, cf. Section [14X11.4-3[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "A7" );;[127X[104X
    [4X[25Xgap>[125X [27XProbGenInfoSimple( t );[127X[104X
    [4X[28X[ "A7", 2/5, 2, [ "7A" ], [ 2 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YStatement (b)  can be read off from the permutation characters, and the fact
  that  the  two classes of maximal subgroups that contain elements of order [22X7[122X
  consist of groups of the structure [22XL_2(7)[122X, see [CCN+85, p. 10].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XOrdersClassRepresentatives( t );[127X[104X
    [4X[28X[ 1, 2, 3, 3, 4, 5, 6, 7, 7 ][128X[104X
    [4X[25Xgap>[125X [27Xprim:= PrimitivePermutationCharacters( t );[127X[104X
    [4X[28X[ Character( CharacterTable( "A7" ), [ 7, 3, 4, 1, 1, 2, 0, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "A7" ), [ 15, 3, 0, 3, 1, 0, 0, 1, 1 ] ),[128X[104X
    [4X[28X  Character( CharacterTable( "A7" ), [ 15, 3, 0, 3, 1, 0, 0, 1, 1 ] ),[128X[104X
    [4X[28X  Character( CharacterTable( "A7" ), [ 21, 5, 6, 0, 1, 1, 2, 0, 0 ] ),[128X[104X
    [4X[28X  Character( CharacterTable( "A7" ), [ 35, 7, 5, 2, 1, 0, 1, 0, 0 ] ) [128X[104X
    [4X[28X ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  statement (c),  we  compute that for all nonidentity elements [22Xs ∈ S[122X and
  involutions  [22Xg  ∈  S[122X, [22XP(g,s) ≥ 2/5[122X holds, with equality if and only if [22Xs[122X has
  order [22X7[122X. We actually compute, for class representatives [22Xs[122X, the proportion of
  involutions [22Xg[122X such that [22X⟨ g, s ⟩ ‡ S[122X holds.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:= AlternatingGroup( 7 );;[127X[104X
    [4X[25Xgap>[125X [27Xinv:= (g.1^3*g.2)^3;[127X[104X
    [4X[28X(2,6)(3,7)[128X[104X
    [4X[25Xgap>[125X [27Xccl:= List( ConjugacyClasses( g ), Representative );;[127X[104X
    [4X[25Xgap>[125X [27XSortParallel( List( ccl, Order ), ccl );[127X[104X
    [4X[25Xgap>[125X [27XList( ccl, Order );[127X[104X
    [4X[28X[ 1, 2, 3, 3, 4, 5, 6, 7, 7 ][128X[104X
    [4X[25Xgap>[125X [27XSize( ConjugacyClass( g, inv ) );[127X[104X
    [4X[28X105[128X[104X
    [4X[25Xgap>[125X [27Xprop:= List( ccl, r -> RatioOfNongenerationTransPermGroup( g, inv, r ) );[127X[104X
    [4X[28X[ 1, 1, 1, 1, 89/105, 17/21, 19/35, 2/5, 2/5 ][128X[104X
    [4X[25Xgap>[125X [27XMinimum( prop );[127X[104X
    [4X[28X2/5[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  statement (d),  we use the random approach described in Section [14X11.3-3[114X.
  By  the  character-theoretic  bounds,  it  suffices  to  consider triples of
  elements in the classes [10X2A[110X or [10X3B[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XOrdersClassRepresentatives( t );[127X[104X
    [4X[28X[ 1, 2, 3, 3, 4, 5, 6, 7, 7 ][128X[104X
    [4X[25Xgap>[125X [27Xspos:= Position( OrdersClassRepresentatives( t ), 7 );;[127X[104X
    [4X[25Xgap>[125X [27XSizesCentralizers( t );[127X[104X
    [4X[28X[ 2520, 24, 36, 9, 4, 5, 12, 7, 7 ][128X[104X
    [4X[25Xgap>[125X [27XApproxP( prim, spos );[127X[104X
    [4X[28X[ 0, 2/5, 0, 2/5, 2/15, 0, 0, 2/15, 2/15 ][128X[104X
    [4X[25Xgap>[125X [27Xs:= (1,2,3,4,5,6,7);;[127X[104X
    [4X[25Xgap>[125X [27X3B:= (1,2,3)(4,5,6);;[127X[104X
    [4X[25Xgap>[125X [27XC3B:= ConjugacyClass( g, 3B );;[127X[104X
    [4X[25Xgap>[125X [27XSize( C3B );[127X[104X
    [4X[28X280[128X[104X
    [4X[25Xgap>[125X [27XResetGlobalRandomNumberGenerators();[127X[104X
    [4X[25Xgap>[125X [27Xfor triple in UnorderedTuples( [ inv, 3B ], 3 ) do[127X[104X
    [4X[25X>[125X [27X     if RandomCheckUniformSpread( g, triple, s, 80 ) <> true then[127X[104X
    [4X[25X>[125X [27X       Print( "#E  nongeneration!\n" );[127X[104X
    [4X[25X>[125X [27X     fi;[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YWe get no output, so the uniform spread of [22XS[122X is at least three.[133X
  
  [33X[0;0YAlternatively,  we  can  use the lemma from Section [14X11.2-2[114X; this approach is
  technically  more  involved but faster. We work with the diagonal product of
  the  two  degree  [22X15[122X  representations  of  [22XS[122X,  which is constructed from the
  information stored in the [5XGAP[105X Library of Tables of Marks.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtom:= TableOfMarks( "A7" );;[127X[104X
    [4X[25Xgap>[125X [27Xa7:= UnderlyingGroup( tom );;[127X[104X
    [4X[25Xgap>[125X [27Xtommaxes:= MaximalSubgroupsTom( tom );[127X[104X
    [4X[28X[ [ 39, 38, 37, 36, 35 ], [ 7, 15, 15, 21, 35 ] ][128X[104X
    [4X[25Xgap>[125X [27Xindex15:= List( tommaxes[1]{ [ 2, 3 ] },[127X[104X
    [4X[25X>[125X [27X                   i -> RepresentativeTom( tom, i ) );[127X[104X
    [4X[28X[ Group([ (1,3)(2,7), (1,5,7)(3,4,6) ]), [128X[104X
    [4X[28X  Group([ (1,4)(2,3), (2,4,6)(3,5,7) ]) ][128X[104X
    [4X[25Xgap>[125X [27Xdeg15:= List( index15, s -> RightTransversal( a7, s ) );;[127X[104X
    [4X[25Xgap>[125X [27Xreps:= List( deg15, l -> Action( a7, l, OnRight ) );[127X[104X
    [4X[28X[ Group([ (1,5,7)(2,9,10)(3,11,4)(6,12,8)(13,14,15), (1,8,15,5,12)[128X[104X
    [4X[28X      (2,13,11,3,10)(4,14,9,7,6) ]), [128X[104X
    [4X[28X  Group([ (1,2,3)(4,6,5)(7,8,9)(10,12,11)(13,15,14), (1,12,3,13,10)[128X[104X
    [4X[28X      (2,9,15,4,11)(5,6,14,7,8) ]) ][128X[104X
    [4X[25Xgap>[125X [27Xg:= DiagonalProductOfPermGroups( reps );;[127X[104X
    [4X[25Xgap>[125X [27XResetGlobalRandomNumberGenerators();[127X[104X
    [4X[25Xgap>[125X [27Xrepeat s:= Random( g );[127X[104X
    [4X[25X>[125X [27X   until Order( s ) = 7;[127X[104X
    [4X[25Xgap>[125X [27XNrMovedPoints( s );[127X[104X
    [4X[28X28[128X[104X
    [4X[25Xgap>[125X [27Xmpg:= MovedPoints( g );;[127X[104X
    [4X[25Xgap>[125X [27Xfixs:= Difference( mpg, MovedPoints( s ) );;[127X[104X
    [4X[25Xgap>[125X [27Xorb_s:= Orbit( g, fixs, OnSets );;[127X[104X
    [4X[25Xgap>[125X [27XLength( orb_s );[127X[104X
    [4X[28X120[128X[104X
    [4X[25Xgap>[125X [27XSizesCentralizers( t );[127X[104X
    [4X[28X[ 2520, 24, 36, 9, 4, 5, 12, 7, 7 ][128X[104X
    [4X[25Xgap>[125X [27Xrepeat 2a:= Random( g ); until Order( 2a ) = 2;[127X[104X
    [4X[25Xgap>[125X [27Xrepeat 3b:= Random( g );[127X[104X
    [4X[25X>[125X [27X   until Order( 3b ) = 3 and Size( Centralizer( g, 3b ) ) = 9;[127X[104X
    [4X[25Xgap>[125X [27Xorb2a:= Orbit( g, Difference( mpg, MovedPoints( 2a ) ), OnSets );;[127X[104X
    [4X[25Xgap>[125X [27Xorb3b:= Orbit( g, Difference( mpg, MovedPoints( 3b ) ), OnSets );;[127X[104X
    [4X[25Xgap>[125X [27Xorb2aor3b:= Union( orb2a, orb3b );;[127X[104X
    [4X[25Xgap>[125X [27XTripleWithProperty( [ [ orb2a[1], orb3b[1] ], orb2aor3b, orb2aor3b ],[127X[104X
    [4X[25X>[125X [27X       l -> ForAll( orb_s,[127X[104X
    [4X[25X>[125X [27X                f -> not IsEmpty( Intersection( Union( l ), f ) ) ) );[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIt  remains to show that for any choice of [22Xs ∈ S[122X, a quadruple of elements in
  [22XS^×[122X  exists  such  that  [22Xs[122X generates a proper subgroup of [22XS[122X together with at
  least one of these elements.[133X
  
  [33X[0;0YFirst  we observe (without using [5XGAP[105X) that there is a pair of [22X3[122X-cycles whose
  fixed   points   cover   the   seven   points  of  the  natural  permutation
  representation. This implies the statement for all elements [22Xs ∈ S[122X that fix a
  point  in  this  representation. So it remains to consider elements [22Xs[122X of the
  orders six and seven.[133X
  
  [33X[0;0YFor   the   order  seven  element,  the  above  setup  and  the  lemma  from
  Section [14X11.2-2[114X can be used.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XQuadrupleWithProperty( [ [ orb2a[1] ], orb2a, orb2a, orb2a ],[127X[104X
    [4X[25X>[125X [27X       l -> ForAll( orb_s,[127X[104X
    [4X[25X>[125X [27X                f -> not IsEmpty( Intersection( Union( l ), f ) ) ) );[127X[104X
    [4X[28X[ [ 2, 5, 12, 18, 19, 26 ], [ 7, 8, 9, 16, 21, 25 ], [128X[104X
    [4X[28X  [ 1, 6, 10, 17, 20, 27 ], [ 13, 14, 15, 28, 29, 30 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  the  order  six  element,  we use the diagonal product of the primitive
  permutation representations of the degrees [22X21[122X and [22X35[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xhas6A:= List( tommaxes[1]{ [ 4, 5 ] },[127X[104X
    [4X[25X>[125X [27X                 i -> RepresentativeTom( tom, i ) );[127X[104X
    [4X[28X[ Group([ (1,2)(3,7), (2,6,5,4)(3,7) ]), [128X[104X
    [4X[28X  Group([ (2,3)(5,7), (1,2)(4,5,6,7), (2,3)(5,6) ]) ][128X[104X
    [4X[25Xgap>[125X [27Xtrans:= List( has6A, s -> RightTransversal( a7, s ) );;[127X[104X
    [4X[25Xgap>[125X [27Xreps:= List( trans, l -> Action( a7, l, OnRight ) );[127X[104X
    [4X[28X[ Group([ (1,16,12)(2,17,13)(3,18,11)(4,19,14)(15,20,21), (1,4,7,9,10)[128X[104X
    [4X[28X      (2,5,8,3,6)(11,12,15,14,13)(16,20,19,17,18) ]), [128X[104X
    [4X[28X  Group([ (2,16,6)(3,17,7)(4,18,8)(5,19,9)(10,20,26)(11,21,27)[128X[104X
    [4X[28X      (12,22,28)(13,23,29)(14,24,30)(15,25,31), (1,2,3,4,5)[128X[104X
    [4X[28X      (6,10,13,15,9)(7,11,14,8,12)(16,20,23,25,19)(17,21,24,18,22)[128X[104X
    [4X[28X      (26,32,35,31,28)(27,33,29,34,30) ]) ][128X[104X
    [4X[25Xgap>[125X [27Xg:= DiagonalProductOfPermGroups( reps );;[127X[104X
    [4X[25Xgap>[125X [27Xrepeat s:= Random( g );[127X[104X
    [4X[25X>[125X [27X   until Order( s ) = 6;[127X[104X
    [4X[25Xgap>[125X [27XNrMovedPoints( s );[127X[104X
    [4X[28X53[128X[104X
    [4X[25Xgap>[125X [27Xmpg:= MovedPoints( g );;[127X[104X
    [4X[25Xgap>[125X [27Xfixs:= Difference( mpg, MovedPoints( s ) );;[127X[104X
    [4X[25Xgap>[125X [27Xorb_s:= Orbit( g, fixs, OnSets );;[127X[104X
    [4X[25Xgap>[125X [27XLength( orb_s );[127X[104X
    [4X[28X105[128X[104X
    [4X[25Xgap>[125X [27Xrepeat 3a:= Random( g );[127X[104X
    [4X[25X>[125X [27X   until Order( 3a ) = 3 and Size( Centralizer( g, 3a ) ) = 36;[127X[104X
    [4X[25Xgap>[125X [27Xorb3a:= Orbit( g, Difference( mpg, MovedPoints( 3a ) ), OnSets );;[127X[104X
    [4X[25Xgap>[125X [27XLength( orb3a );[127X[104X
    [4X[28X35[128X[104X
    [4X[25Xgap>[125X [27XTripleWithProperty( [ [ orb3a[1] ], orb3a, orb3a ],[127X[104X
    [4X[25X>[125X [27X       l -> ForAll( orb_s,[127X[104X
    [4X[25X>[125X [27X                f -> not IsEmpty( Intersection( Union( l ), f ) ) ) );[127X[104X
    [4X[28X[ [ 1, 4, 6, 12, 14, 15, 34, 37, 40, 43, 49 ], [128X[104X
    [4X[28X  [ 1, 4, 6, 16, 19, 20, 27, 30, 33, 44, 49 ], [128X[104X
    [4X[28X  [ 2, 3, 4, 5, 7, 9, 26, 47, 48, 50, 53 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo  we  have  found  not only a quadruple but even a triple of [22X3[122X-cycles that
  excludes candidates [22Xs[122X of order six.[133X
  
  
  [1X11.5-5 [33X[0;0Y[22XL_d(q)[122X[101X[1X[133X[101X
  
  [33X[0;0YIn  the  treatment  of  small dimensional linear groups [22XS = SL(d,q)[122X, [BGK08]
  uses  a  Singer  element  [22Xs[122X  of  order  [22X(q^d-1)/(q-1)[122X.  (So the order of the
  corresponding element in [22XPSL(d,q) = (q^d-1)/[(q-1) gcd(d,q-1)][122X.) By [Ber00],
  [22XMM(S,s)[122X  consists of extension field type subgroups, except in the cases [22Xd =
  2[122X, [22Xq ∈ { 2, 5, 7, 9 }[122X, and [22X(d,q) = (3,4)[122X. These subgroups have the structure
  [22XGL(d/p,q^p):α_q  ∩  S[122X,  for  prime  divisors  [22Xp[122X  of [22Xd[122X, where [22Xα_q[122X denotes the
  Frobenius  automorphism  that  acts on matrices by raising each entry to the
  [22Xq[122X-th  power.  (If  [22Xq[122X is a prime then we have [22XGL(d/p,q^p):α_q = ΓL(d/p,q^p)[122X.)
  Since  [22Xs[122X  acts irreducibly, it is contained in at most one conjugate of each
  class of extension field type subgroups (cf. [BGK08, Lemma 2.12]).[133X
  
  [33X[0;0YFirst we write a [5XGAP[105X function [10XRelativeSigmaL[110X that takes a positive integer [22Xd[122X
  and  a  basis  [22XB[122X  of  the  field extension of degree [22Xn[122X over the field with [22Xq[122X
  elements, and returns the group [22XGL(d,q^n):α_q[122X, as a subgroup of [22XGL(dn,q)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XRelativeSigmaL:= function( d, B )[127X[104X
    [4X[25X>[125X [27X    local n, F, q, glgens, diag, pi, frob, i;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    n:= Length( B );[127X[104X
    [4X[25X>[125X [27X    F:= LeftActingDomain( UnderlyingLeftModule( B ) );[127X[104X
    [4X[25X>[125X [27X    q:= Size( F );[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    # Create the generating matrices inside the linear subgroup.[127X[104X
    [4X[25X>[125X [27X    glgens:= List( GeneratorsOfGroup( SL( d, q^n ) ),[127X[104X
    [4X[25X>[125X [27X                   m -> BlownUpMat( B, m ) );[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    # Create the matrix of a diagonal part that maps to determinant 1.[127X[104X
    [4X[25X>[125X [27X    diag:= IdentityMat( d*n, F );[127X[104X
    [4X[25X>[125X [27X    diag{ [ 1 .. n ] }{ [ 1 .. n ] }:= BlownUpMat( B, [ [ Z(q^n)^(q-1) ] ] );[127X[104X
    [4X[25X>[125X [27X    Add( glgens, diag );[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    # Create the matrix that realizes the Frobenius action,[127X[104X
    [4X[25X>[125X [27X    # and adjust the determinant.[127X[104X
    [4X[25X>[125X [27X    pi:= List( B, b -> Coefficients( B, b^q ) );[127X[104X
    [4X[25X>[125X [27X    frob:= NullMat( d*n, d*n, F );[127X[104X
    [4X[25X>[125X [27X    for i in [ 0 .. d-1 ] do[127X[104X
    [4X[25X>[125X [27X      frob{ [ 1 .. n ] + i*n }{ [ 1 .. n ] + i*n }:= pi;[127X[104X
    [4X[25X>[125X [27X    od;[127X[104X
    [4X[25X>[125X [27X    diag:= IdentityMat( d*n, F );[127X[104X
    [4X[25X>[125X [27X    diag{ [ 1 .. n ] }{ [ 1 .. n ] }:= BlownUpMat( B, [ [ Z(q^n) ] ] );[127X[104X
    [4X[25X>[125X [27X    diag:= diag^LogFFE( Inverse( Determinant( frob ) ), Determinant( diag ) );[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    # Return the result.[127X[104X
    [4X[25X>[125X [27X    return Group( Concatenation( glgens, [ diag * frob ] ) );[127X[104X
    [4X[25X>[125X [27Xend;;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe   next   function   computes  [22Xσ(SL(d,q),s)[122X,  by  computing  the  sum  of
  [22Xμ(g,S/(GL(d/p,q^p):α_q  ∩  S))[122X,  for  prime  divisors [22Xp[122X of [22Xd[122X, and taking the
  maximum  over  [22Xg  ∈  S^×[122X.  The  computations  take  place  in  a permutation
  representation of [22XPSL(d,q)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XApproxPForSL:= function( d, q )[127X[104X
    [4X[25X>[125X [27X    local G, epi, PG, primes, maxes, names, ccl;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    # Check whether this is an admissible case (see [Be00]).[127X[104X
    [4X[25X>[125X [27X    if ( d = 2 and q in [ 2, 5, 7, 9 ] ) or ( d = 3 and q = 4 ) then[127X[104X
    [4X[25X>[125X [27X      return fail;[127X[104X
    [4X[25X>[125X [27X    fi;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    # Create the group SL(d,q), and the map to PSL(d,q).[127X[104X
    [4X[25X>[125X [27X    G:= SL( d, q );[127X[104X
    [4X[25X>[125X [27X    epi:= ActionHomomorphism( G, NormedRowVectors( GF(q)^d ), OnLines );[127X[104X
    [4X[25X>[125X [27X    PG:= ImagesSource( epi );[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    # Create the subgroups corresponding to the prime divisors of `d'.[127X[104X
    [4X[25X>[125X [27X    primes:= PrimeDivisors( d );[127X[104X
    [4X[25X>[125X [27X    maxes:= List( primes, p -> RelativeSigmaL( d/p,[127X[104X
    [4X[25X>[125X [27X                                 Basis( AsField( GF(q), GF(q^p) ) ) ) );[127X[104X
    [4X[25X>[125X [27X    names:= List( primes, p -> Concatenation( "GL(", String( d/p ), ",",[127X[104X
    [4X[25X>[125X [27X                                 String( q^p ), ").", String( p ) ) );[127X[104X
    [4X[25X>[125X [27X    if 2 < q then[127X[104X
    [4X[25X>[125X [27X      names:= List( names, name -> Concatenation( name, " cap G" ) );[127X[104X
    [4X[25X>[125X [27X    fi;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    # Compute the conjugacy classes of prime order elements in the maxes.[127X[104X
    [4X[25X>[125X [27X    # (In order to avoid computing all conjugacy classes of these subgroups,[127X[104X
    [4X[25X>[125X [27X    # we work in Sylow subgroups.)[127X[104X
    [4X[25X>[125X [27X    ccl:= List( List( maxes, x -> ImagesSet( epi, x ) ),[127X[104X
    [4X[25X>[125X [27X            M -> ClassesOfPrimeOrder( M, PrimeDivisors( Size( M ) ),[127X[104X
    [4X[25X>[125X [27X                                      TrivialSubgroup( M ) ) );[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    return [ names, UpperBoundFixedPointRatios( PG, ccl, true )[1] ];[127X[104X
    [4X[25X>[125X [27Xend;;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  apply  this  function  to  the  cases  that  are  interesting in [BGK08,
  Section 5.12].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpairs:= [ [ 3, 2 ], [ 3, 3 ], [ 4, 2 ], [ 4, 3 ], [ 4, 4 ],[127X[104X
    [4X[25X>[125X [27X           [ 6, 2 ], [ 6, 3 ], [ 6, 4 ], [ 6, 5 ], [ 8, 2 ], [ 10, 2 ] ];;[127X[104X
    [4X[25Xgap>[125X [27Xarray:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xfor pair in pairs do[127X[104X
    [4X[25X>[125X [27X     d:= pair[1];  q:= pair[2];[127X[104X
    [4X[25X>[125X [27X     approx:= ApproxPForSL( d, q );[127X[104X
    [4X[25X>[125X [27X     Add( array, [ Concatenation( "SL(", String(d), ",", String(q), ")" ),[127X[104X
    [4X[25X>[125X [27X                   (q^d-1)/(q-1),[127X[104X
    [4X[25X>[125X [27X                   approx[1], approx[2] ] );[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27Xoldsize:= SizeScreen();;[127X[104X
    [4X[25Xgap>[125X [27XSizeScreen( [ 80 ] );;[127X[104X
    [4X[25Xgap>[125X [27XPrintFormattedArray( array );[127X[104X
    [4X[28X   SL(3,2)    7                             [ "GL(1,8).3" ]             1/4[128X[104X
    [4X[28X   SL(3,3)   13                      [ "GL(1,27).3 cap G" ]            1/24[128X[104X
    [4X[28X   SL(4,2)   15                             [ "GL(2,4).2" ]            3/14[128X[104X
    [4X[28X   SL(4,3)   40                       [ "GL(2,9).2 cap G" ]         53/1053[128X[104X
    [4X[28X   SL(4,4)   85                      [ "GL(2,16).2 cap G" ]           1/108[128X[104X
    [4X[28X   SL(6,2)   63                [ "GL(3,4).2", "GL(2,8).3" ]       365/55552[128X[104X
    [4X[28X   SL(6,3)  364   [ "GL(3,9).2 cap G", "GL(2,27).3 cap G" ] 22843/123845436[128X[104X
    [4X[28X   SL(6,4) 1365  [ "GL(3,16).2 cap G", "GL(2,64).3 cap G" ]         1/85932[128X[104X
    [4X[28X   SL(6,5) 3906 [ "GL(3,25).2 cap G", "GL(2,125).3 cap G" ]        1/484220[128X[104X
    [4X[28X   SL(8,2)  255                             [ "GL(4,4).2" ]          1/7874[128X[104X
    [4X[28X  SL(10,2) 1023               [ "GL(5,4).2", "GL(2,32).5" ]        1/129794[128X[104X
    [4X[25Xgap>[125X [27XSizeScreen( oldsize );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  only missing case for [BGK08] is [22XS = L_3(4)[122X, for which [22XMM(S,s)[122X consists
  of  three  groups of the type [22XL_3(2)[122X (see [CCN+85, p. 23]). The group [22XL_3(4)[122X
  has  been  considered already in Section [14X11.4-3[114X, where [22Xσ(S,s) = 1/5[122X has been
  proved. Also the cases [22XSL(3,3)[122X, [22XSL(4,2) ≅ A_8[122X, and [22XSL(4,3)[122X have been handled
  there.[133X
  
  [33X[0;0YAn alternative character-theoretic proof for [22XS = L_6(2)[122X looks as follows. In
  this  case,  the  subgroups  in [22XMM(S,s)[122X have the types [22XΓL(3,4) ≅ GL(3,4).2 ≅
  3.L_3(4).3.2_2[122X and [22XΓL(2,8) ≅ GL(2,8).3 ≅ (7 × L_2(8)).3[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "L6(2)" );;[127X[104X
    [4X[25Xgap>[125X [27Xs1:= CharacterTable( "3.L3(4).3.2_2" );;[127X[104X
    [4X[25Xgap>[125X [27Xs2:= CharacterTable( "(7xL2(8)).3" );;[127X[104X
    [4X[25Xgap>[125X [27XSigmaFromMaxes( t, "63A", [ s1, s2 ], [ 1, 1 ] );[127X[104X
    [4X[28X365/55552[128X[104X
  [4X[32X[104X
  
  
  [1X11.5-6 [33X[0;0Y[22X∗[122X[101X[1X [22XL_d(q)[122X[101X[1X with prime [22Xd[122X[101X[1X[133X[101X
  
  [33X[0;0YFor  [22XS  =  SL(d,q)[122X with [13Xprime[113X dimension [22Xd[122X, and [22Xs ∈ S[122X a Singer cycle, we have
  [22XMM(S,s) = { M }[122X, where [22XM = N_S(⟨ s ⟩) ≅ ΓL(1,q^d) ∩ S[122X. So[133X
  
  
  [24X[33X[0;6Yσ(g,s) = μ(g,S/M) = |g^S ∩ M|/|g^S| < |M|/|g^S| ≤ (q^d-1) ⋅ d/|g^S|[133X[124X
  
  [33X[0;0Yholds  for  any  [22Xg  ∈  S  ∖  Z(S)[122X,  which implies [22Xσ( S, s ) < max{ (q^d-1) ⋅
  d/|g^S|;  g ∈ S ∖ Z(S) }[122X. The right hand side of this inequality is returned
  by  the  following  function.  In [BGK08, Lemma 3.8], the global upper bound
  [22X1/q^d[122X is derived for primes [22Xd ≥ 5[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XUpperBoundForSL:= function( d, q )[127X[104X
    [4X[25X>[125X [27X    local G, Msize, ccl;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    if not IsPrimeInt( d ) then[127X[104X
    [4X[25X>[125X [27X      Error( "<d> must be a prime" );[127X[104X
    [4X[25X>[125X [27X    fi;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    G:= SL( d, q );[127X[104X
    [4X[25X>[125X [27X    Msize:= (q^d-1) * d;[127X[104X
    [4X[25X>[125X [27X    ccl:= Filtered( ConjugacyClasses( G ),[127X[104X
    [4X[25X>[125X [27X                    c ->     Msize mod Order( Representative( c ) ) = 0[127X[104X
    [4X[25X>[125X [27X                         and Size( c ) <> 1 );[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    return Msize / Minimum( List( ccl, Size ) );[127X[104X
    [4X[25X>[125X [27Xend;;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  interesting values are [22X(d,q)[122X with [22Xd ∈ { 5, 7, 11 }[122X and [22Xq ∈ { 2, 3, 4 }[122X,
  and perhaps also [22X(d,q) ∈ { (3,2), (3,3) }[122X. (Here we exclude [22XSL(11,4)[122X because
  writing  down the conjugacy classes of this group would exceed the permitted
  memory.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XNrConjugacyClasses( SL(11,4) );[127X[104X
    [4X[28X1397660[128X[104X
    [4X[25Xgap>[125X [27Xpairs:= [ [ 3, 2 ], [ 3, 3 ], [ 5, 2 ], [ 5, 3 ], [ 5, 4 ],[127X[104X
    [4X[25X>[125X [27X             [ 7, 2 ], [ 7, 3 ], [ 7, 4 ],[127X[104X
    [4X[25X>[125X [27X             [ 11, 2 ], [ 11, 3 ] ];;[127X[104X
    [4X[25Xgap>[125X [27Xarray:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xfor pair in pairs do[127X[104X
    [4X[25X>[125X [27X     d:= pair[1];  q:= pair[2];[127X[104X
    [4X[25X>[125X [27X     approx:= UpperBoundForSL( d, q );[127X[104X
    [4X[25X>[125X [27X     Add( array, [ Concatenation( "SL(", String(d), ",", String(q), ")" ),[127X[104X
    [4X[25X>[125X [27X                   (q^d-1)/(q-1),[127X[104X
    [4X[25X>[125X [27X                   approx ] );[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27XPrintFormattedArray( array );[127X[104X
    [4X[28X   SL(3,2)     7                                   7/8[128X[104X
    [4X[28X   SL(3,3)    13                                   3/4[128X[104X
    [4X[28X   SL(5,2)    31                              31/64512[128X[104X
    [4X[28X   SL(5,3)   121                                 10/81[128X[104X
    [4X[28X   SL(5,4)   341                                15/256[128X[104X
    [4X[28X   SL(7,2)   127                             7/9142272[128X[104X
    [4X[28X   SL(7,3)  1093                                14/729[128X[104X
    [4X[28X   SL(7,4)  5461                               21/4096[128X[104X
    [4X[28X  SL(11,2)  2047 2047/34112245508649716682268134604800[128X[104X
    [4X[28X  SL(11,3) 88573                              22/59049[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  exact  values are clearly better than the above bounds. We compute them
  for  [22XL_5(2)[122X  and  [22XL_7(2)[122X.  In the latter case, the class fusion of the [22X127:7[122X
  type  subgroup [22XM[122X is not uniquely determined by the character tables; here we
  use  the  additional  information  that  the  elements  of order [22X7[122X in [22XM[122X have
  centralizer  order [22X49[122X in [22XL_7(2)[122X. (See Section [14X11.4-3[114X for the examples with [22Xd
  = 3[122X.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSigmaFromMaxes( CharacterTable( "L5(2)" ), "31A",[127X[104X
    [4X[25X>[125X [27X       [ CharacterTable( "31:5" ) ], [ 1 ] );[127X[104X
    [4X[28X1/5376[128X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "L7(2)" );;[127X[104X
    [4X[25Xgap>[125X [27Xs:= CharacterTable( "P:Q", [ 127, 7 ] );;[127X[104X
    [4X[25Xgap>[125X [27Xpi:= PossiblePermutationCharacters( s, t );;[127X[104X
    [4X[25Xgap>[125X [27XLength( pi );[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27Xord7:= PositionsProperty( OrdersClassRepresentatives( t ), x -> x = 7 );[127X[104X
    [4X[28X[ 38, 45, 76, 77, 83 ][128X[104X
    [4X[25Xgap>[125X [27Xsizes:= SizesCentralizers( t ){ ord7 };[127X[104X
    [4X[28X[ 141120, 141120, 3528, 3528, 49 ][128X[104X
    [4X[25Xgap>[125X [27XList( pi, x -> x[83] );[127X[104X
    [4X[28X[ 42, 0 ][128X[104X
    [4X[25Xgap>[125X [27Xspos:= Position( OrdersClassRepresentatives( t ), 127 );;[127X[104X
    [4X[25Xgap>[125X [27XMaximum( ApproxP( pi{ [ 1 ] }, spos ) );[127X[104X
    [4X[28X1/4388290560[128X[104X
  [4X[32X[104X
  
  
  [1X11.5-7 [33X[0;0YAutomorphic Extensions of [22XL_d(q)[122X[101X[1X[133X[101X
  
  [33X[0;0YFor  the following values of [22Xd[122X and [22Xq[122X, automorphic extensions [22XG[122X of [22XL_d(q)[122X had
  to be checked for [BGK08, Section 5.12].[133X
  
  
  [24X[33X[0;6Y(d,q) ∈ { (3,4), (6,2), (6,3), (6,4), (6,5), (10,2) }[133X[124X
  
  [33X[0;0YThe  first  case has been treated in Section [14X11.4-4[114X. For the other cases, we
  compute [22Xσ^'(G,s)[122X below.[133X
  
  [33X[0;0YIn  any  case,  the  extension  by a [13Xgraph[113X automorphism occurs, which can be
  described by mapping each matrix in [22XSL(d,q)[122X to its inverse transpose. If [22Xq >
  2[122X,  also  extensions  by  [13Xdiagonal[113X automorphisms occur, which are induced by
  conjugation  with elements in [22XGL(d,q)[122X. If [22Xq[122X is nonprime then also extensions
  by  [13Xfield[113X automorphisms occur, which can be described by powering the matrix
  entries  by  roots  of  [22Xq[122X. Finally, products (of prime order) of these three
  kinds of automorphisms have to be considered.[133X
  
  [33X[0;0YWe  start with the extension [22XG[122X of [22XS = SL(d,q)[122X by a graph automorphism. [22XG[122X can
  be  embedded  into  [22XGL(2d,q)[122X  by  representing  the  matrix [22XA ∈ S[122X as a block
  diagonal  matrix with diagonal blocks equal to [22XA[122X and [22XA^-tr[122X, and representing
  the  graph  automorphism  by  a permutation matrix that interchanges the two
  blocks.  In  order  to construct the field extension type subgroups of [22XG[122X, we
  have  to  choose  the  basis  of  the field extension in such a way that the
  subgroup  is normalized by the permutation matrix; a sufficient condition is
  that  the matrices of the [22XFF_q[122X-linear mappings induced by the basis elements
  are symmetric.[133X
  
  [33X[0;0Y(We do not give a function that computes a basis with this property from the
  parameters  [22Xd[122X  and  [22Xq[122X.  Instead,  we  only write down the bases that we will
  need.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSymmetricBasis:= function( q, n )[127X[104X
    [4X[25X>[125X [27X    local vectors, B, issymmetric;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    if   q = 2 and n = 2 then[127X[104X
    [4X[25X>[125X [27X      vectors:= [ Z(2)^0, Z(2^2) ];[127X[104X
    [4X[25X>[125X [27X    elif q = 2 and n = 3 then[127X[104X
    [4X[25X>[125X [27X      vectors:= [ Z(2)^0, Z(2^3), Z(2^3)^5 ];[127X[104X
    [4X[25X>[125X [27X    elif q = 2 and n = 5 then[127X[104X
    [4X[25X>[125X [27X      vectors:= [ Z(2)^0, Z(2^5), Z(2^5)^4, Z(2^5)^25, Z(2^5)^26 ];[127X[104X
    [4X[25X>[125X [27X    elif q = 3 and n = 2 then[127X[104X
    [4X[25X>[125X [27X      vectors:= [ Z(3)^0, Z(3^2) ];[127X[104X
    [4X[25X>[125X [27X    elif q = 3 and n = 3 then[127X[104X
    [4X[25X>[125X [27X      vectors:= [ Z(3)^0, Z(3^3)^2, Z(3^3)^7 ];[127X[104X
    [4X[25X>[125X [27X    elif q = 4 and n = 2 then[127X[104X
    [4X[25X>[125X [27X      vectors:= [ Z(2)^0, Z(2^4)^3 ];[127X[104X
    [4X[25X>[125X [27X    elif q = 4 and n = 3 then[127X[104X
    [4X[25X>[125X [27X      vectors:= [ Z(2)^0, Z(2^3), Z(2^3)^5 ];[127X[104X
    [4X[25X>[125X [27X    elif q = 5 and n = 2 then[127X[104X
    [4X[25X>[125X [27X      vectors:= [ Z(5)^0, Z(5^2)^2 ];[127X[104X
    [4X[25X>[125X [27X    elif q = 5 and n = 3 then[127X[104X
    [4X[25X>[125X [27X      vectors:= [ Z(5)^0, Z(5^3)^9, Z(5^3)^27 ];[127X[104X
    [4X[25X>[125X [27X    else[127X[104X
    [4X[25X>[125X [27X      Error( "sorry, no basis for <q> and <n> stored" );[127X[104X
    [4X[25X>[125X [27X    fi;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    B:= Basis( AsField( GF(q), GF(q^n) ), vectors );[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    # Check that the basis really has the required property.[127X[104X
    [4X[25X>[125X [27X    issymmetric:= M -> M = TransposedMat( M );[127X[104X
    [4X[25X>[125X [27X    if not ForAll( B, b -> issymmetric( BlownUpMat( B, [ [ b ] ] ) ) ) then[127X[104X
    [4X[25X>[125X [27X      Error( "wrong basis!" );[127X[104X
    [4X[25X>[125X [27X    fi;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    # Return the result.[127X[104X
    [4X[25X>[125X [27X    return B;[127X[104X
    [4X[25X>[125X [27Xend;;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  later  examples, we will need similar embeddings of matrices. Therefore,
  we  provide  a  more general function [10XEmbeddedMatrix[110X that takes a field [10XF[110X, a
  matrix  [10Xmat[110X, and a function [10Xfunc[110X, and returns a block diagonal matrix over [10XF[110X
  whose diagonal blocks are [10Xmat[110X and [10Xfunc( mat )[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XBindGlobal( "EmbeddedMatrix", function( F, mat, func )[127X[104X
    [4X[25X>[125X [27X  local d, result;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X  d:= Length( mat );[127X[104X
    [4X[25X>[125X [27X  result:= NullMat( 2*d, 2*d, F );[127X[104X
    [4X[25X>[125X [27X  result{ [ 1 .. d ] }{ [ 1 .. d ] }:= mat;[127X[104X
    [4X[25X>[125X [27X  result{ [ d+1 .. 2*d ] }{ [ d+1 .. 2*d ] }:= func( mat );[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X  return result;[127X[104X
    [4X[25X>[125X [27Xend );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  following function is similar to [10XApproxPForSL[110X, the differences are that
  the  group [22XG[122X in question is not [22XSL(d,q)[122X but the extension of this group by a
  graph automorphism, and that [22Xσ^'(G,s)[122X is computed not [22Xσ(G,s)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XApproxPForOuterClassesInExtensionOfSLByGraphAut:= function( d, q )[127X[104X
    [4X[25X>[125X [27X    local embedG, swap, G, orb, epi, PG, Gprime, primes, maxes, ccl, names;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    # Check whether this is an admissible case (see [Be00],[127X[104X
    [4X[25X>[125X [27X    # note that a graph automorphism exists only for `d > 2').[127X[104X
    [4X[25X>[125X [27X    if d = 2 or ( d = 3 and q = 4 ) then[127X[104X
    [4X[25X>[125X [27X      return fail;[127X[104X
    [4X[25X>[125X [27X    fi;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    # Provide a function that constructs a block diagonal matrix.[127X[104X
    [4X[25X>[125X [27X    embedG:= mat -> EmbeddedMatrix( GF( q ), mat,[127X[104X
    [4X[25X>[125X [27X                                    M -> TransposedMat( M^-1 ) );[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    # Create the matrix that exchanges the two blocks.[127X[104X
    [4X[25X>[125X [27X    swap:= NullMat( 2*d, 2*d, GF(q) );[127X[104X
    [4X[25X>[125X [27X    swap{ [ 1 .. d ] }{ [ d+1 .. 2*d ] }:= IdentityMat( d, GF(q) );[127X[104X
    [4X[25X>[125X [27X    swap{ [ d+1 .. 2*d ] }{ [ 1 .. d ] }:= IdentityMat( d, GF(q) );[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    # Create the group SL(d,q).2, and the map to the projective group.[127X[104X
    [4X[25X>[125X [27X    G:= ClosureGroupDefault( Group( List( GeneratorsOfGroup( SL( d, q ) ),[127X[104X
    [4X[25X>[125X [27X                                          embedG ) ),[127X[104X
    [4X[25X>[125X [27X                      swap );[127X[104X
    [4X[25X>[125X [27X    orb:= Orbit( G, One( G )[1], OnLines );[127X[104X
    [4X[25X>[125X [27X    epi:= ActionHomomorphism( G, orb, OnLines );[127X[104X
    [4X[25X>[125X [27X    PG:= ImagesSource( epi );[127X[104X
    [4X[25X>[125X [27X    Gprime:= DerivedSubgroup( PG );[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    # Create the subgroups corresponding to the prime divisors of `d'.[127X[104X
    [4X[25X>[125X [27X    primes:= PrimeDivisors( d );[127X[104X
    [4X[25X>[125X [27X    maxes:= List( primes,[127X[104X
    [4X[25X>[125X [27X              p -> ClosureGroupDefault( Group( List( GeneratorsOfGroup([127X[104X
    [4X[25X>[125X [27X                         RelativeSigmaL( d/p, SymmetricBasis( q, p ) ) ),[127X[104X
    [4X[25X>[125X [27X                         embedG ) ),[127X[104X
    [4X[25X>[125X [27X                     swap ) );[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    # Compute conjugacy classes of outer involutions in the maxes.[127X[104X
    [4X[25X>[125X [27X    # (In order to avoid computing all conjugacy classes of these subgroups,[127X[104X
    [4X[25X>[125X [27X    # we work in the Sylow $2$ subgroups.)[127X[104X
    [4X[25X>[125X [27X    maxes:= List( maxes, M -> ImagesSet( epi, M ) );[127X[104X
    [4X[25X>[125X [27X    ccl:= List( maxes, M -> ClassesOfPrimeOrder( M, [ 2 ], Gprime ) );[127X[104X
    [4X[25X>[125X [27X    names:= List( primes, p -> Concatenation( "GL(", String( d/p ), ",",[127X[104X
    [4X[25X>[125X [27X                                   String( q^p ), ").", String( p ) ) );[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    return [ names, UpperBoundFixedPointRatios( PG, ccl, true )[1] ];[127X[104X
    [4X[25X>[125X [27Xend;;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YAnd these are the results for the groups we are interested in (and others).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XApproxPForOuterClassesInExtensionOfSLByGraphAut( 4, 3 );[127X[104X
    [4X[28X[ [ "GL(2,9).2" ], 17/117 ][128X[104X
    [4X[25Xgap>[125X [27XApproxPForOuterClassesInExtensionOfSLByGraphAut( 4, 4 );[127X[104X
    [4X[28X[ [ "GL(2,16).2" ], 73/1008 ][128X[104X
    [4X[25Xgap>[125X [27XApproxPForOuterClassesInExtensionOfSLByGraphAut( 6, 2 );[127X[104X
    [4X[28X[ [ "GL(3,4).2", "GL(2,8).3" ], 41/1984 ][128X[104X
    [4X[25Xgap>[125X [27XApproxPForOuterClassesInExtensionOfSLByGraphAut( 6, 3 );[127X[104X
    [4X[28X[ [ "GL(3,9).2", "GL(2,27).3" ], 541/352836 ][128X[104X
    [4X[25Xgap>[125X [27XApproxPForOuterClassesInExtensionOfSLByGraphAut( 6, 4 );[127X[104X
    [4X[28X[ [ "GL(3,16).2", "GL(2,64).3" ], 3265/12570624 ][128X[104X
    [4X[25Xgap>[125X [27XApproxPForOuterClassesInExtensionOfSLByGraphAut( 6, 5 );[127X[104X
    [4X[28X[ [ "GL(3,25).2", "GL(2,125).3" ], 13001/195250000 ][128X[104X
    [4X[25Xgap>[125X [27XApproxPForOuterClassesInExtensionOfSLByGraphAut( 8, 2 );[127X[104X
    [4X[28X[ [ "GL(4,4).2" ], 367/1007872 ][128X[104X
    [4X[25Xgap>[125X [27XApproxPForOuterClassesInExtensionOfSLByGraphAut( 10, 2 );[127X[104X
    [4X[28X[ [ "GL(5,4).2", "GL(2,32).5" ], 609281/476346056704 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we  consider diagonal automorphisms. We modify the approach for [22XSL(d,q)[122X
  by constructing the field extension type subgroups of [22XGL(d,q) ...[122X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XRelativeGammaL:= function( d, B )[127X[104X
    [4X[25X>[125X [27X    local n, F, q, diag;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    n:= Length( B );[127X[104X
    [4X[25X>[125X [27X    F:= LeftActingDomain( UnderlyingLeftModule( B ) );[127X[104X
    [4X[25X>[125X [27X    q:= Size( F );[127X[104X
    [4X[25X>[125X [27X    diag:= IdentityMat( d * n, F );[127X[104X
    [4X[25X>[125X [27X    diag{[ 1 .. n ]}{[ 1 .. n ]}:= BlownUpMat( B, [ [ Z(q^n) ] ] );[127X[104X
    [4X[25X>[125X [27X    return ClosureGroup( RelativeSigmaL( d, B ),  diag );[127X[104X
    [4X[25X>[125X [27Xend;;[127X[104X
  [4X[32X[104X
  
  [33X[0;0Y[22X...[122X and counting the elements of prime order outside the simple group.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XApproxPForOuterClassesInGL:= function( d, q )[127X[104X
    [4X[25X>[125X [27X    local G, epi, PG, Gprime, primes, maxes, names;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    # Check whether this is an admissible case (see [Be00]).[127X[104X
    [4X[25X>[125X [27X    if ( d = 2 and q in [ 2, 5, 7, 9 ] ) or ( d = 3 and q = 4 ) then[127X[104X
    [4X[25X>[125X [27X      return fail;[127X[104X
    [4X[25X>[125X [27X    fi;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    # Create the group GL(d,q), and the map to PGL(d,q).[127X[104X
    [4X[25X>[125X [27X    G:= GL( d, q );[127X[104X
    [4X[25X>[125X [27X    epi:= ActionHomomorphism( G, NormedRowVectors( GF(q)^d ), OnLines );[127X[104X
    [4X[25X>[125X [27X    PG:= ImagesSource( epi );[127X[104X
    [4X[25X>[125X [27X    Gprime:= ImagesSet( epi, SL( d, q ) );[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    # Create the subgroups corresponding to the prime divisors of `d'.[127X[104X
    [4X[25X>[125X [27X    primes:= PrimeDivisors( d );[127X[104X
    [4X[25X>[125X [27X    maxes:= List( primes, p -> RelativeGammaL( d/p,[127X[104X
    [4X[25X>[125X [27X                                   Basis( AsField( GF(q), GF(q^p) ) ) ) );[127X[104X
    [4X[25X>[125X [27X    maxes:= List( maxes, M -> ImagesSet( epi, M ) );[127X[104X
    [4X[25X>[125X [27X    names:= List( primes, p -> Concatenation( "M(", String( d/p ), ",",[127X[104X
    [4X[25X>[125X [27X                                   String( q^p ), ")" ) );[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    return [ names,[127X[104X
    [4X[25X>[125X [27X             UpperBoundFixedPointRatios( PG, List( maxes,[127X[104X
    [4X[25X>[125X [27X                 M -> ClassesOfPrimeOrder( M,[127X[104X
    [4X[25X>[125X [27X                          PrimeDivisors( Index( PG, Gprime ) ), Gprime ) ),[127X[104X
    [4X[25X>[125X [27X                 true )[1] ];[127X[104X
    [4X[25X>[125X [27Xend;;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YHere are the required results.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XApproxPForOuterClassesInGL( 6, 3 );[127X[104X
    [4X[28X[ [ "M(3,9)", "M(2,27)" ], 41/882090 ][128X[104X
    [4X[25Xgap>[125X [27XApproxPForOuterClassesInGL( 4, 3 );[127X[104X
    [4X[28X[ [ "M(2,9)" ], 0 ][128X[104X
    [4X[25Xgap>[125X [27XApproxPForOuterClassesInGL( 6, 4 );[127X[104X
    [4X[28X[ [ "M(3,16)", "M(2,64)" ], 1/87296 ][128X[104X
    [4X[25Xgap>[125X [27XApproxPForOuterClassesInGL( 6, 5 );[127X[104X
    [4X[28X[ [ "M(3,25)", "M(2,125)" ], 821563/756593750000 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0Y(Note  that  the extension field type subgroup in [22XPGL(4,3) = L_4(3).2_1[122X is a
  [13Xnon-split[113X extension of its intersection with [22XL_4(3)[122X, hence the zero value.)[133X
  
  [33X[0;0YConcerning  extensions  by  Frobenius  automorphisms,  only the case [22X(d,q) =
  (6,4)[122X  is  interesting  in [BGK08].  In  fact,  we would not need to compute
  anything  for  the  extension  [22XG[122X  of  [22XS  = SL(6,4)[122X by the Frobenius map that
  squares  each  matrix  entry.  This  is  because  [22XMM^'(G,s)[122X  consists of the
  normalizers of the two subgroups of the types [22XSL(3,16)[122X and [22XSL(2,64)[122X, and the
  former maximal subgroup is a [13Xnon-split[113X extension of its intersection with [22XS[122X,
  so  only  one  maximal  subgroup  can  contribute to [22Xσ^'(G,s)[122X, which is thus
  smaller than [22X1/2[122X, by [BGK08, Prop. 2.6].[133X
  
  [33X[0;0YHowever,  it  is easy enough to compute the exact value of [22Xσ^'(G,s)[122X. We work
  with  the  projective  action  of  [22XS[122X  on its natural module, and compute the
  permutation  induced  by  the  Frobenius  map as the Frobenius action on the
  normed row vectors.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmatgrp:= SL(6,4);;[127X[104X
    [4X[25Xgap>[125X [27Xdom:= NormedRowVectors( GF(4)^6 );;[127X[104X
    [4X[25Xgap>[125X [27XGprime:= Action( matgrp, dom, OnLines );;[127X[104X
    [4X[25Xgap>[125X [27Xpi:= PermList( List( dom, v -> Position( dom, List( v, x -> x^2 ) ) ) );;[127X[104X
    [4X[25Xgap>[125X [27XG:= ClosureGroup( Gprime, pi );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThen we compute the maximal subgroups, the classes of outer involutions, and
  the bound, similar to the situation with graph automorphisms.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmaxes:= List( [ 2, 3 ], p -> Normalizer( G,[127X[104X
    [4X[25X>[125X [27X             Action( RelativeSigmaL( 6/p,[127X[104X
    [4X[25X>[125X [27X               Basis( AsField( GF(4), GF(4^p) ) ) ), dom, OnLines ) ) );;[127X[104X
    [4X[25Xgap>[125X [27Xccl:= List( maxes, M -> ClassesOfPrimeOrder( M, [ 2 ], Gprime ) );;[127X[104X
    [4X[25Xgap>[125X [27XList( ccl, Length );[127X[104X
    [4X[28X[ 0, 1 ][128X[104X
    [4X[25Xgap>[125X [27XUpperBoundFixedPointRatios( G, ccl, true );[127X[104X
    [4X[28X[ 1/34467840, true ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  [22X(d,q)  = (6,4)[122X, we have to consider also the extension [22XG[122X of [22XS = SL(6,4)[122X
  by the product [22Xα[122X of the Frobenius map and the graph automorphism. We use the
  same  approach as for the graph automorphism, i. e., we embed [22XSL(6,4)[122X into a
  [22X12[122X-dimensional  group of [22X6 × 6[122X block matrices, where the second block is the
  image of the first block under [22Xα[122X, and describe [22Xα[122X by the transposition of the
  two blocks.[133X
  
  [33X[0;0YFirst  we  construct  the  projective  actions  of  [22XS[122X  and  [22XG[122X on an orbit of
  [22X1[122X-spaces.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XembedFG:= function( F, mat )[127X[104X
    [4X[25X>[125X [27X     return EmbeddedMatrix( F, mat,[127X[104X
    [4X[25X>[125X [27X                M -> List( TransposedMat( M^-1 ),[127X[104X
    [4X[25X>[125X [27X                           row -> List( row, x -> x^2 ) ) );[127X[104X
    [4X[25X>[125X [27X   end;;[127X[104X
    [4X[25Xgap>[125X [27Xd:= 6;;  q:= 4;;[127X[104X
    [4X[25Xgap>[125X [27Xalpha:= NullMat( 2*d, 2*d, GF(q) );;[127X[104X
    [4X[25Xgap>[125X [27Xalpha{ [ 1 .. d ] }{ [ d+1 .. 2*d ] }:= IdentityMat( d, GF(q) );;[127X[104X
    [4X[25Xgap>[125X [27Xalpha{ [ d+1 .. 2*d ] }{ [ 1 .. d ] }:= IdentityMat( d, GF(q) );;[127X[104X
    [4X[25Xgap>[125X [27XGprime:= Group( List( GeneratorsOfGroup( SL(d,q) ),[127X[104X
    [4X[25X>[125X [27X                         mat -> embedFG( GF(q), mat ) ) );;[127X[104X
    [4X[25Xgap>[125X [27XG:= ClosureGroupDefault( Gprime, alpha );;[127X[104X
    [4X[25Xgap>[125X [27Xorb:= Orbit( G, One( G )[1], OnLines );;[127X[104X
    [4X[25Xgap>[125X [27XG:= Action( G, orb, OnLines );;[127X[104X
    [4X[25Xgap>[125X [27XGprime:= Action( Gprime, orb, OnLines );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YNext  we  construct the maximal subgroups, the classes of outer involutions,
  and the bound.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmaxes:= List( PrimeDivisors( d ), p -> Group( List( GeneratorsOfGroup([127X[104X
    [4X[25X>[125X [27X             RelativeSigmaL( d/p, Basis( AsField( GF(q), GF(q^p) ) ) ) ),[127X[104X
    [4X[25X>[125X [27X               mat -> embedFG( GF(q), mat ) ) ) );;[127X[104X
    [4X[25Xgap>[125X [27Xmaxes:= List( maxes, x -> Action( x, orb, OnLines ) );;[127X[104X
    [4X[25Xgap>[125X [27Xmaxes:= List( maxes, x -> Normalizer( G, x ) );;[127X[104X
    [4X[25Xgap>[125X [27Xccl:= List( maxes, M -> ClassesOfPrimeOrder( M, [ 2 ], Gprime ) );;[127X[104X
    [4X[25Xgap>[125X [27XList( ccl, Length );[127X[104X
    [4X[28X[ 0, 1 ][128X[104X
    [4X[25Xgap>[125X [27XUpperBoundFixedPointRatios( G, ccl, true );[127X[104X
    [4X[28X[ 1/10792960, true ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  only  missing  cases  are  the extensions of [22XSL(6,3)[122X and [22XSL(6,5)[122X by the
  involutory  outer  automorphism that acts as the product of a diagonal and a
  graph automorphism.[133X
  
  [33X[0;0YIn the case [22XS = SL(6,3)[122X, we can directly write down the extension [22XG[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xd:= 6;;  q:= 3;;[127X[104X
    [4X[25Xgap>[125X [27Xdiag:= IdentityMat( d, GF(q) );;[127X[104X
    [4X[25Xgap>[125X [27Xdiag[1][1]:= Z(q);;[127X[104X
    [4X[25Xgap>[125X [27XembedDG:= mat -> EmbeddedMatrix( GF(q), mat,[127X[104X
    [4X[25X>[125X [27X                                    M -> TransposedMat( M^-1 )^diag );;[127X[104X
    [4X[25Xgap>[125X [27XGprime:= Group( List( GeneratorsOfGroup( SL(d,q) ), embedDG ) );;[127X[104X
    [4X[25Xgap>[125X [27Xalpha:= NullMat( 2*d, 2*d, GF(q) );;[127X[104X
    [4X[25Xgap>[125X [27Xalpha{ [ 1 .. d ] }{ [ d+1 .. 2*d ] }:= IdentityMat( d, GF(q) );;[127X[104X
    [4X[25Xgap>[125X [27Xalpha{ [ d+1 .. 2*d ] }{ [ 1 .. d ] }:= IdentityMat( d, GF(q) );;[127X[104X
    [4X[25Xgap>[125X [27XG:= ClosureGroupDefault( Gprime, alpha );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  maximal  subgroups  are  constructed  as  the  normalizers  in [22XG[122X of the
  extension   field   type   subgroups  in  [22XS[122X.  We  work  with  a  permutation
  representation of [22XG[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmaxes:= List( PrimeDivisors( d ), p -> Group( List( GeneratorsOfGroup([127X[104X
    [4X[25X>[125X [27X             RelativeSigmaL( d/p, Basis( AsField( GF(q), GF(q^p) ) ) ) ),[127X[104X
    [4X[25X>[125X [27X               embedDG ) ) );;[127X[104X
    [4X[25Xgap>[125X [27Xorb:= Orbit( G, One( G )[1], OnLines );;[127X[104X
    [4X[25Xgap>[125X [27XG:= Action( G, orb, OnLines );;[127X[104X
    [4X[25Xgap>[125X [27XGprime:= Action( Gprime, orb, OnLines );;[127X[104X
    [4X[25Xgap>[125X [27Xmaxes:= List( maxes, M -> Normalizer( G, Action( M, orb, OnLines ) ) );;[127X[104X
    [4X[25Xgap>[125X [27Xccl:= List( maxes, M -> ClassesOfPrimeOrder( M, [ 2 ], Gprime ) );;[127X[104X
    [4X[25Xgap>[125X [27XList( ccl, Length );[127X[104X
    [4X[28X[ 1, 1 ][128X[104X
    [4X[25Xgap>[125X [27XUpperBoundFixedPointRatios( G, ccl, true );[127X[104X
    [4X[28X[ 25/352836, true ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  [22XS  = SL(6,5)[122X, this approach does not work because we cannot realize the
  diagonal  involution  by  an  involutory  matrix.  Instead,  we consider the
  extension  of  [22XGL(6,5) ≅ 2.(2 × L_6(5)).2[122X by the graph automorphism [22Xα[122X, which
  can be embedded into [22XGL(12,5)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xd:= 6;;  q:= 5;;[127X[104X
    [4X[25Xgap>[125X [27XembedG:= mat -> EmbeddedMatrix( GF(q),[127X[104X
    [4X[25X>[125X [27X                                   mat, M -> TransposedMat( M^-1 ) );;[127X[104X
    [4X[25Xgap>[125X [27XGprime:= Group( List( GeneratorsOfGroup( SL(d,q) ), embedG ) );;[127X[104X
    [4X[25Xgap>[125X [27Xmaxes:= List( PrimeDivisors( d ), p -> Group( List( GeneratorsOfGroup([127X[104X
    [4X[25X>[125X [27X             RelativeSigmaL( d/p, Basis( AsField( GF(q), GF(q^p) ) ) ) ),[127X[104X
    [4X[25X>[125X [27X               embedG ) ) );;[127X[104X
    [4X[25Xgap>[125X [27Xdiag:= IdentityMat( d, GF(q) );;[127X[104X
    [4X[25Xgap>[125X [27Xdiag[1][1]:= Z(q);;[127X[104X
    [4X[25Xgap>[125X [27Xdiag:= embedG( diag );;[127X[104X
    [4X[25Xgap>[125X [27Xalpha:= NullMat( 2*d, 2*d, GF(q) );;[127X[104X
    [4X[25Xgap>[125X [27Xalpha{ [ 1 .. d ] }{ [ d+1 .. 2*d ] }:= IdentityMat( d, GF(q) );;[127X[104X
    [4X[25Xgap>[125X [27Xalpha{ [ d+1 .. 2*d ] }{ [ 1 .. d ] }:= IdentityMat( d, GF(q) );;[127X[104X
    [4X[25Xgap>[125X [27XG:= ClosureGroupDefault( Gprime, alpha * diag );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we  switch to the permutation action of this group on the [22X1[122X-dimensional
  subspaces,  thus  factoring out the cyclic normal subgroup of order four. In
  this  action,  the  involutory  diagonal  automorphism  is represented by an
  involution, and we can proceed as above.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xorb:= Orbit( G, One( G )[1], OnLines );;[127X[104X
    [4X[25Xgap>[125X [27XGprime:= Action( Gprime, orb, OnLines );;[127X[104X
    [4X[25Xgap>[125X [27XG:= Action( G, orb, OnLines );;[127X[104X
    [4X[25Xgap>[125X [27Xmaxes:= List( maxes, M -> Action( M, orb, OnLines ) );;[127X[104X
    [4X[25Xgap>[125X [27Xextmaxes:= List( maxes, M -> Normalizer( G, M ) );;[127X[104X
    [4X[25Xgap>[125X [27Xccl:= List( extmaxes, M -> ClassesOfPrimeOrder( M, [ 2 ], Gprime ) );;[127X[104X
    [4X[25Xgap>[125X [27XList( ccl, Length );[127X[104X
    [4X[28X[ 2, 1 ][128X[104X
    [4X[25Xgap>[125X [27XUpperBoundFixedPointRatios( G, ccl, true );[127X[104X
    [4X[28X[ 3863/6052750000, true ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  the same way, we can recheck the values for the extensions of [22XSL(6,5)[122X by
  the diagonal or by the graph automorphism.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xdiag:= Permutation( diag, orb, OnLines );;[127X[104X
    [4X[25Xgap>[125X [27XG:= ClosureGroupDefault( Gprime, diag );;[127X[104X
    [4X[25Xgap>[125X [27Xextmaxes:= List( maxes, M -> Normalizer( G, M ) );;[127X[104X
    [4X[25Xgap>[125X [27Xccl:= List( extmaxes, M -> ClassesOfPrimeOrder( M, [ 2 ], Gprime ) );;[127X[104X
    [4X[25Xgap>[125X [27XList( ccl, Length );[127X[104X
    [4X[28X[ 3, 1 ][128X[104X
    [4X[25Xgap>[125X [27XUpperBoundFixedPointRatios( G, ccl, true );[127X[104X
    [4X[28X[ 821563/756593750000, true ][128X[104X
    [4X[25Xgap>[125X [27Xalpha:= Permutation( alpha, orb, OnLines );;[127X[104X
    [4X[25Xgap>[125X [27XG:= ClosureGroupDefault( Gprime, alpha );;[127X[104X
    [4X[25Xgap>[125X [27Xextmaxes:= List( maxes, M -> Normalizer( G, M ) );;[127X[104X
    [4X[25Xgap>[125X [27Xccl:= List( extmaxes, M -> ClassesOfPrimeOrder( M, [ 2 ], Gprime ) );;[127X[104X
    [4X[25Xgap>[125X [27XList( ccl, Length );[127X[104X
    [4X[28X[ 2, 2 ][128X[104X
    [4X[25Xgap>[125X [27XUpperBoundFixedPointRatios( G, ccl, true );[127X[104X
    [4X[28X[ 13001/195250000, true ][128X[104X
  [4X[32X[104X
  
  [33X[0;0Ygap> t2:= CharacterTable( "L6(2).2" );; gap> map:= InverseMap( GetFusionMap(
  t, t2 ) );; gap> torso:= List( Concatenation( prim ), pi -> CompositionMaps(
  pi, map ) );; gap> ext:= List( torso, x -> PermChars( t2, rec( torso:= x ) )
  );  [  [  Character( CharacterTable( "L6(2).2" ), [ 55552, 0, 128, 256, 337,
  112,  22, 0, 0, 16, 0, 16, 2, 17, 0, 0, 8, 2, 4, 28, 0, 0, 0, 4, 1, 0, 1, 0,
  0, 4, 0, 2, 2, 2, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1120, 192, 32, 0, 0, 40, 13,
  0,  4,  6,  0,  4,  4,  4, 0, 2, 8, 5, 0, 2, 0, 0, 0, 0, 1, 0, 1, 0 ] ) ], [
  Character(  CharacterTable(  "L6(2).2" ), [ 1904640, 0, 0, 512, 960, 0, 120,
  0,  0,  0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 73, 24, 3, 0, 0, 15, 0, 0, 0, 0, 1, 0,
  0,  0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 960, 960, 0, 0, 0, 0, 24, 0, 12, 12,
  0,  0,  0,  0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 3, 0, 0, 0 ] ) ] ] gap> sigma:=
  ApproxP( Concatenation( ext ), > Position( OrdersClassRepresentatives( t2 ),
  63   )   );;   gap>   Maximum(   sigma{   Difference(  PositionsProperty(  >
  OrdersClassRepresentatives(       t2       ),      IsPrimeInt      ),      >
  ClassPositionsOfDerivedSubgroup( t2 ) ) } ); 41/1984 -->[133X
  
  
  [1X11.5-8 [33X[0;0Y[22XL_3(2)[122X[101X[1X[133X[101X
  
  [33X[0;0YWe show that [22XS = L_3(2) = SL(3,2)[122X satisfies the following.[133X
  
  [8X(a)[108X
        [33X[0;6Y[22Xσ(S)  =  1/4[122X,  and this value is attained exactly for [22Xσ(S,s)[122X with [22Xs[122X of
        order [22X7[122X.[133X
  
  [8X(b)[108X
        [33X[0;6YFor [22Xs[122X of order [22X7[122X, [22XMM(S,s)[122X consists of one group of the type [22X7:3[122X.[133X
  
  [8X(c)[108X
        [33X[0;6Y[22XP(S)  =  1/4[122X,  and this value is attained exactly for [22XP(S,s)[122X with [22Xs[122X of
        order [22X7[122X.[133X
  
  [8X(d)[108X
        [33X[0;6YThe  uniform  spread  of [22XS[122X is at exactly three, with [22Xs[122X of order [22X7[122X, and
        the spread of [22XS[122X is exactly four. (This had been left open in [BW75].)[133X
  
  [33X[0;0Y(Note that in this example, the spread and the uniform spread differ.)[133X
  
  [33X[0;0YStatement (a)   follows   from   inspection  of  the  primitive  permutation
  characters, cf. Section [14X11.4-3[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "L3(2)" );;[127X[104X
    [4X[25Xgap>[125X [27XProbGenInfoSimple( t );[127X[104X
    [4X[28X[ "L3(2)", 1/4, 3, [ "7A" ], [ 1 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YStatement (b)  can be read off from the permutation characters, and the fact
  that  the unique class of maximal subgroups that contain elements of order [22X7[122X
  consists of groups of the structure [22X7:3[122X, see [CCN+85, p. 3].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XOrdersClassRepresentatives( t );[127X[104X
    [4X[28X[ 1, 2, 3, 4, 7, 7 ][128X[104X
    [4X[25Xgap>[125X [27XPrimitivePermutationCharacters( t );[127X[104X
    [4X[28X[ Character( CharacterTable( "L3(2)" ), [ 7, 3, 1, 1, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "L3(2)" ), [ 7, 3, 1, 1, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "L3(2)" ), [ 8, 0, 2, 0, 1, 1 ] ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor   the   other   statements,   we  will  use  the  primitive  permutation
  representations  on  [22X7[122X  and  [22X8[122X points of [22XS[122X (computed from the [5XGAP[105X Library of
  Tables of Marks), and their diagonal products of the degrees [22X14[122X and [22X15[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtom:= TableOfMarks( "L3(2)" );;[127X[104X
    [4X[25Xgap>[125X [27Xg:= UnderlyingGroup( tom );[127X[104X
    [4X[28XGroup([ (2,4)(5,7), (1,2,3)(4,5,6) ])[128X[104X
    [4X[25Xgap>[125X [27Xmx:= MaximalSubgroupsTom( tom );[127X[104X
    [4X[28X[ [ 14, 13, 12 ], [ 7, 7, 8 ] ][128X[104X
    [4X[25Xgap>[125X [27Xmaxes:= List( mx[1], i -> RepresentativeTom( tom, i ) );;[127X[104X
    [4X[25Xgap>[125X [27Xtr:= List( maxes, s -> RightTransversal( g, s ) );;[127X[104X
    [4X[25Xgap>[125X [27Xacts:= List( tr, x -> Action( g, x, OnRight ) );;[127X[104X
    [4X[25Xgap>[125X [27Xg7:= acts[1];[127X[104X
    [4X[28XGroup([ (3,4)(6,7), (1,3,2)(4,6,5) ])[128X[104X
    [4X[25Xgap>[125X [27Xg8:= acts[3];[127X[104X
    [4X[28XGroup([ (1,6)(2,5)(3,8)(4,7), (1,7,3)(2,5,8) ])[128X[104X
    [4X[25Xgap>[125X [27Xg14:= DiagonalProductOfPermGroups( acts{ [ 1, 2 ] } );[127X[104X
    [4X[28XGroup([ (3,4)(6,7)(11,13)(12,14), (1,3,2)(4,6,5)(8,11,9)(10,12,13) ])[128X[104X
    [4X[25Xgap>[125X [27Xg15:= DiagonalProductOfPermGroups( acts{ [ 2, 3 ] } );[127X[104X
    [4X[28XGroup([ (4,6)(5,7)(8,13)(9,12)(10,15)(11,14), (1,4,2)(3,5,6)(8,14,10)[128X[104X
    [4X[28X  (9,12,15) ])[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFirst  we  compute  that  for all nonidentity elements [22Xs ∈ S[122X and order three
  elements [22Xg ∈ S[122X, [22XP(g,s) ≥ 1/4[122X holds, with equality if and only if [22Xs[122X has order
  [22X7[122X;   this   implies   statement (c).   We   actually   compute,   for  class
  representatives [22Xs[122X, the proportion of order three elements [22Xg[122X such that [22X⟨ g, s
  ⟩ ‡ S[122X holds.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xccl:= List( ConjugacyClasses( g7 ), Representative );;[127X[104X
    [4X[25Xgap>[125X [27XSortParallel( List( ccl, Order ), ccl );[127X[104X
    [4X[25Xgap>[125X [27XList( ccl, Order );[127X[104X
    [4X[28X[ 1, 2, 3, 4, 7, 7 ][128X[104X
    [4X[25Xgap>[125X [27XSize( ConjugacyClass( g7, ccl[3] ) );[127X[104X
    [4X[28X56[128X[104X
    [4X[25Xgap>[125X [27Xprop:= List( ccl,[127X[104X
    [4X[25X>[125X [27X                r -> RatioOfNongenerationTransPermGroup( g7, ccl[3], r ) );[127X[104X
    [4X[28X[ 1, 5/7, 19/28, 2/7, 1/4, 1/4 ][128X[104X
    [4X[25Xgap>[125X [27XMinimum( prop );[127X[104X
    [4X[28X1/4[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we  show  that the uniform spread of [22XS[122X is less than four. In any of the
  primitive  permutation  representations  of  degree  seven,  we  find  three
  involutions  whose sets of fixed points cover the seven points. The elements
  [22Xs[122X of order different from [22X7[122X in [22XS[122X fix a point in this representation, so each
  such  [22Xs[122X  generates  a  proper  subgroup  of [22XS[122X together with one of the three
  involutions.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xx:= g7.1;[127X[104X
    [4X[28X(3,4)(6,7)[128X[104X
    [4X[25Xgap>[125X [27Xfix:= Difference( MovedPoints( g7 ), MovedPoints( x ) );[127X[104X
    [4X[28X[ 1, 2, 5 ][128X[104X
    [4X[25Xgap>[125X [27Xorb:= Orbit( g7, fix, OnSets );[127X[104X
    [4X[28X[ [ 1, 2, 5 ], [ 1, 3, 4 ], [ 2, 3, 6 ], [ 2, 4, 7 ], [ 1, 6, 7 ], [128X[104X
    [4X[28X  [ 3, 5, 7 ], [ 4, 5, 6 ] ][128X[104X
    [4X[25Xgap>[125X [27XUnion( orb{ [ 1, 2, 5 ] } ) = [ 1 .. 7 ];[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo  we  still  have  to  exclude  elements  [22Xs[122X  of  order [22X7[122X. In the primitive
  permutation  representation  of  [22XS[122X on eight points, we find four elements of
  order  three whose sets of fixed points cover the set of all points that are
  moved  by [22XS[122X, so with each element of order seven in [22XS[122X, one of them generates
  an intransitive group.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xthree:= g8.2;[127X[104X
    [4X[28X(1,7,3)(2,5,8)[128X[104X
    [4X[25Xgap>[125X [27Xfix:= Difference( MovedPoints( g8 ), MovedPoints( three ) );[127X[104X
    [4X[28X[ 4, 6 ][128X[104X
    [4X[25Xgap>[125X [27Xorb:= Orbit( g8, fix, OnSets );;[127X[104X
    [4X[25Xgap>[125X [27XQuadrupleWithProperty( [ [ fix ], orb, orb, orb ],[127X[104X
    [4X[25X>[125X [27X       list -> Union( list ) = [ 1 .. 8 ] );[127X[104X
    [4X[28X[ [ 4, 6 ], [ 1, 7 ], [ 3, 8 ], [ 2, 5 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YTogether  with  statement (a),  this  proves that the uniform spread of [22XS[122X is
  exactly three, with [22Xs[122X of order seven.[133X
  
  [33X[0;0YEach  element  of  [22XS[122X  fixes  a point in the permutation representation on [22X15[122X
  points.  So  for  proving  that  the  spread  of  [22XS[122X is less than five, it is
  sufficient  to find a quintuple of elements whose sets of fixed points cover
  all  [22X15[122X  points.  (From  the permutation characters it is clear that four of
  these elements must have order three, and the fifth must be an involution.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xx:= g15.1;[127X[104X
    [4X[28X(4,6)(5,7)(8,13)(9,12)(10,15)(11,14)[128X[104X
    [4X[25Xgap>[125X [27Xfixx:= Difference( MovedPoints( g15 ), MovedPoints( x ) );[127X[104X
    [4X[28X[ 1, 2, 3 ][128X[104X
    [4X[25Xgap>[125X [27Xorbx:= Orbit( g15, fixx, OnSets );[127X[104X
    [4X[28X[ [ 1, 2, 3 ], [ 1, 4, 5 ], [ 1, 6, 7 ], [ 2, 4, 6 ], [ 3, 4, 7 ], [128X[104X
    [4X[28X  [ 3, 5, 6 ], [ 2, 5, 7 ] ][128X[104X
    [4X[25Xgap>[125X [27Xy:= g15.2;[127X[104X
    [4X[28X(1,4,2)(3,5,6)(8,14,10)(9,12,15)[128X[104X
    [4X[25Xgap>[125X [27Xfixy:= Difference( MovedPoints( g15 ), MovedPoints( y ) );[127X[104X
    [4X[28X[ 7, 11, 13 ][128X[104X
    [4X[25Xgap>[125X [27Xorby:= Orbit( g15, fixy, OnSets );;[127X[104X
    [4X[25Xgap>[125X [27XQuadrupleWithProperty( [ [ fixy ], orby, orby, orby ],[127X[104X
    [4X[25X>[125X [27X       l -> Difference( [ 1 .. 15 ], Union( l ) ) in orbx );[127X[104X
    [4X[28X[ [ 7, 11, 13 ], [ 5, 8, 14 ], [ 1, 10, 15 ], [ 3, 9, 12 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YIt  remains  to  show  that  the  spread  of  [22XS[122X  is  (at least) four. By the
  consideration of permutation characters, we know that we can find a suitable
  order seven element for all quadruples in question except perhaps quadruples
  of order three elements. We show that for each such case, we can choose [22Xs[122X of
  order four. Since [22XMM(S,s)[122X consists of two subgroups of the type [22XS_4[122X, we work
  with the representation on [22X14[122X points.)[133X
  
  [33X[0;0YFirst  we  compute [22Xs[122X and the [22XS[122X-orbit of its fixed points, and the [22XS[122X-orbit of
  the fixed points of an element [22Xx[122X of order three. Then we prove that for each
  quadruple of conjugates of [22Xx[122X, the union of their fixed points intersects the
  fixed points of at least one conjugate of [22Xs[122X trivially.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XResetGlobalRandomNumberGenerators();[127X[104X
    [4X[25Xgap>[125X [27Xrepeat s:= Random( g14 );[127X[104X
    [4X[25X>[125X [27X   until Order( s ) = 4;[127X[104X
    [4X[25Xgap>[125X [27Xs;[127X[104X
    [4X[28X(1,3)(2,6,7,5)(9,11,10,12)(13,14)[128X[104X
    [4X[25Xgap>[125X [27Xfixs:= Difference( MovedPoints( g14 ), MovedPoints( s ) );[127X[104X
    [4X[28X[ 4, 8 ][128X[104X
    [4X[25Xgap>[125X [27Xorbs:= Orbit( g14, fixs, OnSets );;[127X[104X
    [4X[25Xgap>[125X [27XLength( orbs );[127X[104X
    [4X[28X21[128X[104X
    [4X[25Xgap>[125X [27Xthree:= g14.2;[127X[104X
    [4X[28X(1,3,2)(4,6,5)(8,11,9)(10,12,13)[128X[104X
    [4X[25Xgap>[125X [27Xfix:= Difference( MovedPoints( g14 ), MovedPoints( three ) );[127X[104X
    [4X[28X[ 7, 14 ][128X[104X
    [4X[25Xgap>[125X [27Xorb:= Orbit( g14, fix, OnSets );;[127X[104X
    [4X[25Xgap>[125X [27XLength( orb );[127X[104X
    [4X[28X28[128X[104X
    [4X[25Xgap>[125X [27XQuadrupleWithProperty( [ [ fix ], orb, orb, orb ],[127X[104X
    [4X[25X>[125X [27X       l -> ForAll( orbs, o -> not IsEmpty( Intersection( o,[127X[104X
    [4X[25X>[125X [27X                       Union( l ) ) ) ) );[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  [33X[0;0YBy the lemma from Section [14X11.2-2[114X, we are done.[133X
  
  
  [1X11.5-9 [33X[0;0Y[22XM_11[122X[101X[1X[133X[101X
  
  [33X[0;0YWe show that [22XS = M_11[122X satisfies the following.[133X
  
  [8X(a)[108X
        [33X[0;6Y[22Xσ(S)  =  1/3[122X,  and this value is attained exactly for [22Xσ(S,s)[122X with [22Xs[122X of
        order [22X11[122X.[133X
  
  [8X(b)[108X
        [33X[0;6YFor [22Xs[122X of order [22X11[122X, [22XMM(S,s)[122X consists of one group of the type [22XL_2(11)[122X.[133X
  
  [8X(c)[108X
        [33X[0;6Y[22XP(S)  =  1/3[122X,  and this value is attained exactly for [22XP(S,s)[122X with [22Xs[122X of
        order [22X11[122X.[133X
  
  [8X(d)[108X
        [33X[0;6YBoth  the  uniform spread and the spread of [22XS[122X is exactly three, with [22Xs[122X
        of order [22X11[122X.[133X
  
  [33X[0;0YStatement (a)   follows   from   inspection  of  the  primitive  permutation
  characters, cf. Section [14X11.4-1[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "M11" );;[127X[104X
    [4X[25Xgap>[125X [27XProbGenInfoSimple( t );[127X[104X
    [4X[28X[ "M11", 1/3, 2, [ "11A" ], [ 1 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YStatement (b)  can be read off from the permutation characters, and the fact
  that the unique class of maximal subgroups that contain elements of order [22X11[122X
  consists of groups of the structure [22XL_2(11)[122X, see [CCN+85, p. 18].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XOrdersClassRepresentatives( t );[127X[104X
    [4X[28X[ 1, 2, 3, 4, 5, 6, 8, 8, 11, 11 ][128X[104X
    [4X[25Xgap>[125X [27XPrimitivePermutationCharacters( t );[127X[104X
    [4X[28X[ Character( CharacterTable( "M11" ),[128X[104X
    [4X[28X  [ 11, 3, 2, 3, 1, 0, 1, 1, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "M11" ),[128X[104X
    [4X[28X  [ 12, 4, 3, 0, 2, 1, 0, 0, 1, 1 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "M11" ),[128X[104X
    [4X[28X  [ 55, 7, 1, 3, 0, 1, 1, 1, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "M11" ),[128X[104X
    [4X[28X  [ 66, 10, 3, 2, 1, 1, 0, 0, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "M11" ),[128X[104X
    [4X[28X  [ 165, 13, 3, 1, 0, 1, 1, 1, 0, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XMaxes( t );[127X[104X
    [4X[28X[ "A6.2_3", "L2(11)", "3^2:Q8.2", "A5.2", "2.S4" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor   the   other   statements,   we  will  use  the  primitive  permutation
  representations  of  [22XS[122X on [22X11[122X and [22X12[122X points (which are fetched from the [5XAtlas[105X
  of Group Representations [WWT+]), and their diagonal product.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgens11:= OneAtlasGeneratingSet( "M11", NrMovedPoints, 11 );[127X[104X
    [4X[28Xrec( charactername := "1a+10a", constituents := [ 1, 2 ], [128X[104X
    [4X[28X  contents := "core", [128X[104X
    [4X[28X  generators := [ (2,10)(4,11)(5,7)(8,9), (1,4,3,8)(2,5,6,9) ], [128X[104X
    [4X[28X  groupname := "M11", id := "", [128X[104X
    [4X[28X  identifier := [ "M11", [ "M11G1-p11B0.m1", "M11G1-p11B0.m2" ], 1, [128X[104X
    [4X[28X      11 ], isPrimitive := true, maxnr := 1, p := 11, rankAction := 2,[128X[104X
    [4X[28X  repname := "M11G1-p11B0", repnr := 1, size := 7920, [128X[104X
    [4X[28X  stabilizer := "A6.2_3", standardization := 1, transitivity := 4, [128X[104X
    [4X[28X  type := "perm" )[128X[104X
    [4X[25Xgap>[125X [27Xg11:= GroupWithGenerators( gens11.generators );;[127X[104X
    [4X[25Xgap>[125X [27Xgens12:= OneAtlasGeneratingSet( "M11", NrMovedPoints, 12 );;[127X[104X
    [4X[25Xgap>[125X [27Xg12:= GroupWithGenerators( gens12.generators );;[127X[104X
    [4X[25Xgap>[125X [27Xg23:= DiagonalProductOfPermGroups( [ g11, g12 ] );[127X[104X
    [4X[28XGroup([ (2,10)(4,11)(5,7)(8,9)(12,17)(13,20)(16,18)(19,21), (1,4,3,8)[128X[104X
    [4X[28X  (2,5,6,9)(12,17,18,15)(13,19)(14,20)(16,22,23,21) ])[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFirst we compute that for all nonidentity elements [22Xs ∈ S[122X and involutions [22Xg ∈
  S[122X,  [22XP(g,s)  ≥  1/3[122X  holds, with equality if and only if [22Xs[122X has order [22X11[122X; this
  implies statement (c). We actually compute, for class representatives [22Xs[122X, the
  proportion of involutions [22Xg[122X such that [22X⟨ g, s ⟩ ‡ S[122X holds.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xinv:= g11.1;[127X[104X
    [4X[28X(2,10)(4,11)(5,7)(8,9)[128X[104X
    [4X[25Xgap>[125X [27Xccl:= List( ConjugacyClasses( g11 ), Representative );;[127X[104X
    [4X[25Xgap>[125X [27XSortParallel( List( ccl, Order ), ccl );[127X[104X
    [4X[25Xgap>[125X [27XList( ccl, Order );[127X[104X
    [4X[28X[ 1, 2, 3, 4, 5, 6, 8, 8, 11, 11 ][128X[104X
    [4X[25Xgap>[125X [27XSize( ConjugacyClass( g11, inv ) );[127X[104X
    [4X[28X165[128X[104X
    [4X[25Xgap>[125X [27Xprop:= List( ccl,[127X[104X
    [4X[25X>[125X [27X                r -> RatioOfNongenerationTransPermGroup( g11, inv, r ) );[127X[104X
    [4X[28X[ 1, 1, 1, 149/165, 25/33, 31/55, 23/55, 23/55, 1/3, 1/3 ][128X[104X
    [4X[25Xgap>[125X [27XMinimum( prop );[127X[104X
    [4X[28X1/3[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  the  first part of statement (d), we have to deal only with the case of
  triples of involutions.[133X
  
  [33X[0;0YThe  [22X11[122X-cycle  [22Xs[122X is contained in exactly one maximal subgroup of [22XS[122X, of index
  [22X12[122X.  By  Corollary 1  in  Section [14X11.2-2[114X,  it  is enough to show that in the
  primitive degree [22X12[122X representation of [22XS[122X, the fixed points of no triple [22X(x_1,
  x_2,  x_3)[122X  of  involutions  in  [22XS[122X  can cover all twelve points; equivalenly
  (considering  complements),  we  show  that there is no triple such that the
  intersection of the sets of [13Xmoved[113X points is empty.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xinv:= g12.1;[127X[104X
    [4X[28X(1,6)(2,9)(5,7)(8,10)[128X[104X
    [4X[25Xgap>[125X [27Xmoved:= MovedPoints( inv );[127X[104X
    [4X[28X[ 1, 2, 5, 6, 7, 8, 9, 10 ][128X[104X
    [4X[25Xgap>[125X [27Xorb12:= Orbit( g12, moved, OnSets );;[127X[104X
    [4X[25Xgap>[125X [27XLength( orb12 );[127X[104X
    [4X[28X165[128X[104X
    [4X[25Xgap>[125X [27XTripleWithProperty( [ orb12{[1]}, orb12, orb12 ],[127X[104X
    [4X[25X>[125X [27X       list -> IsEmpty( Intersection( list ) ) );[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThis implies that the uniform spread of [22XS[122X is at least three.[133X
  
  [33X[0;0YNow  we  show  that  there is a quadruple consisting of one element of order
  three  and  three  involutions  whose  fixed  points cover all points in the
  degree  [22X23[122X representation constructed above; since the permutation character
  of  this  representation  is strictly positive, this implies that [22XS[122X does not
  have  spread  four,  by  Corollary 2  in  Section [14X11.2-2[114X, and we have proved
  statement (d).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xinv:= g23.1;[127X[104X
    [4X[28X(2,10)(4,11)(5,7)(8,9)(12,17)(13,20)(16,18)(19,21)[128X[104X
    [4X[25Xgap>[125X [27Xmoved:= MovedPoints( inv );[127X[104X
    [4X[28X[ 2, 4, 5, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21 ][128X[104X
    [4X[25Xgap>[125X [27Xorb23:= Orbit( g23, moved, OnSets );;[127X[104X
    [4X[25Xgap>[125X [27Xthree:= ( g23.1*g23.2^2 )^2;[127X[104X
    [4X[28X(2,6,10)(4,8,7)(5,9,11)(12,17,23)(15,18,16)(19,21,22)[128X[104X
    [4X[25Xgap>[125X [27Xmovedthree:= MovedPoints( three );;[127X[104X
    [4X[25Xgap>[125X [27XQuadrupleWithProperty( [ [ movedthree ], orb23, orb23, orb23 ],[127X[104X
    [4X[25X>[125X [27X       list -> IsEmpty( Intersection( list ) ) );[127X[104X
    [4X[28X[ [ 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 16, 17, 18, 19, 21, 22, 23 ],[128X[104X
    [4X[28X  [ 1, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 16, 17, 18, 20, 21 ], [128X[104X
    [4X[28X  [ 1, 2, 3, 4, 5, 6, 7, 11, 12, 13, 14, 15, 18, 19, 20, 23 ], [128X[104X
    [4X[28X  [ 1, 2, 3, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 20, 22, 23 ] ][128X[104X
  [4X[32X[104X
  
  
  [1X11.5-10 [33X[0;0Y[22XM_12[122X[101X[1X[133X[101X
  
  [33X[0;0YWe show that [22XS = M_12[122X satisfies the following.[133X
  
  [8X(a)[108X
        [33X[0;6Y[22Xσ(S)  =  1/3[122X,  and this value is attained exactly for [22Xσ(S,s)[122X with [22Xs[122X of
        order [22X10[122X.[133X
  
  [8X(b)[108X
        [33X[0;6YFor  [22Xs ∈ S[122X of order [22X10[122X, [22XMM(S,s)[122X consists of two nonconjugate subgroups
        of the type [22XA_6.2^2[122X, and one group of the type [22X2 × S_5[122X.[133X
  
  [8X(c)[108X
        [33X[0;6Y[22XP(S)  = 31/99[122X, and this value is attained exactly for [22XP(S,s)[122X with [22Xs[122X of
        order [22X10[122X.[133X
  
  [8X(d)[108X
        [33X[0;6YThe uniform spread of [22XS[122X is at least three, with [22Xs[122X of order [22X10[122X.[133X
  
  [8X(e)[108X
        [33X[0;6Y[22Xσ^'(Aut(S), s) = 4/99[122X.[133X
  
  [33X[0;0YStatement (a)   follows   from   inspection  of  the  primitive  permutation
  characters, cf. Section [14X11.4-1[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "M12" );;[127X[104X
    [4X[25Xgap>[125X [27XProbGenInfoSimple( t );[127X[104X
    [4X[28X[ "M12", 1/3, 2, [ "10A" ], [ 3 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YStatement (b)  can be read off from the permutation characters, and the fact
  that the only classes of maximal subgroups that contain elements of order [22X10[122X
  consist  of  groups of the structures [22XA_6.2^2[122X (two classes) and [22X2 × S_5[122X (one
  class), see [CCN+85, p. 33].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xspos:= Position( OrdersClassRepresentatives( t ), 10 );[127X[104X
    [4X[28X13[128X[104X
    [4X[25Xgap>[125X [27Xprim:= PrimitivePermutationCharacters( t );;[127X[104X
    [4X[25Xgap>[125X [27XList( prim, x -> x{ [ 1, spos ] } );[127X[104X
    [4X[28X[ [ 12, 0 ], [ 12, 0 ], [ 66, 1 ], [ 66, 1 ], [ 144, 0 ], [ 220, 0 ], [128X[104X
    [4X[28X  [ 220, 0 ], [ 396, 1 ], [ 495, 0 ], [ 495, 0 ], [ 1320, 0 ] ][128X[104X
    [4X[25Xgap>[125X [27XMaxes( t );[127X[104X
    [4X[28X[ "M11", "M12M2", "A6.2^2", "M12M4", "L2(11)", "3^2.2.S4", "M12M7", [128X[104X
    [4X[28X  "2xS5", "M8.S4", "4^2:D12", "A4xS3" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  statement (c)  (which  implies  statement (d)),  we  use  the primitive
  permutation representation on [22X12[122X points.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:= MathieuGroup( 12 );[127X[104X
    [4X[28XGroup([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6), (1,12)(2,11)[128X[104X
    [4X[28X  (3,6)(4,8)(5,9)(7,10) ])[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFirst we show that for [22Xs[122X of order [22X10[122X, [22XP(S,s) = 31/99[122X holds.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xapprox:= ApproxP( prim, spos );[127X[104X
    [4X[28X[ 0, 3/11, 1/3, 1/11, 1/132, 13/99, 13/99, 13/396, 1/132, 1/33, 1/33, [128X[104X
    [4X[28X  1/33, 13/396, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27X2B:= g.2^2;[127X[104X
    [4X[28X(3,11)(4,5)(6,10)(7,8)[128X[104X
    [4X[25Xgap>[125X [27XSize( ConjugacyClass( g, 2B ) );[127X[104X
    [4X[28X495[128X[104X
    [4X[25Xgap>[125X [27XResetGlobalRandomNumberGenerators();[127X[104X
    [4X[25Xgap>[125X [27Xrepeat s:= Random( g );[127X[104X
    [4X[25X>[125X [27X   until Order( s ) = 10;[127X[104X
    [4X[25Xgap>[125X [27Xprop:= RatioOfNongenerationTransPermGroup( g, 2B, s );[127X[104X
    [4X[28X31/99[128X[104X
    [4X[25Xgap>[125X [27XFiltered( approx, x -> x >= prop );[127X[104X
    [4X[28X[ 1/3 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNext  we  show  that for [22Xs[122X of order different from [22X10[122X, [22XP(g,s)[122X is larger than
  [22X31/99[122X  for  suitable  [22Xg  ∈ S^×[122X. Except for [22Xs[122X in the class [10X6A[110X (which fixes no
  point  in  the  degree  [22X12[122X representation), it suffices to consider [22Xg[122X in the
  class [10X2B[110X (with four fixed points).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xx:= g.2^2;[127X[104X
    [4X[28X(3,11)(4,5)(6,10)(7,8)[128X[104X
    [4X[25Xgap>[125X [27Xccl:= List( ConjugacyClasses( g ), Representative );;[127X[104X
    [4X[25Xgap>[125X [27XSortParallel( List( ccl, Order ), ccl );[127X[104X
    [4X[25Xgap>[125X [27Xprop:= List( ccl, r -> RatioOfNongenerationTransPermGroup( g, x, r ) );;[127X[104X
    [4X[25Xgap>[125X [27XSortedList( prop );[127X[104X
    [4X[28X[ 7/55, 31/99, 5/9, 5/9, 39/55, 383/495, 383/495, 43/55, 29/33, 1, 1, [128X[104X
    [4X[28X  1, 1, 1, 1 ][128X[104X
    [4X[25Xgap>[125X [27Xbad:= Filtered( prop, x -> x < 31/99 );[127X[104X
    [4X[28X[ 7/55 ][128X[104X
    [4X[25Xgap>[125X [27Xpos:= Position( prop, bad[1] );;[127X[104X
    [4X[25Xgap>[125X [27X[ Order( ccl[ pos ] ), NrMovedPoints( ccl[ pos ] ) ];[127X[104X
    [4X[28X[ 6, 12 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  the  remaining  case,  we choose [22Xg[122X in the class [10X2A[110X (which is fixed point
  free).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xx:= g.3;[127X[104X
    [4X[28X(1,12)(2,11)(3,6)(4,8)(5,9)(7,10)[128X[104X
    [4X[25Xgap>[125X [27Xs:= ccl[ pos ];;[127X[104X
    [4X[25Xgap>[125X [27Xprop:= RatioOfNongenerationTransPermGroup( g, x, s );[127X[104X
    [4X[28X17/33[128X[104X
    [4X[25Xgap>[125X [27Xprop > 31/99;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YStatement (e) has been shown already in Section [14X11.4-2[114X.[133X
  
  
  [1X11.5-11 [33X[0;0Y[22XO_7(3)[122X[101X[1X[133X[101X
  
  [33X[0;0YWe show that [22XS = O_7(3)[122X satisfies the following.[133X
  
  [8X(a)[108X
        [33X[0;6Y[22Xσ(S)  =  199/351[122X, and this value is attained exactly for [22Xσ(S,s)[122X with [22Xs[122X
        of order [22X14[122X.[133X
  
  [8X(b)[108X
        [33X[0;6YFor  [22Xs  ∈  S[122X  of  order  [22X14[122X, [22XMM(S,s)[122X consists of one group of the type
        [22X2.U_4(3).2_2 = Ω^-(6,3).2[122X and two nonconjugate groups of the type [22XS_9[122X.[133X
  
  [8X(c)[108X
        [33X[0;6Y[22XP(S)  =  155/351[122X, and this value is attained exactly for [22XP(S,s)[122X with [22Xs[122X
        of order [22X14[122X.[133X
  
  [8X(d)[108X
        [33X[0;6YThe uniform spread of [22XS[122X is at least three, with [22Xs[122X of order [22X14[122X.[133X
  
  [8X(e)[108X
        [33X[0;6Y[22Xσ^'(Aut(S), s) = 1/3[122X.[133X
  
  [33X[0;0YCurrently  [5XGAP[105X  provides  neither  the table of marks of [22XS[122X nor all character
  tables   of   its  maximal  subgroups.  First  we  compute  those  primitive
  permutation  characters  of  [22XS[122X  that  have the degrees [22X351[122X (point stabilizer
  [22X2.U_4(3).2_2[122X),  [22X364[122X  (point  stabilizer [22X3^5:U_4(2).2[122X), [22X378[122X (point stabilizer
  [22XL_4(3).2_2[122X),  [22X1080[122X  (point  stabilizer  [22XG_2(3)[122X,  two  classes),  [22X1120[122X (point
  stabilizer [22X3^3+3:L_3(3)[122X), [22X3159[122X (point stabilizer [22XS_6(2)[122X, two classes), [22X12636[122X
  (point  stabilizer  [22XS_9[122X,  two  classes),  [22X22113[122X  (point  stabilizer  [22X(2^2  ×
  U_4(2)).2[122X,  which  extends  to [22XD_8 × U_4(2).2[122X in [22XO_7(3).2[122X), and [22X28431[122X (point
  stabilizer [22X2^6:A_7[122X).[133X
  
  [33X[0;0Y(So  we  ignore  the  primitive  permutation characters of the degrees [22X3640[122X,
  [22X265356[122X,  and [22X331695[122X. Note that the orders of the corresponding subgroups are
  not divisible by [22X7[122X.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "O7(3)" );;[127X[104X
    [4X[25Xgap>[125X [27Xsomeprim:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xpi:= PossiblePermutationCharacters([127X[104X
    [4X[25X>[125X [27X            CharacterTable( "2.U4(3).2_2" ), t );;  Length( pi );[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27XAppend( someprim, pi );[127X[104X
    [4X[25Xgap>[125X [27Xpi:= PermChars( t, rec( torso:= [ 364 ] ) );;  Length( pi );[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27XAppend( someprim, pi );[127X[104X
    [4X[25Xgap>[125X [27Xpi:= PossiblePermutationCharacters([127X[104X
    [4X[25X>[125X [27X            CharacterTable( "L4(3).2_2" ), t );;  Length( pi );[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27XAppend( someprim, pi );[127X[104X
    [4X[25Xgap>[125X [27Xpi:= PossiblePermutationCharacters( CharacterTable( "G2(3)" ), t );[127X[104X
    [4X[28X[ Character( CharacterTable( "O7(3)" ),[128X[104X
    [4X[28X  [ 1080, 0, 0, 24, 108, 0, 0, 0, 27, 18, 9, 0, 12, 4, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 12, 0, 0, 0, 0, 0, 3, 6, 0, 3, 2, 2, 2, 0, 0, 0, 3, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 4, 0, 3, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "O7(3)" ),[128X[104X
    [4X[28X  [ 1080, 0, 0, 24, 108, 0, 0, 27, 0, 18, 9, 0, 12, 4, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 12, 0, 0, 0, 0, 3, 0, 0, 6, 3, 2, 2, 2, 0, 0, 3, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 4, 3, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XAppend( someprim, pi );[127X[104X
    [4X[25Xgap>[125X [27Xpi:= PermChars( t, rec( torso:= [ 1120 ] ) );;  Length( pi );[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27XAppend( someprim, pi );[127X[104X
    [4X[25Xgap>[125X [27Xpi:= PossiblePermutationCharacters( CharacterTable( "S6(2)" ), t );[127X[104X
    [4X[28X[ Character( CharacterTable( "O7(3)" ),[128X[104X
    [4X[28X  [ 3159, 567, 135, 39, 0, 81, 0, 0, 27, 27, 0, 15, 3, 3, 7, 4, 0, [128X[104X
    [4X[28X      27, 0, 0, 0, 0, 0, 9, 3, 0, 9, 0, 3, 9, 3, 0, 2, 1, 1, 0, 0, 0, [128X[104X
    [4X[28X      3, 0, 2, 0, 0, 0, 3, 0, 0, 3, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "O7(3)" ),[128X[104X
    [4X[28X  [ 3159, 567, 135, 39, 0, 81, 0, 27, 0, 27, 0, 15, 3, 3, 7, 4, 0, [128X[104X
    [4X[28X      27, 0, 0, 0, 0, 0, 9, 3, 0, 9, 3, 0, 3, 9, 0, 2, 1, 1, 0, 0, 3, [128X[104X
    [4X[28X      0, 0, 2, 0, 0, 0, 3, 0, 3, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XAppend( someprim, pi );[127X[104X
    [4X[25Xgap>[125X [27Xpi:= PossiblePermutationCharacters( CharacterTable( "S9" ), t );[127X[104X
    [4X[28X[ Character( CharacterTable( "O7(3)" ),[128X[104X
    [4X[28X  [ 12636, 1296, 216, 84, 0, 81, 0, 0, 108, 27, 0, 6, 0, 12, 10, 1, [128X[104X
    [4X[28X      0, 27, 0, 0, 0, 0, 0, 9, 3, 0, 9, 0, 12, 9, 3, 0, 1, 0, 2, 0, [128X[104X
    [4X[28X      0, 0, 3, 1, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, [128X[104X
    [4X[28X      1 ] ), Character( CharacterTable( "O7(3)" ),[128X[104X
    [4X[28X  [ 12636, 1296, 216, 84, 0, 81, 0, 108, 0, 27, 0, 6, 0, 12, 10, 1, [128X[104X
    [4X[28X      0, 27, 0, 0, 0, 0, 0, 9, 3, 0, 9, 12, 0, 3, 9, 0, 1, 0, 2, 0, [128X[104X
    [4X[28X      0, 3, 0, 1, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, [128X[104X
    [4X[28X      1 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XAppend( someprim, pi );[127X[104X
    [4X[25Xgap>[125X [27Xt2:= CharacterTable( "O7(3).2" );;[127X[104X
    [4X[25Xgap>[125X [27Xs2:= CharacterTable( "Dihedral", 8 ) * CharacterTable( "U4(2).2" );[127X[104X
    [4X[28XCharacterTable( "Dihedral(8)xU4(2).2" )[128X[104X
    [4X[25Xgap>[125X [27Xpi:= PossiblePermutationCharacters( s2, t2 );;  Length( pi );[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27Xpi:= RestrictedClassFunctions( pi, t );;[127X[104X
    [4X[25Xgap>[125X [27XAppend( someprim, pi );[127X[104X
    [4X[25Xgap>[125X [27Xpi:= PossiblePermutationCharacters([127X[104X
    [4X[25X>[125X [27X            CharacterTable( "2^6:A7" ), t );;  Length( pi );[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27XAppend( someprim, pi );[127X[104X
    [4X[25Xgap>[125X [27XList( someprim, x -> x[1] );[127X[104X
    [4X[28X[ 351, 364, 378, 1080, 1080, 1120, 3159, 3159, 12636, 12636, 22113, [128X[104X
    [4X[28X  28431 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that in the three cases where two possible permutation characters were
  found,  there  are  in  fact  two classes of subgroups that induce different
  permutation  characters.  For  the subgroups of the types [22XG_2(3)[122X and [22XS_6(2)[122X,
  this  is  stated in [CCN+85, p. 109], and for the subgroups of the type [22XS_9[122X,
  this  follows  from  the  fact  that  each  [22XS_9[122X  type subgroup in [22XS[122X contains
  elements  in  exactly one of the classes [10X3D[110X or [10X3E[110X, and these two classes are
  fused by the outer automorphism of [22XS[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xcl:= PositionsProperty( AtlasClassNames( t ),[127X[104X
    [4X[25X>[125X [27X                           x -> x in [ "3D", "3E" ] );[127X[104X
    [4X[28X[ 8, 9 ][128X[104X
    [4X[25Xgap>[125X [27XList( Filtered( someprim, x -> x[1] = 12636 ), pi -> pi{ cl } );[127X[104X
    [4X[28X[ [ 0, 108 ], [ 108, 0 ] ][128X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( t, t2 ){ cl };[127X[104X
    [4X[28X[ 8, 8 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we  compute  the  lower  bounds  for  [22Xσ( S, s^' )[122X that are given by the
  sublist [10Xsomeprim[110X of the primitive permutation characters.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xspos:= Position( OrdersClassRepresentatives( t ), 14 );[127X[104X
    [4X[28X52[128X[104X
    [4X[25Xgap>[125X [27XMaximum( ApproxP( someprim, spos ) );[127X[104X
    [4X[28X199/351[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThis  shows  that  [22Xσ(  S, s ) = 199/351[122X holds. For statement (a), we have to
  show that choosing [22Xs^'[122X from another class than [10X14A[110X yields a larger value for
  [22Xσ( S, s^' )[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xapprox:= List( [ 1 .. NrConjugacyClasses( t ) ],[127X[104X
    [4X[25X>[125X [27X      i -> Maximum( ApproxP( someprim, i ) ) );;[127X[104X
    [4X[25Xgap>[125X [27XPositionsProperty( approx, x -> x <= 199/351 );[127X[104X
    [4X[28X[ 52 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YStatement (b) can be read off from the permutation characters.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpos:= PositionsProperty( someprim, x -> x[ spos ] <> 0 );[127X[104X
    [4X[28X[ 1, 9, 10 ][128X[104X
    [4X[25Xgap>[125X [27XList( someprim{ pos }, x -> x{ [ 1, spos ] } );[127X[104X
    [4X[28X[ [ 351, 1 ], [ 12636, 1 ], [ 12636, 1 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  statement (c),  we  first  compute  [22XP(g,  s)[122X for [22Xg[122X in the class [10X2A[110X, via
  explicit  computations  with the group. For dealing with this case, we first
  construct  a  faithful permutation representation of [22XO_7(3)[122X from the natural
  matrix representation of [22XSO(7,3)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xso73:= SpecialOrthogonalGroup( 7, 3 );;[127X[104X
    [4X[25Xgap>[125X [27Xo73:= DerivedSubgroup( so73 );;[127X[104X
    [4X[25Xgap>[125X [27Xorbs:= OrbitsDomain( o73, Elements( GF(3)^7 ) );;[127X[104X
    [4X[25Xgap>[125X [27XSet( orbs, Length );[127X[104X
    [4X[28X[ 1, 702, 728, 756 ][128X[104X
    [4X[25Xgap>[125X [27Xg:= Action( o73, First( orbs, x -> Length( x ) = 702 ) );;[127X[104X
    [4X[25Xgap>[125X [27XSize( g ) = Size( t );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YA  [10X2A[110X element [22Xg[122X can be found as the [22X7[122X-th power of any element of order [22X14[122X in
  [22XS[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XResetGlobalRandomNumberGenerators();[127X[104X
    [4X[25Xgap>[125X [27Xrepeat s:= Random( g );[127X[104X
    [4X[25X>[125X [27X   until Order( s ) = 14;[127X[104X
    [4X[25Xgap>[125X [27X2A:= s^7;;[127X[104X
    [4X[25Xgap>[125X [27Xbad:= RatioOfNongenerationTransPermGroup( g, 2A, s );[127X[104X
    [4X[28X155/351[128X[104X
    [4X[25Xgap>[125X [27Xbad > 1/3;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xapprox:= ApproxP( someprim, spos );;[127X[104X
    [4X[25Xgap>[125X [27XPositionsProperty( approx, x -> x >= 1/3 );[127X[104X
    [4X[28X[ 2 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThis  shows  that  [22XP(g,s)  =  155/351  >  1/3[122X. Since [22Xσ( g, s ) < 1/3[122X for all
  nonidentity  [22Xg[122X  not  in  the  class  [10X2A[110X,  we  have  [22XP( S, s ) = 155/351[122X. For
  statement (c),  it  remains  to show that [22XP( S, s^' )[122X is larger than [22X155/351[122X
  whenever  [22Xs^'[122X is not of order [22X14[122X. First we compute [22XP( g, s^' )[122X, for [22Xg[122X in the
  class [10X2A[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xconsider:= RepresentativesMaximallyCyclicSubgroups( t );[127X[104X
    [4X[28X[ 18, 19, 25, 26, 27, 30, 31, 32, 34, 35, 38, 39, 41, 42, 43, 44, 45, [128X[104X
    [4X[28X  46, 47, 48, 49, 50, 52, 53, 54, 56, 57, 58 ][128X[104X
    [4X[25Xgap>[125X [27XLength( consider );[127X[104X
    [4X[28X28[128X[104X
    [4X[25Xgap>[125X [27Xconsider:= ClassesPerhapsCorrespondingToTableColumns( g, t, consider );;[127X[104X
    [4X[25Xgap>[125X [27XLength( consider );[127X[104X
    [4X[28X31[128X[104X
    [4X[25Xgap>[125X [27Xconsider:= List( consider, Representative );;[127X[104X
    [4X[25Xgap>[125X [27XSortParallel( List( consider, Order ), consider );[127X[104X
    [4X[25Xgap>[125X [27Xapp2A:= List( consider, c ->[127X[104X
    [4X[25X>[125X [27X      RatioOfNongenerationTransPermGroup( g, 2A, c ) );;[127X[104X
    [4X[25Xgap>[125X [27XSortedList( app2A );[127X[104X
    [4X[28X[ 1/3, 1/3, 155/351, 191/351, 67/117, 23/39, 23/39, 85/117, 10/13, [128X[104X
    [4X[28X  10/13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, [128X[104X
    [4X[28X  1 ][128X[104X
    [4X[25Xgap>[125X [27Xtest:= PositionsProperty( app2A, x -> x <= 155/351 );;[127X[104X
    [4X[25Xgap>[125X [27XList( test, i -> Order( consider[i] ) );[127X[104X
    [4X[28X[ 13, 13, 14 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe see that only for [22Xs^'[122X in one of the two (algebraically conjugate) classes
  of  element  order  [22X13[122X, [22XP( S, s^' )[122X has a chance to be smaller than [22X155/351[122X.
  This  possibility  is now excluded by counting elements in the class [10X3A[110X that
  do not generate [22XS[122X together with [22Xs^'[122X of order [22X13[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XC3A:= First( ConjugacyClasses( g ),[127X[104X
    [4X[25X>[125X [27X              c -> Order( Representative( c ) ) = 3 and Size( c ) = 7280 );;[127X[104X
    [4X[25Xgap>[125X [27Xrepeat ss:= Random( g );[127X[104X
    [4X[25X>[125X [27X   until Order( ss ) = 13;[127X[104X
    [4X[25Xgap>[125X [27Xbad:= RatioOfNongenerationTransPermGroup( g, Representative( C3A ), ss );[127X[104X
    [4X[28X17/35[128X[104X
    [4X[25Xgap>[125X [27Xbad > 155/351;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we  show  statement (d): For each triple [22X(x_1, x_2, x_3)[122X of nonidentity
  elements in [22XS[122X, there is an element [22Xs[122X in the class [10X14A[110X such that [22X⟨ x_i, s ⟩ =
  S[122X  holds  for  [22X1  ≤ i ≤ 3[122X. We can read off from the character-theoretic data
  that  only  those triples have to be checked for which at least two elements
  are  contained  in  the  class  [10X2A[110X, and the third element lies in one of the
  classes [10X2A[110X, [10X2B[110X, [10X3B[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xapprox:= ApproxP( someprim, spos );;[127X[104X
    [4X[25Xgap>[125X [27Xmax:= Maximum( approx{ [ 3 .. Length( approx ) ] } );[127X[104X
    [4X[28X59/351[128X[104X
    [4X[25Xgap>[125X [27X155 + 2*59 < 351;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xthird:= PositionsProperty( approx, x -> 2 * 155/351 + x >= 1 );[127X[104X
    [4X[28X[ 2, 3, 6 ][128X[104X
    [4X[25Xgap>[125X [27XClassNames( t ){ third };[127X[104X
    [4X[28X[ "2a", "2b", "3b" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  can  find  elements  in  the  classes  [10X2B[110X  and [10X3B[110X as powers of arbitrary
  elements of the orders [22X20[122X and [22X15[122X, respectively.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xord20:= PositionsProperty( OrdersClassRepresentatives( t ),[127X[104X
    [4X[25X>[125X [27X                              x -> x = 20 );[127X[104X
    [4X[28X[ 58 ][128X[104X
    [4X[25Xgap>[125X [27XPowerMap( t, 10 ){ ord20 };[127X[104X
    [4X[28X[ 3 ][128X[104X
    [4X[25Xgap>[125X [27Xrepeat x:= Random( g );[127X[104X
    [4X[25X>[125X [27X   until Order( x ) = 20;[127X[104X
    [4X[25Xgap>[125X [27X2B:= x^10;;[127X[104X
    [4X[25Xgap>[125X [27XC2B:= ConjugacyClass( g, 2B );;[127X[104X
    [4X[25Xgap>[125X [27Xord15:= PositionsProperty( OrdersClassRepresentatives( t ),[127X[104X
    [4X[25X>[125X [27X                              x -> x = 15 );[127X[104X
    [4X[28X[ 53 ][128X[104X
    [4X[25Xgap>[125X [27XPowerMap( t, 10 ){ ord15 };[127X[104X
    [4X[28X[ 6 ][128X[104X
    [4X[25Xgap>[125X [27Xrepeat x:= Random( g );[127X[104X
    [4X[25X>[125X [27X   until Order( x ) = 15;[127X[104X
    [4X[25Xgap>[125X [27X3B:= x^5;;[127X[104X
    [4X[25Xgap>[125X [27XC3B:= ConjugacyClass( g, 3B );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  existence  of  [22Xs[122X  can  be  shown  with the random approach described in
  Section [14X11.3-3[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xrepeat s:= Random( g );[127X[104X
    [4X[25X>[125X [27X   until Order( s ) = 14;[127X[104X
    [4X[25Xgap>[125X [27XRandomCheckUniformSpread( g, [ 2A, 2A, 2A ], s, 50 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XRandomCheckUniformSpread( g, [ 2B, 2A, 2A ], s, 50 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XRandomCheckUniformSpread( g, [ 3B, 2A, 2A ], s, 50 );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFinally,  we  show statement (e). Let [22XG = Aut(S) = S.2[122X. By [CCN+85, p. 109],
  [22XMM^'(G,s)[122X  consists  of  the extension of the [22X2.U_4(3).2_1[122X type subgroup. We
  compute the extension of the permutation character.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xprim:= someprim{ [ 1 ] };[127X[104X
    [4X[28X[ Character( CharacterTable( "O7(3)" ),[128X[104X
    [4X[28X  [ 351, 127, 47, 15, 27, 45, 36, 0, 0, 9, 0, 15, 3, 3, 7, 6, 19, 19, [128X[104X
    [4X[28X      10, 11, 12, 8, 3, 5, 3, 6, 1, 0, 0, 3, 3, 0, 1, 1, 1, 6, 3, 0, [128X[104X
    [4X[28X      0, 2, 2, 0, 3, 0, 3, 3, 0, 0, 1, 0, 0, 1, 0, 4, 4, 1, 2, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27Xspos:= Position( AtlasClassNames( t ), "14A" );;[127X[104X
    [4X[25Xgap>[125X [27Xt2:= CharacterTable( "O7(3).2" );;[127X[104X
    [4X[25Xgap>[125X [27Xmap:= InverseMap( GetFusionMap( t, t2 ) );;[127X[104X
    [4X[25Xgap>[125X [27Xtorso:= List( prim, pi -> CompositionMaps( pi, map ) );;[127X[104X
    [4X[25Xgap>[125X [27Xext:= List( torso, x -> PermChars( t2, rec( torso:= x ) ) );[127X[104X
    [4X[28X[ [ Character( CharacterTable( "O7(3).2" ),[128X[104X
    [4X[28X      [ 351, 127, 47, 15, 27, 45, 36, 0, 9, 0, 15, 3, 3, 7, 6, 19, [128X[104X
    [4X[28X          19, 10, 11, 12, 8, 3, 5, 3, 6, 1, 0, 3, 0, 1, 1, 1, 6, 3, [128X[104X
    [4X[28X          0, 2, 2, 0, 3, 0, 3, 3, 0, 1, 0, 0, 1, 0, 4, 1, 2, 0, 117, [128X[104X
    [4X[28X          37, 21, 45, 1, 13, 5, 1, 9, 9, 18, 15, 1, 7, 9, 6, 4, 0, 3, [128X[104X
    [4X[28X          0, 3, 3, 6, 2, 2, 9, 6, 1, 3, 1, 4, 1, 2, 1, 1, 0, 3, 1, 0, [128X[104X
    [4X[28X          0, 0, 0, 1, 1, 0, 0 ] ) ] ][128X[104X
    [4X[25Xgap>[125X [27Xapprox:= ApproxP( Concatenation( ext ),[127X[104X
    [4X[25X>[125X [27X       Position( AtlasClassNames( t2 ), "14A" ) );;[127X[104X
    [4X[25Xgap>[125X [27XMaximum( approx{ Difference([127X[104X
    [4X[25X>[125X [27X     PositionsProperty( OrdersClassRepresentatives( t2 ), IsPrimeInt ),[127X[104X
    [4X[25X>[125X [27X     ClassPositionsOfDerivedSubgroup( t2 ) ) } );[127X[104X
    [4X[28X1/3[128X[104X
  [4X[32X[104X
  
  
  [1X11.5-12 [33X[0;0Y[22XO_8^+(2)[122X[101X[1X[133X[101X
  
  [33X[0;0YWe show that [22XS = O_8^+(2) = Ω^+(8,2)[122X satisfies the following.[133X
  
  [8X(a)[108X
        [33X[0;6Y[22Xσ(S)  =  334/315[122X, and this value is attained exactly for [22Xσ(S,s)[122X with [22Xs[122X
        of order [22X15[122X.[133X
  
  [8X(b)[108X
        [33X[0;6YFor  [22Xs  ∈  S[122X  of  order  [22X15[122X, [22XMM(S,s)[122X consists of one group of the type
        [22XS_6(2)[122X, two conjugate groups of the type [22X2^6:A_8[122X, two conjugate groups
        of  the  type [22XA_9[122X, and one group of each of the types [22X(3 × U_4(2)):2 =
        (3 × Ω^-(6,2)):2[122X and [22X(A_5 × A_5):2^2 = (Ω^-(4,2) × Ω^-(4,2)):2^2[122X.[133X
  
  [8X(c)[108X
        [33X[0;6Y[22XP(S)  = 29/42[122X, and this value is attained exactly for [22XP(S,s)[122X with [22Xs[122X of
        order [22X15[122X.[133X
  
  [8X(d)[108X
        [33X[0;6YLet [22Xx, y ∈ S[122X such that [22Xx, y, x y[122X lie in the unique involution class of
        length  [22X1575[122X  of  [22XS[122X.  (This  is  the class [10X2A[110X.) Then each element in [22XS[122X
        together with one of [22Xx[122X, [22Xy[122X, [22Xx y[122X generates a proper subgroup of [22XS[122X.[133X
  
  [8X(e)[108X
        [33X[0;6YBoth  the spread and the uniform spread of [22XS[122X is exactly two, with [22Xs[122X of
        order [22X15[122X.[133X
  
  [8X(f)[108X
        [33X[0;6YFor  each choice of [22Xs ∈ S[122X, there is an extension [22XS.2[122X such that for any
        element [22Xg[122X in the (outer) class [10X2F[110X, [22X⟨ s, g ⟩[122X does not contain [22XS[122X.[133X
  
  [8X(g)[108X
        [33X[0;6YFor  an  element  [22Xs[122X  of  order  [22X15[122X  in [22XS[122X, either [22XS[122X is the only maximal
        subgroup  of [22XS.2[122X that contains [22Xs[122X, or the maximal subgroups of [22XS.2[122X that
        contain  [22Xs[122X  are  [22XS[122X  and  the  extensions  of  the  subgroups listed in
        statement (b);  these  groups  have the structures [22XS_6(2) × 2[122X, [22X2^6:S_8[122X
        (twice), [22XS_9[122X (twice), [22XS_3 × U_4(2).2[122X, and [22XS_5 ≀ 2[122X.[133X
  
  [8X(h)[108X
        [33X[0;6YFor  [22Xs  ∈  S[122X of order [22X15[122X and arbitrary [22Xg ∈ S.3 ∖ S[122X, we have [22X⟨ s, g ⟩ =
        S.3[122X.[133X
  
  [8X(i)[108X
        [33X[0;6YIf  [22Xx[122X, [22Xy[122X are nonidentity elements in [22XAut(S)[122X then there is an element [22Xs[122X
        of order [22X15[122X in [22XS[122X such that [22XS ⊆ ⟨ x, s ⟩ ∩ ⟨ y, s ⟩[122X.[133X
  
  [33X[0;0YStatement (a)   follows   from   inspection  of  the  primitive  permutation
  characters, cf. Section [14X11.4-3[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "O8+(2)" );;[127X[104X
    [4X[25Xgap>[125X [27XProbGenInfoSimple( t );[127X[104X
    [4X[28X[ "O8+(2)", 334/315, 0, [ "15A", "15B", "15C" ], [ 7, 7, 7 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YStatement (b)  can be read off from the permutation characters, and the fact
  that the only classes of maximal subgroups that contain elements of order [22X15[122X
  consist of groups of the structures as claimed, see [CCN+85, p. 85].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xprim:= PrimitivePermutationCharacters( t );;[127X[104X
    [4X[25Xgap>[125X [27Xspos:= Position( OrdersClassRepresentatives( t ), 15 );;[127X[104X
    [4X[25Xgap>[125X [27XList( Filtered( prim, x -> x[ spos ] <> 0 ), l -> l{ [ 1, spos ] } );[127X[104X
    [4X[28X[ [ 120, 1 ], [ 135, 2 ], [ 960, 2 ], [ 1120, 1 ], [ 12096, 1 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor the remaining statements, we take a primitive permutation representation
  on  [22X120[122X  points,  and  assume  that the permutation character is [10X1a+35a+84a[110X.
  (See [CCN+85,  p. 85],  note  that the three classes of maximal subgroups of
  index [22X120[122X in [22XS[122X are conjugate under triality.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmatgroup:= DerivedSubgroup( GeneralOrthogonalGroup( 1, 8, 2 ) );;[127X[104X
    [4X[25Xgap>[125X [27Xpoints:= NormedRowVectors( GF(2)^8 );;[127X[104X
    [4X[25Xgap>[125X [27Xorbs:= OrbitsDomain( matgroup, points );;[127X[104X
    [4X[25Xgap>[125X [27XList( orbs, Length );[127X[104X
    [4X[28X[ 135, 120 ][128X[104X
    [4X[25Xgap>[125X [27Xg:= Action( matgroup, orbs[2] );;[127X[104X
    [4X[25Xgap>[125X [27XSize( g );[127X[104X
    [4X[28X174182400[128X[104X
    [4X[25Xgap>[125X [27Xpi:= Sum( Irr( t ){ [ 1, 3, 7 ] } );[127X[104X
    [4X[28XCharacter( CharacterTable( "O8+(2)" ),[128X[104X
    [4X[28X [ 120, 24, 32, 0, 0, 8, 36, 0, 0, 3, 6, 12, 4, 8, 0, 0, 0, 10, 0, 0, [128X[104X
    [4X[28X  12, 0, 0, 8, 0, 0, 3, 6, 0, 0, 2, 0, 0, 2, 1, 2, 2, 3, 0, 0, 2, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 3, 2, 0, 0, 1, 0, 0 ] )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn order to show statement (c), we first observe that for [22Xs[122X in the class [10X15A[110X
  and  [22Xg[122X [13Xnot[113X in one of the classes [10X2A[110X, [10X2B[110X, [10X3A[110X, [22Xσ(g,s) < 1/3[122X holds, and for the
  exceptional three classes, we have [22Xσ(g,s) > 1/2[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xapprox:= ApproxP( prim, spos );;[127X[104X
    [4X[25Xgap>[125X [27Xtestpos:= PositionsProperty( approx, x -> x >= 1/3 );[127X[104X
    [4X[28X[ 2, 3, 7 ][128X[104X
    [4X[25Xgap>[125X [27XAtlasClassNames( t ){ testpos };[127X[104X
    [4X[28X[ "2A", "2B", "3A" ][128X[104X
    [4X[25Xgap>[125X [27Xapprox{ testpos };[127X[104X
    [4X[28X[ 254/315, 334/315, 1093/1120 ][128X[104X
    [4X[25Xgap>[125X [27XForAll( approx{ testpos }, x -> x > 1/2 );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow we compute the values [22XP(g,s)[122X, for [22Xs[122X in the class [10X15A[110X and [22Xg[122X in one of the
  classes [10X2A[110X, [10X2B[110X, [10X3A[110X.[133X
  
  [33X[0;0YBy our choice of the character of the permutation representation we use, the
  class  [10X15A[110X  is  determined  as the unique class of element order [22X15[122X with one
  fixed  point.  (Note  that  the  three  classes of element order [22X15[122X in [22XS[122X are
  conjugate  under triality.) A [10X2A[110X element can be found as the fourth power of
  any element of order [22X8[122X in [22XS[122X, a [10X3A[110X element can be found as the fifth power of
  a  [10X15A[110X  element,  and  a  [10X2B[110X  element  can be found as the sixth power of an
  element of order [22X12[122X, with [22X32[122X fixed points.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XResetGlobalRandomNumberGenerators();[127X[104X
    [4X[25Xgap>[125X [27Xrepeat s:= Random( g );[127X[104X
    [4X[25X>[125X [27X   until Order( s ) = 15 and NrMovedPoints( g ) = 1 + NrMovedPoints( s );[127X[104X
    [4X[25Xgap>[125X [27X3A:= s^5;;[127X[104X
    [4X[25Xgap>[125X [27Xrepeat x:= Random( g ); until Order( x ) = 8;[127X[104X
    [4X[25Xgap>[125X [27X2A:= x^4;;[127X[104X
    [4X[25Xgap>[125X [27Xrepeat x:= Random( g ); until Order( x ) = 12 and[127X[104X
    [4X[25X>[125X [27X     NrMovedPoints( g ) = 32 + NrMovedPoints( x^6 );[127X[104X
    [4X[25Xgap>[125X [27X2B:= x^6;;[127X[104X
    [4X[25Xgap>[125X [27Xprop15A:= List( [ 2A, 2B, 3A ],[127X[104X
    [4X[25X>[125X [27X                   x -> RatioOfNongenerationTransPermGroup( g, x, s ) );[127X[104X
    [4X[28X[ 23/35, 29/42, 149/224 ][128X[104X
    [4X[25Xgap>[125X [27XMaximum( prop15A );[127X[104X
    [4X[28X29/42[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThis  means  that for [22Xs[122X in the class [10X15A[110X, we have [22XP( S, s ) = 29/42[122X, and the
  same holds for all [22Xs[122X of order [22X15[122X since the three classes of element order [22X15[122X
  are conjugate under triality. Now we show that for [22Xs[122X of order different from
  [22X15[122X,  the  value [22XP(g,s)[122X is larger than [22X29/42[122X, for [22Xg[122X in one of the classes [10X2A[110X,
  [10X2B[110X, [10X3A[110X, or their images under triality. This implies statement (c).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtest:= List( [ 2A, 2B, 3A ], x -> ConjugacyClass( g, x ) );;[127X[104X
    [4X[25Xgap>[125X [27Xccl:= ConjugacyClasses( g );;[127X[104X
    [4X[25Xgap>[125X [27Xconsider:= Filtered( ccl, c -> Size( c ) in List( test, Size ) );;[127X[104X
    [4X[25Xgap>[125X [27XLength( consider );[127X[104X
    [4X[28X7[128X[104X
    [4X[25Xgap>[125X [27Xfilt:= Filtered( ccl, c -> ForAll( consider, cc ->[127X[104X
    [4X[25X>[125X [27X      RatioOfNongenerationTransPermGroup( g, Representative( cc ),[127X[104X
    [4X[25X>[125X [27X          Representative( c ) ) <= 29/42 ) );;[127X[104X
    [4X[25Xgap>[125X [27XLength( filt );[127X[104X
    [4X[28X3[128X[104X
    [4X[25Xgap>[125X [27XList( filt, c -> Order( Representative( c ) ) );[127X[104X
    [4X[28X[ 15, 15, 15 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow we show statement (d). First we observe that all those Klein four groups
  in  [22XS[122X  whose  involutions  lie in the class [10X2A[110X are conjugate in [22XS[122X. Note that
  this  is  the  unique  class  of length [22X1575[122X in [22XS[122X, and also the unique class
  whose   elements  have  [22X24[122X  fixed  points  in  the  degree  [22X120[122X  permutation
  representation.[133X
  
  [33X[0;0YFor  that,  we  use  the  character  table  of [22XS[122X to read off that [22XS[122X contains
  exactly  [22X14175[122X  such  subgroups,  and  we  use the group to compute one such
  subgroup and its normalizer of index [22X14175[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSizesConjugacyClasses( t );[127X[104X
    [4X[28X[ 1, 1575, 3780, 3780, 3780, 56700, 2240, 2240, 2240, 89600, 268800, [128X[104X
    [4X[28X  37800, 340200, 907200, 907200, 907200, 2721600, 580608, 580608, [128X[104X
    [4X[28X  580608, 100800, 100800, 100800, 604800, 604800, 604800, 806400, [128X[104X
    [4X[28X  806400, 806400, 806400, 2419200, 2419200, 2419200, 7257600, [128X[104X
    [4X[28X  24883200, 5443200, 5443200, 6451200, 6451200, 6451200, 8709120, [128X[104X
    [4X[28X  8709120, 8709120, 1209600, 1209600, 1209600, 4838400, 7257600, [128X[104X
    [4X[28X  7257600, 7257600, 11612160, 11612160, 11612160 ][128X[104X
    [4X[25Xgap>[125X [27XNrPolyhedralSubgroups( t, 2, 2, 2 );[127X[104X
    [4X[28Xrec( number := 14175, type := "V4" )[128X[104X
    [4X[25Xgap>[125X [27Xrepeat x:= Random( g );[127X[104X
    [4X[25X>[125X [27X   until     Order( x ) mod 2 = 0[127X[104X
    [4X[25X>[125X [27X         and NrMovedPoints( x^( Order(x)/2 ) ) = 120 - 24;[127X[104X
    [4X[25Xgap>[125X [27Xx:= x^( Order(x)/2 );;[127X[104X
    [4X[25Xgap>[125X [27Xrepeat y:= x^Random( g );[127X[104X
    [4X[25X>[125X [27X   until NrMovedPoints( x*y ) = 120 - 24;[127X[104X
    [4X[25Xgap>[125X [27Xv4:= SubgroupNC( g, [ x, y ] );;[127X[104X
    [4X[25Xgap>[125X [27Xn:= Normalizer( g, v4 );;[127X[104X
    [4X[25Xgap>[125X [27XIndex( g, n );[127X[104X
    [4X[28X14175[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe verify that the triple has the required property.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmaxorder:= RepresentativesMaximallyCyclicSubgroups( t );;[127X[104X
    [4X[25Xgap>[125X [27Xmaxorderreps:= List( ClassesPerhapsCorrespondingToTableColumns( g, t,[127X[104X
    [4X[25X>[125X [27X       maxorder ), Representative );;[127X[104X
    [4X[25Xgap>[125X [27XLength( maxorderreps );[127X[104X
    [4X[28X28[128X[104X
    [4X[25Xgap>[125X [27XCommonGeneratorWithGivenElements( g, maxorderreps, [ x, y, x*y ] );[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  the  simple  group [22XS[122X, it remains to show statement (e). We want to show
  that  for  any  choice  of  two  nonidentity elements [22Xx[122X, [22Xy[122X in [22XS[122X, there is an
  element  [22Xs[122X in the class [10X15A[110X such that [22X⟨ s, x ⟩ = ⟨ s, y ⟩ = S[122X holds. Only [22Xx[122X,
  [22Xy[122X  in  the  classes  given  by  the  list [10Xtestpos[110X must be considered, by the
  estimates [22Xσ(g,s)[122X.[133X
  
  [33X[0;0YWe  replace  the  values  [22Xσ(g,s)[122X by the exact values [22XP(g,s)[122X, for [22Xg[122X in one of
  these  three classes. Each of the three classes is determined by its element
  order and its number of fixed points.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xreps:= List( ccl, Representative );;[127X[104X
    [4X[25Xgap>[125X [27Xbading:= List( testpos, i -> Filtered( reps,[127X[104X
    [4X[25X>[125X [27X       r -> Order( r ) = OrdersClassRepresentatives( t )[i] and[127X[104X
    [4X[25X>[125X [27X            NrMovedPoints( r ) = 120 - pi[i] ) );;[127X[104X
    [4X[25Xgap>[125X [27XList( bading, Length );[127X[104X
    [4X[28X[ 1, 1, 1 ][128X[104X
    [4X[25Xgap>[125X [27Xbading:= List( bading, x -> x[1] );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  each  pair  [22X(C_1,  C_2)[122X of classes represented by this list, we have to
  show  that  for  any  choice  of elements [22Xx ∈ C_1[122X, [22Xy ∈ C_2[122X there is [22Xs[122X in the
  class  [10X15A[110X  such  that  [22X⟨ s, x ⟩ = ⟨ s, y ⟩ = S[122X holds. This is done with the
  random approach that is described in Section [14X11.3-3[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfor pair in UnorderedTuples( bading, 2 ) do[127X[104X
    [4X[25X>[125X [27X     test:= RandomCheckUniformSpread( g, pair, s, 80 );[127X[104X
    [4X[25X>[125X [27X     if test <> true then[127X[104X
    [4X[25X>[125X [27X       Error( test );[127X[104X
    [4X[25X>[125X [27X     fi;[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YWe get no error message, so statement (e) holds.[133X
  
  [33X[0;0YNow  we  turn  to  the  automorphic  extensions  of  [22XS[122X.  First  we compute a
  permutation  representation of [22XSO^+(8,2) ≅ S.2[122X and an element [22Xg[122X in the class
  [10X2F[110X, which is the unique conjugacy class of size [22X120[122X in [22XS.2[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmatgrp:= SO(1,8,2);;[127X[104X
    [4X[25Xgap>[125X [27Xg2:= Image( IsomorphismPermGroup( matgrp ) );;[127X[104X
    [4X[25Xgap>[125X [27XIsTransitive( g2, MovedPoints( g2 ) );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xrepeat x:= Random( g2 ); until Order( x ) = 14;[127X[104X
    [4X[25Xgap>[125X [27X2F:= x^7;;[127X[104X
    [4X[25Xgap>[125X [27XSize( ConjugacyClass( g2, 2F ) );[127X[104X
    [4X[28X120[128X[104X
  [4X[32X[104X
  
  [33X[0;0YOnly  for [22Xs[122X in six conjugacy classes of [22XS[122X, there is a nonzero probability to
  have [22XS.2 = ⟨ g, s ⟩[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xder:= DerivedSubgroup( g2 );;[127X[104X
    [4X[25Xgap>[125X [27Xcclreps:= List( ConjugacyClasses( der ), Representative );;[127X[104X
    [4X[25Xgap>[125X [27Xnongen:= List( cclreps,[127X[104X
    [4X[25X>[125X [27X              x -> RatioOfNongenerationTransPermGroup( g2, 2F, x ) );;[127X[104X
    [4X[25Xgap>[125X [27Xgoodpos:= PositionsProperty( nongen, x -> x < 1 );;[127X[104X
    [4X[25Xgap>[125X [27Xinvariants:= List( goodpos, i -> [ Order( cclreps[i] ),[127X[104X
    [4X[25X>[125X [27X     Size( Centralizer( g2, cclreps[i] ) ), nongen[i] ] );;[127X[104X
    [4X[25Xgap>[125X [27XSortedList( invariants );[127X[104X
    [4X[28X[ [ 10, 20, 1/3 ], [ 10, 20, 1/3 ], [ 12, 24, 2/5 ], [ 12, 24, 2/5 ], [128X[104X
    [4X[28X  [ 15, 15, 0 ], [ 15, 15, 0 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[22XS[122X  contains  three  classes of element order [22X10[122X, which are conjugate in [22XS.3[122X.
  For  a  fixed extension of the type [22XS.2[122X, the element [22Xs[122X can be chosen only in
  two  of  these three classes, which means that there is another group of the
  type [22XS.2[122X (more precisely, another subgroup of index three in [22XS.S_3[122X) in which
  this choice of [22Xs[122X is not suitable –note that the general aim is to find [22Xs ∈ S[122X
  uniformly for all automorphic extensions of [22XS[122X. Analogous statements hold for
  the other possibilities for [22Xs[122X, so statement (f) follows.[133X
  
  [33X[0;0YStatement (g) follows from the list of maximal subgroups in [CCN+85, p. 85].[133X
  
  [33X[0;0YStatement (h)  follows  from the fact that [22XS[122X is the only maximal subgroup of
  [22XS.3[122X  that  contains  elements  of order [22X15[122X, according to the list of maximal
  subgroups  in [CCN+85,  p. 85].  Alternatively,  if we do not want to assume
  this  information, we can use explicit computations, as follows. All we have
  to check is that any element in the classes [10X3F[110X and [10X3G[110X generates [22XS.3[122X together
  with a fixed element of order [22X15[122X in [22XS[122X.[133X
  
  [33X[0;0YWe  compute a permutation representation of [22XS.3[122X as the derived subgroup of a
  subgroup  of  the type [22XS.S_3[122X inside the sporadic simple Fischer group [22XFi_22[122X;
  these  subgroups  lie  in  the  fourth  class of maximal subgroups of [22XFi_22[122X,
  see [CCN+85,  p. 163].  An  element in the class [10X3F[110X of [22XS.3[122X can be found as a
  power of an order [22X21[122X element, and an element in the class [10X3G[110X can be found as
  the fourth power of a [10X12P[110X element.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xaut:= Group( AtlasGenerators( "Fi22", 1, 4 ).generators );;[127X[104X
    [4X[25Xgap>[125X [27XSize( aut ) = 6 * Size( t );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xg3:= DerivedSubgroup( aut );;[127X[104X
    [4X[25Xgap>[125X [27Xorbs:= OrbitsDomain( g3, MovedPoints( g3 ) );;[127X[104X
    [4X[25Xgap>[125X [27XList( orbs, Length );[127X[104X
    [4X[28X[ 3150, 360 ][128X[104X
    [4X[25Xgap>[125X [27Xg3:= Action( g3, orbs[2] );;[127X[104X
    [4X[25Xgap>[125X [27Xrepeat s:= Random( g3 ); until Order( s ) = 15;[127X[104X
    [4X[25Xgap>[125X [27Xrepeat x:= Random( g3 ); until Order( x ) = 21;[127X[104X
    [4X[25Xgap>[125X [27X3F:= x^7;;[127X[104X
    [4X[25Xgap>[125X [27XRatioOfNongenerationTransPermGroup( g3, 3F, s );[127X[104X
    [4X[28X0[128X[104X
    [4X[25Xgap>[125X [27Xrepeat x:= Random( g3 );[127X[104X
    [4X[25X>[125X [27X   until Order( x ) = 12 and Size( Centralizer( g3, x^4 ) ) = 648;[127X[104X
    [4X[25Xgap>[125X [27X3G:= x^4;;[127X[104X
    [4X[25Xgap>[125X [27XRatioOfNongenerationTransPermGroup( g3, 3G, s );[127X[104X
    [4X[28X0[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFinally,  consider  statement (i).  It  implies  that [BGK08, Corollary 1.5]
  holds  for  [22XΩ^+(8,2)[122X, with [22Xs[122X of order [22X15[122X. Note that by part (f), [22Xs[122X [13Xcannot be
  chosen  in  a  prescribed  conjugacy  class[113X  of [22XS[122X that is independent of the
  elements [22Xx[122X, [22Xy[122X.[133X
  
  [33X[0;0YIf  [22Xx[122X  and  [22Xy[122X  lie  in  [22XS[122X  then  statement (i) follows from part (e), and by
  part (g),  the case that [22Xx[122X or [22Xy[122X lie in [22XS.3 ∖ S[122X is also not a problem. We now
  show that also [22Xx[122X or [22Xy[122X in [22XS.2 ∖ S[122X is not a problem. Here we have to deal with
  the  cases  that [22Xx[122X and [22Xy[122X lie in the same subgroup of index [22X3[122X in [22XAut(S)[122X or in
  different  such subgroups. Actually we show that for each index [22X3[122X subgroup [22XH
  =  S.2  <  Aut(S)[122X,  we can choose [22Xs[122X from two of the three classes of element
  order [22X15[122X in [22XS[122X such that [22XS[122X is the only maximal subgroup of [22XH[122X that contains [22Xs[122X,
  and thus [22X⟨ x, s ⟩[122X contains [22XH[122X, for any choice of [22Xx ∈ H ∖ S[122X.[133X
  
  [33X[0;0YFor  that,  we note that no novelty in [22XS.2[122X contains elements of order [22X15[122X, so
  all maximal subgroups of [22XS.2[122X that contain such elements –besides [22XS[122X– have one
  of  the  indices [22X120, 135, 960, 1120[122X, or [22X12096[122X, and point stabilizers of the
  types  [22XS_6(2)  × 2[122X, [22X2^6:S_8[122X, [22XS_9[122X, [22XS_3 × U_4(2):2[122X, or [22XS_5 ≀ 2[122X. We compute the
  corresponding permutation characters.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt2:= CharacterTable( "O8+(2).2" );;[127X[104X
    [4X[25Xgap>[125X [27Xs:= CharacterTable( "S6(2)" ) * CharacterTable( "Cyclic", 2 );;[127X[104X
    [4X[25Xgap>[125X [27Xpi:= PossiblePermutationCharacters( s, t2 );;[127X[104X
    [4X[25Xgap>[125X [27Xprim:= pi;;[127X[104X
    [4X[25Xgap>[125X [27Xpi:= PermChars( t2, rec( torso:= [ 135 ] ) );;[127X[104X
    [4X[25Xgap>[125X [27XAppend( prim, pi );[127X[104X
    [4X[25Xgap>[125X [27Xpi:= PossiblePermutationCharacters( CharacterTable( "A9.2" ), t2 );;[127X[104X
    [4X[25Xgap>[125X [27XAppend( prim, pi );[127X[104X
    [4X[25Xgap>[125X [27Xs:= CharacterTable( "Dihedral(6)" ) * CharacterTable( "U4(2).2" );;[127X[104X
    [4X[25Xgap>[125X [27Xpi:= PossiblePermutationCharacters( s, t2 );;[127X[104X
    [4X[25Xgap>[125X [27XAppend( prim, pi );[127X[104X
    [4X[25Xgap>[125X [27Xs:= CharacterTableWreathSymmetric( CharacterTable( "S5" ), 2 );;[127X[104X
    [4X[25Xgap>[125X [27Xpi:= PossiblePermutationCharacters( s, t2 );;[127X[104X
    [4X[25Xgap>[125X [27XAppend( prim, pi );[127X[104X
    [4X[25Xgap>[125X [27XLength( prim );[127X[104X
    [4X[28X5[128X[104X
    [4X[25Xgap>[125X [27Xord15:= PositionsProperty( OrdersClassRepresentatives( t2 ),[127X[104X
    [4X[25X>[125X [27X                              x -> x = 15 );[127X[104X
    [4X[28X[ 39, 40 ][128X[104X
    [4X[25Xgap>[125X [27XList( prim, pi -> pi{ ord15 } );[127X[104X
    [4X[28X[ [ 1, 0 ], [ 2, 0 ], [ 2, 0 ], [ 1, 0 ], [ 1, 0 ] ][128X[104X
    [4X[25Xgap>[125X [27XList( ord15, i -> Maximum( ApproxP( prim, i ) ) );[127X[104X
    [4X[28X[ 307/120, 0 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YHere it is appropriate to clean the workspace again.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCleanWorkspace();[127X[104X
  [4X[32X[104X
  
  
  [1X11.5-13 [33X[0;0Y[22XO_8^+(3)[122X[101X[1X[133X[101X
  
  [33X[0;0YWe show that [22XS = O_8^+(3)[122X satisfies the following.[133X
  
  [8X(a)[108X
        [33X[0;6Y[22Xσ(S)  = 863/1820[122X, and this value is attained exactly for [22Xσ(S,s)[122X with [22Xs[122X
        of order [22X20[122X.[133X
  
  [8X(b)[108X
        [33X[0;6YFor  [22Xs ∈ S[122X of order [22X20[122X, [22XMM(S,s)[122X consists of two nonconjugate groups of
        the  type  [22XO_7(3)  =  Ω(7,3)[122X,  two  conjugate  subgroups  of  the type
        [22X3^6:L_4(3)[122X,  two  nonconjugate subgroups of the type [22X(A_4 × U_4(2)):2[122X,
        and  one  subgroup  of each of the types [22X2.U_4(3).(2^2)_122[122X and [22X(A_6 ×
        A_6):2^2[122X.[133X
  
  [8X(c)[108X
        [33X[0;6Y[22XP(S)  =  194/455[122X, and this value is attained exactly for [22XP(S,s)[122X with [22Xs[122X
        of order [22X20[122X.[133X
  
  [8X(d)[108X
        [33X[0;6YThe uniform spread of [22XS[122X is at least three, with [22Xs[122X of order [22X20[122X.[133X
  
  [8X(e)[108X
        [33X[0;6YThe  preimage of [22Xs[122X in the matrix group [22X2.S = Ω^+(8,3)[122X can be chosen of
        order  [22X40[122X, and then the maximal subgroups of [22X2.S[122X containing [22Xs[122X have the
        structures  [22X2.O_7(3)[122X, [22X3^6:2.L_4(3)[122X, [22X4.U_4(3).2^2 = SU(4,3).2^2[122X, [22X2.(A_4
        ×  U_4(2)).2  =  2.(PSp(2,3)  ⊗  PSp(4,3)).2[122X,  and [22X2.(A_6 × A_6):2^2 =
        2.(Ω^-(4,3) × Ω^-(4,3)):2^2[122X, respectively.[133X
  
  [8X(f)[108X
        [33X[0;6YFor  [22Xs  ∈ S[122X of order [22X20[122X, we have [22XP^'(S.2_1, s) ∈ { 83/567, 574/1215 }[122X,
        [22XP^'(S.2_2,  s) ∈ { 0, 1 }[122X (depending on the choice of [22Xs[122X), and [22Xσ^'(S.3,
        s) = 0[122X.[133X
  
        [33X[0;6YFurthermore,  for  any  choice of [22Xs^' ∈ S[122X, we have [22Xσ^'(S.2_2, s^') = 1[122X
        for  some  group  [22XS.2_2[122X. However, if it is allowed to choose [22Xs[122X from an
        [22XAut(S)[122X-class  of  elements  of order [22X20[122X (and not from a fixed [22XS[122X-class)
        then we can achieve [22Xσ(g,s) = 0[122X for any given [22Xg ∈ S.2_2 ∖ S[122X.[133X
  
  [8X(g)[108X
        [33X[0;6YThe maximal subgroups of [22XS.2_1[122X that contain an element of order [22X20[122X are
        either  [22XS[122X  and the extensions of the subgroups listed in statement (b)
        or    they    are    [22XS[122X    and    [22XL_4(3).2^2[122X,   [22X3^6:L_4(3).2[122X   (twice),
        [22X2.U_4(3).(2^2)_122.2[122X, and [22X(A_6 × A_6):2^2.2[122X.[133X
  
        [33X[0;6YIn  the  former case, the groups have the structures [22XO_7(3):2[122X (twice),
        [22X3^6:(L_4(3)    ×    2)[122X    (twice),    [22XS_4    ×    U_4(2).2[122X    (twice),
        [22X2.U_4(3).(2^2)_122.2[122X, and [22X(A_6 × A_6):2^2 × 2[122X.[133X
  
  [33X[0;0YStatement (a)   follows   from   inspection  of  the  primitive  permutation
  characters.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "O8+(3)" );;[127X[104X
    [4X[25Xgap>[125X [27XProbGenInfoSimple( t );[127X[104X
    [4X[28X[ "O8+(3)", 863/1820, 2, [ "20A", "20B", "20C" ], [ 8, 8, 8 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YAlso  statement (b)  follows  from the information provided by the character
  table of [22XS[122X (cf. [CCN+85, p. 140]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xprim:= PrimitivePermutationCharacters( t );;[127X[104X
    [4X[25Xgap>[125X [27Xord:= OrdersClassRepresentatives( t );;[127X[104X
    [4X[25Xgap>[125X [27Xspos:= Position( ord, 20 );;[127X[104X
    [4X[25Xgap>[125X [27Xfilt:= PositionsProperty( prim, x -> x[ spos ] <> 0 );[127X[104X
    [4X[28X[ 1, 2, 7, 15, 18, 19, 24 ][128X[104X
    [4X[25Xgap>[125X [27XMaxes( t ){ filt };[127X[104X
    [4X[28X[ "O7(3)", "O8+(3)M2", "3^6:L4(3)", "2.U4(3).(2^2)_{122}", [128X[104X
    [4X[28X  "(A4xU4(2)):2", "O8+(3)M19", "(A6xA6):2^2" ][128X[104X
    [4X[25Xgap>[125X [27Xprim{ filt }{ [ 1, spos ] };[127X[104X
    [4X[28X[ [ 1080, 1 ], [ 1080, 1 ], [ 1120, 2 ], [ 189540, 1 ], [128X[104X
    [4X[28X  [ 7960680, 1 ], [ 7960680, 1 ], [ 9552816, 1 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  statement (c),  we  first  show that [22XP(S,s) = 194/455[122X holds. Since this
  value  is  larger  than  [22X1/3[122X,  we have to inspect only those classes [22Xg^S[122X for
  which [22Xσ(g,s) ≥ 1/3[122X holds,[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xord:= OrdersClassRepresentatives( t );;[127X[104X
    [4X[25Xgap>[125X [27Xord20:= PositionsProperty( ord, x -> x = 20 );;[127X[104X
    [4X[25Xgap>[125X [27Xcand:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xfor i in ord20 do[127X[104X
    [4X[25X>[125X [27X     approx:= ApproxP( prim, i );[127X[104X
    [4X[25X>[125X [27X     Add( cand, PositionsProperty( approx, x -> x >= 1/3 ) );[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27Xcand;[127X[104X
    [4X[28X[ [ 2, 6, 7, 10 ], [ 3, 6, 8, 11 ], [ 4, 6, 9, 12 ] ][128X[104X
    [4X[25Xgap>[125X [27XAtlasClassNames( t ){ cand[1] };[127X[104X
    [4X[28X[ "2A", "3A", "3B", "3E" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe three possibilities form one orbit under the outer automorphism group of
  [22XS[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt3:= CharacterTable( "O8+(3).3" );;[127X[104X
    [4X[25Xgap>[125X [27Xtfust3:= GetFusionMap( t, t3 );;[127X[104X
    [4X[25Xgap>[125X [27XList( cand, x -> tfust3{ x } );[127X[104X
    [4X[28X[ [ 2, 4, 5, 6 ], [ 2, 4, 5, 6 ], [ 2, 4, 5, 6 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YBy  symmetry,  we may consider only the first possibility, and assume that [22Xs[122X
  is in the class [10X20A[110X.[133X
  
  [33X[0;0YWe  work  with  a permutation representation of degree [22X1080[122X, and assume that
  the  permutation  character  is  [10X1a+260a+819a[110X.  (Note  that  all permutation
  characters of [22XS[122X of degree [22X1080[122X are conjugate under [22XAut(S)[122X.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:= Action( SO(1,8,3), NormedRowVectors( GF(3)^8 ), OnLines );;[127X[104X
    [4X[25Xgap>[125X [27XSize( g );[127X[104X
    [4X[28X9904359628800[128X[104X
    [4X[25Xgap>[125X [27Xg:= DerivedSubgroup( g );;  Size( g );[127X[104X
    [4X[28X4952179814400[128X[104X
    [4X[25Xgap>[125X [27Xorbs:= OrbitsDomain( g, MovedPoints( g ) );;[127X[104X
    [4X[25Xgap>[125X [27XList( orbs, Length );[127X[104X
    [4X[28X[ 1080, 1080, 1120 ][128X[104X
    [4X[25Xgap>[125X [27Xg:= Action( g, orbs[1] );;[127X[104X
    [4X[25Xgap>[125X [27XPositionProperty( Irr( t ), chi -> chi[1] = 819 );[127X[104X
    [4X[28X9[128X[104X
    [4X[25Xgap>[125X [27Xpermchar:= Sum( Irr( t ){ [ 1, 2, 9 ] } );[127X[104X
    [4X[28XCharacter( CharacterTable( "O8+(3)" ),[128X[104X
    [4X[28X [ 1080, 128, 0, 0, 24, 108, 135, 0, 0, 108, 0, 0, 27, 27, 0, 0, 18, [128X[104X
    [4X[28X  9, 12, 16, 0, 0, 4, 15, 0, 0, 20, 0, 0, 12, 11, 0, 0, 20, 0, 0, 15, [128X[104X
    [4X[28X  0, 0, 12, 0, 0, 2, 0, 0, 3, 3, 0, 0, 6, 6, 0, 0, 3, 2, 2, 2, 18, 0, [128X[104X
    [4X[28X  0, 9, 0, 0, 0, 0, 0, 0, 3, 3, 0, 0, 3, 0, 0, 12, 0, 0, 3, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 4, 3, 3, 0, 0, 1, 0, 0, 4, 0, 0, 1, 1, 2, 0, 0, 0, 0, 0, 3, [128X[104X
    [4X[28X  0, 0, 2, 0, 0, 5, 0, 0, 1, 0, 0 ] )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we  show  that  for  [22Xs[122X  in  the  class [10X20A[110X (which fixes one point), the
  proportion  of nongenerating elements [22Xg[122X in one of the classes [10X2A[110X, [10X3A[110X, [10X3B[110X, [10X3E[110X
  has  the  maximum  [22X194/455[122X,  which is attained exactly for [10X3A[110X. (We find a [10X2A[110X
  element as a power of [22Xs[122X, a [10X3A[110X element as a power of any element of order [22X18[122X,
  a  [10X3B[110X  and  a  [10X3E[110X  element  as  elements  with  [22X135[122X  and  [22X108[122X  fixed points,
  respectively, which occur as powers of suitable elements of order [22X15[122X.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpermchar{ ord20 };[127X[104X
    [4X[28X[ 1, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27XAtlasClassNames( t )[ PowerMap( t, 10 )[ ord20[1] ] ];[127X[104X
    [4X[28X"2A"[128X[104X
    [4X[25Xgap>[125X [27Xord18:= PositionsProperty( ord, x -> x = 18 );;[127X[104X
    [4X[25Xgap>[125X [27XSet( AtlasClassNames( t ){ PowerMap( t, 6 ){ ord18 } } );[127X[104X
    [4X[28X[ "3A" ][128X[104X
    [4X[25Xgap>[125X [27Xord15:= PositionsProperty( ord, x -> x = 15 );;[127X[104X
    [4X[25Xgap>[125X [27XPowerMap( t, 5 ){ ord15 };[127X[104X
    [4X[28X[ 7, 8, 9, 10, 11, 12 ][128X[104X
    [4X[25Xgap>[125X [27XAtlasClassNames( t ){ [ 7 .. 12 ] };[127X[104X
    [4X[28X[ "3B", "3C", "3D", "3E", "3F", "3G" ][128X[104X
    [4X[25Xgap>[125X [27Xpermchar{ [ 7 .. 12 ] };[127X[104X
    [4X[28X[ 135, 0, 0, 108, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27Xmp:= NrMovedPoints( g );;[127X[104X
    [4X[25Xgap>[125X [27XResetGlobalRandomNumberGenerators();[127X[104X
    [4X[25Xgap>[125X [27Xrepeat 20A:= Random( g );[127X[104X
    [4X[25X>[125X [27X   until Order( 20A ) = 20 and mp - NrMovedPoints( 20A ) = 1;[127X[104X
    [4X[25Xgap>[125X [27X2A:= 20A^10;;[127X[104X
    [4X[25Xgap>[125X [27Xrepeat x:= Random( g ); until Order( x ) = 18;[127X[104X
    [4X[25Xgap>[125X [27X3A:= x^6;;[127X[104X
    [4X[25Xgap>[125X [27Xrepeat x:= Random( g );[127X[104X
    [4X[25X>[125X [27X   until Order( x ) = 15 and mp - NrMovedPoints( x^5 ) = 135;[127X[104X
    [4X[25Xgap>[125X [27X3B:= x^5;;[127X[104X
    [4X[25Xgap>[125X [27Xrepeat x:= Random( g );[127X[104X
    [4X[25X>[125X [27X   until Order( x ) = 15 and mp - NrMovedPoints( x^5 ) = 108;[127X[104X
    [4X[25Xgap>[125X [27X3E:= x^5;;[127X[104X
    [4X[25Xgap>[125X [27Xnongen:= List( [ 2A, 3A, 3B, 3E ],[127X[104X
    [4X[25X>[125X [27X                  c -> RatioOfNongenerationTransPermGroup( g, c, 20A ) );[127X[104X
    [4X[28X[ 3901/9477, 194/455, 451/1092, 451/1092 ][128X[104X
    [4X[25Xgap>[125X [27XMaximum( nongen );[127X[104X
    [4X[28X194/455[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNext  we  compute  the  values  [22XP(g,s)[122X, for [22Xg[122X is in the class [10X3A[110X and certain
  elements  [22Xs[122X.  It is enough to consider representatives [22Xs[122X of maximally cyclic
  subgroups  in  [22XS[122X,  but  here  we  can do better, as follows. Since [10X3A[110X is the
  unique  class  of length [22X72800[122X, it is fixed under [22XAut(S)[122X, so it is enough to
  consider  one  element  [22Xs[122X from each [22XAut(S)[122X-orbit on the classes of [22XS[122X. We use
  the  class fusion between the character tables of [22XS[122X and [22XAut(S)[122X for computing
  orbit representatives.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmaxorder:= RepresentativesMaximallyCyclicSubgroups( t );;[127X[104X
    [4X[25Xgap>[125X [27XLength( maxorder );[127X[104X
    [4X[28X57[128X[104X
    [4X[25Xgap>[125X [27Xautt:= CharacterTable( "O8+(3).S4" );;[127X[104X
    [4X[25Xgap>[125X [27Xfus:= PossibleClassFusions( t, autt );;[127X[104X
    [4X[25Xgap>[125X [27Xorbreps:= Set( fus, map -> Set( ProjectionMap( map ) ) );[127X[104X
    [4X[28X[ [ 1, 2, 5, 6, 7, 13, 17, 18, 19, 20, 23, 24, 27, 30, 31, 37, 43, [128X[104X
    [4X[28X      46, 50, 54, 55, 56, 57, 58, 64, 68, 72, 75, 78, 84, 85, 89, 95, [128X[104X
    [4X[28X      96, 97, 100, 106, 112 ] ][128X[104X
    [4X[25Xgap>[125X [27Xtotest:= Intersection( maxorder, orbreps[1] );[127X[104X
    [4X[28X[ 43, 50, 54, 56, 57, 64, 68, 75, 78, 84, 85, 89, 95, 97, 100, 106, [128X[104X
    [4X[28X  112 ][128X[104X
    [4X[25Xgap>[125X [27XLength( totest );[127X[104X
    [4X[28X17[128X[104X
    [4X[25Xgap>[125X [27XAtlasClassNames( t ){ totest };[127X[104X
    [4X[28X[ "6Q", "6X", "6B1", "8A", "8B", "9G", "9K", "12A", "12D", "12J", [128X[104X
    [4X[28X  "12K", "12O", "13A", "14A", "15A", "18A", "20A" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThis  means  that  we have to test one element of each of the element orders
  [22X13[122X,  [22X14[122X, [22X15[122X, and [22X18[122X (note that we know already a bound for elements of order
  [22X20[122X),  plus  certain  elements  of  the  orders  [22X6[122X, [22X8[122X, [22X9[122X, and [22X12[122X which can be
  identified  by  their centralizer orders and (for elements of order [22X6[122X and [22X8[122X)
  perhaps the centralizer orders of some powers.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xelementstotest:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xfor elord in [ 13, 14, 15, 18 ] do[127X[104X
    [4X[25X>[125X [27X     repeat s:= Random( g );[127X[104X
    [4X[25X>[125X [27X     until Order( s ) = elord;[127X[104X
    [4X[25X>[125X [27X     Add( elementstotest, s );[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  next  elements  to  be tested are in the classes [10X6B1[110X (centralizer order
  [22X162[122X),  in one of [10X9G[110X–[10X9J[110X (centralizer order [22X729[122X), in one of [10X9K[110X–[10X9N[110X (centralizer
  order  [22X81[122X),  in  one  of [10X12A[110X–[10X12C[110X (centralizer order [22X1728[122X), in one of [10X12D[110X–[10X12I[110X
  (centralizer  order  [22X432[122X), in [10X12J[110X (centralizer order [22X192[122X), in one of [10X12K[110X–[10X12N[110X
  (centralizer order [22X108[122X), and in one of [10X12O[110X–[10X12T[110X (centralizer order [22X72[122X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xordcent:= [ [ 6, 162 ], [ 9, 729 ], [ 9, 81 ], [ 12, 1728 ],[127X[104X
    [4X[25X>[125X [27X               [ 12, 432 ], [ 12, 192 ], [ 12, 108 ], [ 12, 72 ] ];;[127X[104X
    [4X[25Xgap>[125X [27Xcents:= SizesCentralizers( t );;[127X[104X
    [4X[25Xgap>[125X [27Xfor pair in ordcent do[127X[104X
    [4X[25X>[125X [27X     Print( pair, ": ", AtlasClassNames( t ){[127X[104X
    [4X[25X>[125X [27X         Filtered( [ 1 .. Length( ord ) ],[127X[104X
    [4X[25X>[125X [27X                   i -> ord[i] = pair[1] and cents[i] = pair[2] ) }, "\n" );[127X[104X
    [4X[25X>[125X [27X     repeat s:= Random( g );[127X[104X
    [4X[25X>[125X [27X     until Order( s ) = pair[1] and Size( Centralizer( g, s ) ) = pair[2];[127X[104X
    [4X[25X>[125X [27X     Add( elementstotest, s );[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[28X[ 6, 162 ]: [ "6B1" ][128X[104X
    [4X[28X[ 9, 729 ]: [ "9G", "9H", "9I", "9J" ][128X[104X
    [4X[28X[ 9, 81 ]: [ "9K", "9L", "9M", "9N" ][128X[104X
    [4X[28X[ 12, 1728 ]: [ "12A", "12B", "12C" ][128X[104X
    [4X[28X[ 12, 432 ]: [ "12D", "12E", "12F", "12G", "12H", "12I" ][128X[104X
    [4X[28X[ 12, 192 ]: [ "12J" ][128X[104X
    [4X[28X[ 12, 108 ]: [ "12K", "12L", "12M", "12N" ][128X[104X
    [4X[28X[ 12, 72 ]: [ "12O", "12P", "12Q", "12R", "12S", "12T" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  next elements to be tested are in one of the classes [10X6Q[110X–[10X6S[110X (centralizer
  order [22X648[122X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XAtlasClassNames( t ){ Filtered( [ 1 .. Length( ord ) ],[127X[104X
    [4X[25X>[125X [27X       i -> cents[i] = 648 and cents[ PowerMap( t, 2 )[i] ] = 52488[127X[104X
    [4X[25X>[125X [27X                           and cents[ PowerMap( t, 3 )[i] ] = 26127360 ) };[127X[104X
    [4X[28X[ "6Q", "6R", "6S" ][128X[104X
    [4X[25Xgap>[125X [27Xrepeat s:= Random( g );[127X[104X
    [4X[25X>[125X [27X   until Order( s ) = 6 and Size( Centralizer( g, s ) ) = 648[127X[104X
    [4X[25X>[125X [27X     and Size( Centralizer( g, s^2 ) ) = 52488[127X[104X
    [4X[25X>[125X [27X     and Size( Centralizer( g, s^3 ) ) = 26127360;[127X[104X
    [4X[25Xgap>[125X [27XAdd( elementstotest, s );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  next  elements  to be tested are in the class [10X6X[110X–[10X6A1[110X (centralizer order
  [22X648[122X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XAtlasClassNames( t ){ Filtered( [ 1 .. Length( ord ) ],[127X[104X
    [4X[25X>[125X [27X       i -> cents[i] = 648 and cents[ PowerMap( t, 2 )[i] ] = 52488[127X[104X
    [4X[25X>[125X [27X                           and cents[ PowerMap( t, 3 )[i] ] = 331776 ) };[127X[104X
    [4X[28X[ "6X", "6Y", "6Z", "6A1" ][128X[104X
    [4X[25Xgap>[125X [27Xrepeat s:= Random( g );[127X[104X
    [4X[25X>[125X [27X   until Order( s ) = 6 and Size( Centralizer( g, s ) ) = 648[127X[104X
    [4X[25X>[125X [27X     and Size( Centralizer( g, s^2 ) ) = 52488[127X[104X
    [4X[25X>[125X [27X     and Size( Centralizer( g, s^3 ) ) = 331776;[127X[104X
    [4X[25Xgap>[125X [27XAdd( elementstotest, s );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YFinally, we add elements from the classes [10X8A[110X and [10X8B[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XAtlasClassNames( t ){ Filtered( [ 1 .. Length( ord ) ],[127X[104X
    [4X[25X>[125X [27X       i -> ord[i] = 8 and cents[ PowerMap( t, 2 )[i] ] = 13824 ) };[127X[104X
    [4X[28X[ "8A" ][128X[104X
    [4X[25Xgap>[125X [27Xrepeat s:= Random( g );[127X[104X
    [4X[25X>[125X [27X   until Order( s ) = 8 and Size( Centralizer( g, s^2 ) ) = 13824;[127X[104X
    [4X[25Xgap>[125X [27XAdd( elementstotest, s );[127X[104X
    [4X[25Xgap>[125X [27XAtlasClassNames( t ){ Filtered( [ 1 .. Length( ord ) ],[127X[104X
    [4X[25X>[125X [27X       i -> ord[i] = 8 and cents[ PowerMap( t, 2 )[i] ] = 1536 ) };[127X[104X
    [4X[28X[ "8B" ][128X[104X
    [4X[25Xgap>[125X [27Xrepeat s:= Random( g );[127X[104X
    [4X[25X>[125X [27X   until Order( s ) = 8 and Size( Centralizer( g, s^2 ) ) = 1536;[127X[104X
    [4X[25Xgap>[125X [27XAdd( elementstotest, s );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we  compute  the  ratios. It turns out that from these candidates, only
  elements [22Xs[122X of the orders [22X14[122X and [22X15[122X satisfy [22XP(g,s) < 194/455[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xnongen:= List( elementstotest,[127X[104X
    [4X[25X>[125X [27X                  s -> RatioOfNongenerationTransPermGroup( g, 3A, s ) );;[127X[104X
    [4X[25Xgap>[125X [27Xsmaller:= PositionsProperty( nongen, x -> x < 194/455 );[127X[104X
    [4X[28X[ 2, 3 ][128X[104X
    [4X[25Xgap>[125X [27Xnongen{ smaller };[127X[104X
    [4X[28X[ 127/325, 1453/3640 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo  the  only candidates for [22Xs[122X that may be better than order [22X20[122X elements are
  elements  of order [22X14[122X or [22X15[122X. In order to exclude these two possibilities, we
  compute [22XP(g,s)[122X for [22Xs[122X in the class [10X14A[110X and [22Xg = s^7[122X in the class [10X2A[110X, and for [22Xs[122X
  in  the class [10X15A[110X and [22Xg[122X in the class [10X2A[110X, which yields values that are larger
  than [22X194/455[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xrepeat s:= Random( g );[127X[104X
    [4X[25X>[125X [27X   until Order( s ) = 14 and NrMovedPoints( s ) = 1078;[127X[104X
    [4X[25Xgap>[125X [27X2A:= s^7;;[127X[104X
    [4X[25Xgap>[125X [27Xnongen:= RatioOfNongenerationTransPermGroup( g, 2A, s );[127X[104X
    [4X[28X1573/3645[128X[104X
    [4X[25Xgap>[125X [27Xnongen > 194/455;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xrepeat s:= Random( g );[127X[104X
    [4X[25X>[125X [27X   until Order( s ) = 15 and NrMovedPoints( s ) = 1080 - 3;[127X[104X
    [4X[25Xgap>[125X [27Xnongen:= RatioOfNongenerationTransPermGroup( g, 2A, s );[127X[104X
    [4X[28X490/1053[128X[104X
    [4X[25Xgap>[125X [27Xnongen > 194/455;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  statement (d), we show that for each triple of elements in the union of
  the  classes  [10X2A[110X,  [10X3A[110X,  [10X3B[110X,  [10X3E[110X  there  is  an element in the class [10X20A[110X that
  generates [22XS[122X together with each element of the triple.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfor tup in UnorderedTuples( [ 2A, 3A, 3B, 3E ], 3 ) do[127X[104X
    [4X[25X>[125X [27X     cl:= ShallowCopy( tup );[127X[104X
    [4X[25X>[125X [27X     test:= RandomCheckUniformSpread( g, cl, 20A, 100 );[127X[104X
    [4X[25X>[125X [27X     if test <> true then[127X[104X
    [4X[25X>[125X [27X       Error( test );[127X[104X
    [4X[25X>[125X [27X     fi;[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YWe get no error message, so statement (d) is true.[133X
  
  [33X[0;0YFor  statement (e),  first  we show that [22X2.S = Ω^+(8,3)[122X contains elements of
  order [22X40[122X but [22XS[122X does not.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xder:= DerivedSubgroup( SO(1,8,3) );;[127X[104X
    [4X[25Xgap>[125X [27Xrepeat x:= PseudoRandom( der ); until Order( x ) = 40;[127X[104X
    [4X[25Xgap>[125X [27X40 in ord;[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThus elements of order [22X40[122X must arise as preimages of order [22X20[122X elements under
  the  natural  epimorphism  from  [22X2.S[122X to [22XS[122X, which means that we may choose an
  order  [22X40[122X  preimage  [22Xhats[122X  of  [22Xs[122X.  Then  [22XMM(2.S,  hats)[122X  consists of central
  extensions  of  the subgroups listed in statement (b). The perfect subgroups
  [22XO_7(3)[122X,  [22XL_4(3)[122X,  [22X2.U_4(3)[122X,  and  [22XU_4(2)[122X  of these groups must lift to their
  Schur double covers in [22X2.S[122X because otherwise the preimages would not contain
  elements of order [22X40[122X.[133X
  
  [33X[0;0YNext  we consider the preimage of the subgroup [22XU = (A_4 × U_4(2)).2[122X of [22XS[122X. We
  show  that  the  preimages of the two direct factors [22XA_4[122X and [22XU_4(2)[122X in [22XU^' =
  A_4  × U_4(2)[122X are Schur covers. For [22XA_4[122X, this follows from the fact that the
  preimage  of [22XU^'[122X must contain elements of order [22X20[122X, and that [22XU_4(2)[122X does not
  contain elements of order [22X10[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xu42:= CharacterTable( "U4(2)" );;[127X[104X
    [4X[25Xgap>[125X [27XFiltered( OrdersClassRepresentatives( u42 ), x -> x mod 5 = 0 );[127X[104X
    [4X[28X[ 5 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  order  to  show that the [22XU_4(2)[122X type subgroup of [22XU^'[122X lifts to its double
  cover  in  [22X2.S[122X,  we  note  that  the  class [10X2B[110X of [22XU_4(2)[122X lifts to a class of
  elements  of  order  four  in  the  double  cover  [22X2.U_4(2)[122X,  and  that  the
  corresponding  class  of  elements  in  [22XU[122X  is  [22XS[122X-conjugate  to  the class of
  involutions  in  the  direct factor [22XA_4[122X (which is the unique class of length
  three in [22XU[122X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xu:= CharacterTable( Maxes( t )[18] );[127X[104X
    [4X[28XCharacterTable( "(A4xU4(2)):2" )[128X[104X
    [4X[25Xgap>[125X [27X2u42:= CharacterTable( "2.U4(2)" );;[127X[104X
    [4X[25Xgap>[125X [27XOrdersClassRepresentatives( 2u42 )[4];[127X[104X
    [4X[28X4[128X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( 2u42, u42 )[4];[127X[104X
    [4X[28X3[128X[104X
    [4X[25Xgap>[125X [27XOrdersClassRepresentatives( u42 )[3];[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27XList( PossibleClassFusions( u42, u ), x -> x[3] );[127X[104X
    [4X[28X[ 8 ][128X[104X
    [4X[25Xgap>[125X [27XPositionsProperty( SizesConjugacyClasses( u ), x -> x = 3 );[127X[104X
    [4X[28X[ 2 ][128X[104X
    [4X[25Xgap>[125X [27XForAll( PossibleClassFusions( u, t ), x -> x[2] = x[8] );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe last subgroup for which the structure of the preimage has to be shown is
  [22XU  =  (A_6  ×  A_6):2^2[122X. We claim that each of the [22XA_6[122X type subgroups in the
  derived subgroup [22XU^' = A_6 × A_6[122X lifts to its double cover in [22X2.S[122X. Since all
  elements of order [22X20[122X in [22XU[122X lie in [22XU^'[122X, at least one of the two direct factors
  must  lift to its double cover, in order to give rise to an order [22X40[122X element
  in  [22XU[122X.  In  fact  both factors lift to the double cover since the two direct
  factors  are  interchanged  by conjugation in [22XU[122X; the latter follows form tha
  fact that [22XU[122X has no normal subgroup of type [22XA_6[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xu:= CharacterTable( Maxes( t )[24] );[127X[104X
    [4X[28XCharacterTable( "(A6xA6):2^2" )[128X[104X
    [4X[25Xgap>[125X [27XClassPositionsOfDerivedSubgroup( u );[127X[104X
    [4X[28X[ 1 .. 22 ][128X[104X
    [4X[25Xgap>[125X [27XPositionsProperty( OrdersClassRepresentatives( u ), x -> x = 20 );[127X[104X
    [4X[28X[ 8 ][128X[104X
    [4X[25Xgap>[125X [27XList( ClassPositionsOfNormalSubgroups( u ),[127X[104X
    [4X[25X>[125X [27X         x -> Sum( SizesConjugacyClasses( u ){ x } ) );[127X[104X
    [4X[28X[ 1, 129600, 259200, 259200, 259200, 518400 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo statement (e) holds.[133X
  
  [33X[0;0YFor  statement (f),  we have to consider the upward extensions [22XS.2_1[122X, [22XS.2_2[122X,
  and [22XS.3[122X.[133X
  
  [33X[0;0YFirst we look at [22XS.2_1[122X, an extension by an outer automorphism that acts as a
  double  transposition  in  the  outer  automorphism group [22XS_4[122X. Note that the
  symmetry  between  the  three  classes  of element oder [22X20[122X in [22XS[122X is broken in
  [22XS.2_1[122X, two of these classes have square roots in [22XS.2_1[122X, the third has not.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt2:= CharacterTable( "O8+(3).2_1" );;[127X[104X
    [4X[25Xgap>[125X [27Xord20:= PositionsProperty( OrdersClassRepresentatives( t2 ),[127X[104X
    [4X[25X>[125X [27X               x -> x = 20 );;[127X[104X
    [4X[25Xgap>[125X [27Xord20:= Intersection( ord20, ClassPositionsOfDerivedSubgroup( t2 ) );[127X[104X
    [4X[28X[ 84, 85, 86 ][128X[104X
    [4X[25Xgap>[125X [27XList( ord20, x -> x in PowerMap( t2, 2 ) );[127X[104X
    [4X[28X[ false, true, true ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YChanging the viewpoint, we see that for each class of element order [22X20[122X in [22XS[122X,
  there  is  a  group of the type [22XS.2_1[122X in which the elements in this class do
  not  have  square  roots,  and  there are groups of this type in which these
  elements have square roots. So we have to deal with two different cases, and
  we  do  this by first collecting the permutation characters induced from [13Xall[113X
  maximal  subgroups of [22XS.2_1[122X (other than [22XS[122X) that contain elements of order [22X20[122X
  in [22XS[122X, and then considering [22Xs[122X in each of these classes of [22XS[122X.[133X
  
  [33X[0;0YWe  fix  an embedding of [22XS[122X into [22XS.2_1[122X in which the elements in the class [10X20A[110X
  do  not  have  square  roots.  This  situation is given for the stored class
  fusion between the tables in the [5XGAP[105X Character Table Library.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtfust2:= GetFusionMap( t, t2 );;[127X[104X
    [4X[25Xgap>[125X [27Xtfust2{ PositionsProperty( OrdersClassRepresentatives( t ),[127X[104X
    [4X[25X>[125X [27X               x -> x = 20 ) };[127X[104X
    [4X[28X[ 84, 85, 86 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe six different actions of [22XS[122X on the cosets of [22XO_7(3)[122X type subgroups induce
  pairwise  different  permutation  characters  that  form  an orbit under the
  action of [22XAut(S)[122X. Four of these characters cannot extend to [22XS.2_1[122X, the other
  two extend to permutation characters of [22XS.2_1[122X on the cosets of [22XO_7(3).2[122X type
  subgroups; these subgroups contain [10X20A[110X elements.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xprimt2:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xposs:= PossiblePermutationCharacters( CharacterTable( "O7(3)" ), t );;[127X[104X
    [4X[25Xgap>[125X [27Xinvfus:= InverseMap( tfust2 );;[127X[104X
    [4X[25Xgap>[125X [27XList( poss, pi -> ForAll( CompositionMaps( pi, invfus ), IsInt ) );[127X[104X
    [4X[28X[ false, false, false, false, true, true ][128X[104X
    [4X[25Xgap>[125X [27XPossiblePermutationCharacters([127X[104X
    [4X[25X>[125X [27X       CharacterTable( "O7(3)" ) * CharacterTable( "Cyclic", 2 ), t2 );[127X[104X
    [4X[28X[  ][128X[104X
    [4X[25Xgap>[125X [27Xext:= PossiblePermutationCharacters( CharacterTable( "O7(3).2" ), t2 );;[127X[104X
    [4X[25Xgap>[125X [27XList( ext, pi -> pi{ ord20 } );[127X[104X
    [4X[28X[ [ 1, 0, 0 ], [ 1, 0, 0 ] ][128X[104X
    [4X[25Xgap>[125X [27XAppend( primt2, ext );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  novelties  in [22XS.2_1[122X that arise from [22XO_7(3)[122X type subgroups of [22XS[122X have the
  structure  [22XL_4(3).2^2[122X.  These  subgroups contain elements in the classes [10X20B[110X
  and [10X20C[110X of [22XS[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xext:= PossiblePermutationCharacters( CharacterTable( "L4(3).2^2" ), t2 );;[127X[104X
    [4X[25Xgap>[125X [27XList( ext, pi -> pi{ ord20 } );[127X[104X
    [4X[28X[ [ 0, 0, 1 ], [ 0, 1, 0 ] ][128X[104X
    [4X[25Xgap>[125X [27XAppend( primt2, ext );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YNote that from the possible permutation characters of [22XS.2_1[122X on the cosets of
  [22XL_4(3):2  ×  2[122X  type  subgroups, we see that such subgroups must contain [10X20A[110X
  elements,  i. e.,  all  such  subgroups  of  [22XS.2_1[122X  lie inside [22XO_7(3).2[122X type
  subgroups.  This  means  that  the  structure description of these novelties
  in [CCN+85, p. 140] is not correct. The correct structure is [22XL_4(3).2^2[122X.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XList( PossiblePermutationCharacters( CharacterTable( "L4(3).2_2" ) *[127X[104X
    [4X[25X>[125X [27X             CharacterTable( "Cyclic", 2 ), t2 ), pi -> pi{ ord20 } );[127X[104X
    [4X[28X[ [ 1, 0, 0 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YAll  [22X3^6:L_4(3)[122X  type  subgroups  of  [22XS[122X  extend  to  [22XS.2_1[122X. We compute these
  permutation  characters  as the possible permutation characters of the right
  degree.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xext:= PermChars( t2, rec( torso:= [ 1120 ] ) );;[127X[104X
    [4X[25Xgap>[125X [27XList( ext, pi -> pi{ ord20 } );[127X[104X
    [4X[28X[ [ 2, 0, 0 ], [ 0, 0, 2 ], [ 0, 2, 0 ] ][128X[104X
    [4X[25Xgap>[125X [27XAppend( primt2, ext );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YAlso  all  [22X2.U_4(3).2^2[122X  type subgroups of [22XS[122X extend to [22XS.2_1[122X. We compute the
  permutation  characters  as  the extensions of the corresponding permutation
  characters of [22XS[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfilt:= Filtered( prim, x -> x[1] = 189540 );;[127X[104X
    [4X[25Xgap>[125X [27Xcand:= List( filt, x -> CompositionMaps( x, invfus ) );;[127X[104X
    [4X[25Xgap>[125X [27Xext:= Concatenation( List( cand,[127X[104X
    [4X[25X>[125X [27X             pi -> PermChars( t2, rec( torso:= pi ) ) ) );;[127X[104X
    [4X[25Xgap>[125X [27XList( ext, x -> x{ ord20 } );[127X[104X
    [4X[28X[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27XAppend( primt2, ext );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  extensions  of  [22X(A_4  × U_4(2)):2[122X type subgroups of [22XS[122X to [22XS.2_1[122X have the
  type [22XS_4 × U_4(2):2[122X, they contain [10X20A[110X elements.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xext:= PossiblePermutationCharacters( CharacterTable( "Symmetric", 4 ) *[127X[104X
    [4X[25X>[125X [27X             CharacterTable( "U4(2).2" ), t2 );;[127X[104X
    [4X[25Xgap>[125X [27XList( ext, x -> x{ ord20 } );[127X[104X
    [4X[28X[ [ 1, 0, 0 ], [ 1, 0, 0 ] ][128X[104X
    [4X[25Xgap>[125X [27XAppend( primt2, ext );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YAll  [22X(A_6  ×  A_6):2^2[122X  type  subgroups of [22XS[122X extend to [22XS.2_1[122X. We compute the
  permutation  characters  as  the extensions of the corresponding permutation
  characters of [22XS[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfilt:= Filtered( prim, x -> x[1] = 9552816 );;[127X[104X
    [4X[25Xgap>[125X [27Xcand:= List( filt, x -> CompositionMaps( x, InverseMap( tfust2 ) ));;[127X[104X
    [4X[25Xgap>[125X [27Xext:= Concatenation( List( cand,[127X[104X
    [4X[25X>[125X [27X             pi -> PermChars( t2, rec( torso:= pi ) ) ) );;[127X[104X
    [4X[25Xgap>[125X [27XList( ext, x -> x{ ord20 } );[127X[104X
    [4X[28X[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27XAppend( primt2, ext );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  have  found  all relevant permutation characters of [22XS.2_1[122X. This together
  with the list in [CCN+85, p. 140] implies statement (g).[133X
  
  [33X[0;0YNow we compute the bounds [22Xσ^'(S.2_1, s)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLength( primt2 );[127X[104X
    [4X[28X15[128X[104X
    [4X[25Xgap>[125X [27Xapprox:= List( ord20, x -> ApproxP( primt2, x ) );;[127X[104X
    [4X[25Xgap>[125X [27Xouter:= Difference([127X[104X
    [4X[25X>[125X [27X     PositionsProperty( OrdersClassRepresentatives( t2 ), IsPrimeInt ),[127X[104X
    [4X[25X>[125X [27X     ClassPositionsOfDerivedSubgroup( t2 ) );;[127X[104X
    [4X[25Xgap>[125X [27XList( approx, l -> Maximum( l{ outer } ) );[127X[104X
    [4X[28X[ 574/1215, 83/567, 83/567 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNext  we look at [22XS.2_2[122X, an extension by an outer automorphism that acts as a
  transposition  in  the  outer  automorphism  group [22XS_4[122X. Similar to the above
  situation, the symmetry between the three classes of element oder [22X20[122X in [22XS[122X is
  broken also in [22XS.2_2[122X: The first is a conjugacy class of [22XS.2_2[122X, the other two
  classes are fused in [22XS.2_2[122X,[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt2:= CharacterTable( "O8+(3).2_2" );;[127X[104X
    [4X[25Xgap>[125X [27Xord20:= PositionsProperty( OrdersClassRepresentatives( t2 ),[127X[104X
    [4X[25X>[125X [27X               x -> x = 20 );;[127X[104X
    [4X[25Xgap>[125X [27Xord20:= Intersection( ord20, ClassPositionsOfDerivedSubgroup( t2 ) );[127X[104X
    [4X[28X[ 82, 83 ][128X[104X
    [4X[25Xgap>[125X [27Xtfust2:= GetFusionMap( t, t2 );;[127X[104X
    [4X[25Xgap>[125X [27Xtfust2{ PositionsProperty( OrdersClassRepresentatives( t ),[127X[104X
    [4X[25X>[125X [27X               x -> x = 20 ) };[127X[104X
    [4X[28X[ 82, 83, 83 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YLike  in  the case [22XS.2_1[122X, we compute the permutation characters induced from
  [13Xall[113X maximal subgroups of [22XS.2_2[122X (other than [22XS[122X) that contain elements of order
  [22X20[122X in [22XS[122X.[133X
  
  [33X[0;0YWe  fix the embedding of [22XS[122X into [22XS.2_2[122X in which the class [10X20A[110X of [22XS[122X is a class
  of  [22XS.2_2[122X.  This  situation is given for the stored class fusion between the
  tables in the [5XGAP[105X Character Table Library.[133X
  
  [33X[0;0YExactly  two  classes  of  [22XO_7(3)[122X type subgroups in [22XS[122X extend to [22XS.2_2[122X, these
  groups contain [10X20A[110X elements.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xprimt2:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xext:= PermChars( t2, rec( torso:= [ 1080 ] ) );;[127X[104X
    [4X[25Xgap>[125X [27XList( ext, pi -> pi{ ord20 } );[127X[104X
    [4X[28X[ [ 1, 0 ], [ 1, 0 ] ][128X[104X
    [4X[25Xgap>[125X [27XAppend( primt2, ext );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YOnly  one class of [22X3^6:L_4(3)[122X type subgroups extends to [22XS.2_2[122X. (Note that we
  need  not consider the novelties of the type [22X3^3+6:(L_3(3) × 2)[122X, because the
  order of these groups is not divisible by [22X5[122X.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xext:= PermChars( t2, rec( torso:= [ 1120 ] ) );;[127X[104X
    [4X[25Xgap>[125X [27XList( ext, pi -> pi{ ord20 } );[127X[104X
    [4X[28X[ [ 2, 0 ] ][128X[104X
    [4X[25Xgap>[125X [27XAppend( primt2, ext );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YOnly  one  class  of  [22X2.U_4(3).2^2[122X  type subgroups of [22XS[122X extends to [22XS.2_2[122X. We
  compute  the  permutation  character  as  the extension of the corresponding
  permutation characters of [22XS[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfilt:= Filtered( prim, x -> x[1] = 189540 );;[127X[104X
    [4X[25Xgap>[125X [27Xcand:= List( filt, x -> CompositionMaps( x, InverseMap( tfust2 ) ));;[127X[104X
    [4X[25Xgap>[125X [27Xext:= Concatenation( List( cand,[127X[104X
    [4X[25X>[125X [27X             pi -> PermChars( t2, rec( torso:= pi ) ) ) );;[127X[104X
    [4X[25Xgap>[125X [27XList( ext, x -> x{ ord20 } );[127X[104X
    [4X[28X[ [ 1, 0 ] ][128X[104X
    [4X[25Xgap>[125X [27XAppend( primt2, ext );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YTwo classes of [22X(A_4 × U_4(2)):2[122X type subgroups of [22XS[122X extend to [22XS.2_2[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfilt:= Filtered( prim, x -> x[1] = 7960680 );;[127X[104X
    [4X[25Xgap>[125X [27Xcand:= List( filt, x -> CompositionMaps( x, InverseMap( tfust2 ) ));;[127X[104X
    [4X[25Xgap>[125X [27Xext:= Concatenation( List( cand,[127X[104X
    [4X[25X>[125X [27X             pi -> PermChars( t2, rec( torso:= pi ) ) ) );;[127X[104X
    [4X[25Xgap>[125X [27XList( ext, x -> x{ ord20 } );[127X[104X
    [4X[28X[ [ 1, 0 ], [ 1, 0 ] ][128X[104X
    [4X[25Xgap>[125X [27XAppend( primt2, ext );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YExactly  one  class of [22X(A_6 × A_6):2^2[122X type subgroups in [22XS[122X extends to [22XS.2_2[122X,
  and the extensions have the structure [22XS_6 ≀ 2[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xext:= PossiblePermutationCharacters( CharacterTableWreathSymmetric([127X[104X
    [4X[25X>[125X [27X             CharacterTable( "S6" ), 2 ), t2 );;[127X[104X
    [4X[25Xgap>[125X [27XList( ext, x -> x{ ord20 } );[127X[104X
    [4X[28X[ [ 1, 0 ] ][128X[104X
    [4X[25Xgap>[125X [27XAppend( primt2, ext );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  have found all relevant permutation characters of [22XS.2_2[122X, and compute the
  bounds [22Xσ^'(S.2_2, s)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLength( primt2 );[127X[104X
    [4X[28X7[128X[104X
    [4X[25Xgap>[125X [27Xapprox:= List( ord20, x -> ApproxP( primt2, x ) );;[127X[104X
    [4X[25Xgap>[125X [27Xouter:= Difference([127X[104X
    [4X[25X>[125X [27X     PositionsProperty( OrdersClassRepresentatives( t2 ), IsPrimeInt ),[127X[104X
    [4X[25X>[125X [27X     ClassPositionsOfDerivedSubgroup( t2 ) );;[127X[104X
    [4X[25Xgap>[125X [27XList( approx, l -> Maximum( l{ outer } ) );[127X[104X
    [4X[28X[ 14/9, 0 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThis means that there is an extension of the type [22XS.2_2[122X in which [22Xs[122X cannot be
  chosen  such that the bound is less than [22X1/2[122X. More precisely, we have [22Xσ(g,s)
  ≥ 1/2[122X exactly for [22Xg[122X in the unique outer involution class of size [22X1080[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xapprox:= ApproxP( primt2, ord20[1] );;[127X[104X
    [4X[25Xgap>[125X [27Xbad:= Filtered( outer, i -> approx[i] >= 1/2 );[127X[104X
    [4X[28X[ 84 ][128X[104X
    [4X[25Xgap>[125X [27XOrdersClassRepresentatives( t2 ){ bad };[127X[104X
    [4X[28X[ 2 ][128X[104X
    [4X[25Xgap>[125X [27XSizesConjugacyClasses( t2 ){ bad };[127X[104X
    [4X[28X[ 1080 ][128X[104X
    [4X[25Xgap>[125X [27XNumber( SizesConjugacyClasses( t2 ), x -> x = 1080 );[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo  we  compute the proportion of elements in this class that generate [22XS.2_2[122X
  together  with an element [22Xs[122X of order [22X20[122X in [22XS[122X. (As above, we have to consider
  two   conjugacy   classes.)   For  that,  we  first  compute  a  permutation
  representation  of  [22XS.2_2[122X,  using  that  [22XS.2_2[122X  is  isomporphic  to  the two
  subgroups  of  index  [22X2[122X  in [22XPGO^+(8,3) = O_8^+(3).2^2_122[122X that are different
  from [22XPSO^+(8,3) = O_8^+(3).2_1[122X, cf. [CCN+85, p. 140].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgo:= GO(1,8,3);;[127X[104X
    [4X[25Xgap>[125X [27Xso:= SO(1,8,3);;[127X[104X
    [4X[25Xgap>[125X [27Xouterelm:= First( GeneratorsOfGroup( go ), x -> not x in so );;[127X[104X
    [4X[25Xgap>[125X [27Xg2:= ClosureGroup( DerivedSubgroup( so ), outerelm );;[127X[104X
    [4X[25Xgap>[125X [27XSize( g2 );[127X[104X
    [4X[28X19808719257600[128X[104X
    [4X[25Xgap>[125X [27Xdom:= NormedRowVectors( GF(3)^8 );;[127X[104X
    [4X[25Xgap>[125X [27Xorbs:= OrbitsDomain( g2, dom, OnLines );;[127X[104X
    [4X[25Xgap>[125X [27XList( orbs, Length );[127X[104X
    [4X[28X[ 1080, 1080, 1120 ][128X[104X
    [4X[25Xgap>[125X [27Xact:= Action( g2, orbs[1], OnLines );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YAn involution [22Xg[122X can be found as a power of one of the given generators.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XOrder( outerelm );[127X[104X
    [4X[28X26[128X[104X
    [4X[25Xgap>[125X [27Xg:= Permutation( outerelm^13, orbs[1], OnLines );;[127X[104X
    [4X[25Xgap>[125X [27XSize( ConjugacyClass( act, g ) );[127X[104X
    [4X[28X1080[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we  find the candidates for the elements [22Xs[122X, and compute their ratios of
  nongeneration.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xord20;[127X[104X
    [4X[28X[ 82, 83 ][128X[104X
    [4X[25Xgap>[125X [27XSizesCentralizers( t2 ){ ord20 };[127X[104X
    [4X[28X[ 40, 20 ][128X[104X
    [4X[25Xgap>[125X [27Xder:= DerivedSubgroup( act );;[127X[104X
    [4X[25Xgap>[125X [27Xrepeat 20A:= Random( der );[127X[104X
    [4X[25X>[125X [27X   until Order( 20A ) = 20 and Size( Centralizer( act, 20A ) ) = 40;[127X[104X
    [4X[25Xgap>[125X [27XRatioOfNongenerationTransPermGroup( act, g, 20A );[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27Xrepeat 20BC:= Random( der );[127X[104X
    [4X[25X>[125X [27X   until Order( 20BC ) = 20 and Size( Centralizer( act, 20BC ) ) = 20;[127X[104X
    [4X[25Xgap>[125X [27XRatioOfNongenerationTransPermGroup( act, g, 20BC );[127X[104X
    [4X[28X0[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThis means that for [22Xs[122X in one [22XS[122X-class of elements of order [22X20[122X, we have [22XP^'(g,
  s)  =  1[122X, and [22Xs[122X in the other two [22XS[122X-classes of elements of order [22X20[122X generates
  with any conjugate of [22Xg[122X.[133X
  
  [33X[0;0YConcerning  [22XS.2_2[122X,  it  remains to show that we cannot find a better element
  than  [22Xs[122X.  For  that,  we  first  compute  class  representatives  [22Xs^'[122X  in [22XS[122X,
  w.r.t. conjugacy  in  [22XS.2_2[122X,  and  then  compute [22XP^'( s^', g )[122X. (It would be
  enough  to  check  representatives  of classes of maximal element order, but
  computing all classes is easy enough.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xccl:= ConjugacyClasses( act );;[127X[104X
    [4X[25Xgap>[125X [27Xder:= DerivedSubgroup( act );;[127X[104X
    [4X[25Xgap>[125X [27Xreps:= Filtered( List( ccl, Representative ), x -> x in der );;[127X[104X
    [4X[25Xgap>[125X [27XLength( reps );[127X[104X
    [4X[28X83[128X[104X
    [4X[25Xgap>[125X [27Xratios:= List( reps,[127X[104X
    [4X[25X>[125X [27X                  s -> RatioOfNongenerationTransPermGroup( act, g, s ) );;[127X[104X
    [4X[25Xgap>[125X [27Xcand:= PositionsProperty( ratios, x -> x < 1 );;[127X[104X
    [4X[25Xgap>[125X [27Xratios:= ratios{ cand };;[127X[104X
    [4X[25Xgap>[125X [27XSortParallel( ratios, cand );[127X[104X
    [4X[25Xgap>[125X [27Xratios;[127X[104X
    [4X[28X[ 0, 1/10, 1/10, 16/135, 1/3, 1/3, 11/27, 7/15, 7/15 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  [22XS.2_2[122X,  it  remains  to show that there is no element [22Xs^' ∈ S[122X such that
  [22XP^'(  s^'}^x, g ) < 1[122X holds for any [22Xx ∈ Aut(S)[122X and [22Xg ∈ S.2_2[122X. So we are done
  when  we  can  show  that  each class given by [10Xcand[110X is conjugate in [22XS.3[122X to a
  class  outside  [10Xcand[110X.  The  classes  can be identified by element orders and
  centralizer orders.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xinvs:= List( cand,[127X[104X
    [4X[25X>[125X [27X      x -> [ Order( reps[x] ), Size( Centralizer( der, reps[x] ) ) ] );[127X[104X
    [4X[28X[ [ 20, 20 ], [ 18, 108 ], [ 18, 108 ], [ 14, 28 ], [ 15, 45 ], [128X[104X
    [4X[28X  [ 15, 45 ], [ 10, 40 ], [ 12, 72 ], [ 12, 72 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNamely, [10Xcand[110X contains no full [22XS.3[122X-orbit of classes of the element orders [22X20[122X,
  [22X18[122X,  [22X14[122X,  [22X15[122X,  and  [22X10[122X;  also,  [10Xcand[110X does not contain full [22XS.3[122X-orbits on the
  classes [10X12O[110X–[10X12T[110X.[133X
  
  [33X[0;0YFinally, we deal with [22XS.3[122X. The fact that no maximal subgroup of [22XS[122X containing
  an  element  of  order  [22X20[122X  extends  to  [22XS.3[122X follows either from the list of
  maximal  subgroups of [22XS[122X in [CCN+85, p. 140] or directly from the permutation
  characters.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt3:= CharacterTable( "O8+(3).3" );;[127X[104X
    [4X[25Xgap>[125X [27Xtfust3:= GetFusionMap( t, t3 );;[127X[104X
    [4X[25Xgap>[125X [27Xinv:= InverseMap( tfust3 );;[127X[104X
    [4X[25Xgap>[125X [27Xfilt:= PositionsProperty( prim, x -> x[ spos ] <> 0 );;[127X[104X
    [4X[25Xgap>[125X [27XForAll( prim{ filt },[127X[104X
    [4X[25X>[125X [27X           pi -> ForAny( CompositionMaps( pi, inv ), IsList ) );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo  we  have to consider only the classes of novelties in [22XS.3[122X, but the order
  of  none  of  these  groups is divisible by [22X20[122X –again see [CCN+85, p. 140]).
  This  means that [13Xany[113X element in [22XS.3 ∖ S[122X together with an element of order [22X20[122X
  in  [22XS[122X  generates  [22XS.3[122X.  This  is  in fact stronger than statement (f), which
  claims  this property only for elements of prime order in [22XS.3 ∖ S[122X (and their
  roots); note that [22XS.3 ∖ S[122X contains elements of the orders [22X9[122X and [22X27[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xouter:= Difference( [ 1 .. NrConjugacyClasses( t3 ) ],[127X[104X
    [4X[25X>[125X [27X               ClassPositionsOfDerivedSubgroup( t3 ) );[127X[104X
    [4X[28X[ 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, [128X[104X
    [4X[28X  70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, [128X[104X
    [4X[28X  87, 88, 89, 90, 91, 92, 93, 94 ][128X[104X
    [4X[25Xgap>[125X [27XSet( OrdersClassRepresentatives( t3 ){ outer } );[127X[104X
    [4X[28X[ 3, 6, 9, 12, 18, 21, 24, 27, 36, 39 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YBefore we turn to the next computations, we clean the workspace.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCleanWorkspace();[127X[104X
  [4X[32X[104X
  
  
  [1X11.5-14 [33X[0;0Y[22XO^+_8(4)[122X[101X[1X[133X[101X
  
  [33X[0;0YWe show that [22XS = O^+_8(4) = Ω^+(8,4)[122X satisfies the following.[133X
  
  [8X(a)[108X
        [33X[0;6YFor  suitable  [22Xs ∈ S[122X of the type [22X2^- perp 6^-[122X (i. e., [22Xs[122X decomposes the
        natural  [22X8[122X-dimensional  module  for  [22XS[122X  into  an orthogonal sum of two
        irreducible  modules  of  the dimensions [22X2[122X and [22X6[122X, respectively) and of
        order  [22X65[122X,  [22XMM(S,s)[122X  consists  of  exactly three pairwise nonconjugate
        subgroups of the type [22X(5 × O^-_6(4)).2 = (5 × Ω^-(6,4)).2[122X.[133X
  
  [8X(b)[108X
        [33X[0;6Y[22Xσ( S, s ) ≤ 34817 / 1645056[122X.[133X
  
  [8X(c)[108X
        [33X[0;6YIn  the extensions [22XS.2_1[122X and [22XS.3[122X of [22XS[122X by graph automorphisms, there is
        at  most  one  maximal  subgroup  besides  [22XS[122X  that contains [22Xs[122X. For the
        extension  [22XS.2_2[122X of [22XS[122X by a field automorphism, we have [22Xσ^'(S.2_2, s) =
        0[122X.  In  the extension [22XS.2_3[122X of [22XS[122X by the product of an involutory graph
        automorphism  and  a  field  automorphism,  there  is a unique maximal
        subgroup besides [22XS[122X that contains [22Xs[122X.[133X
  
  [33X[0;0YA  safe  source  for  determining  [22XMM(S,s)[122X  is [Kle87]. By inspection of the
  result  matrix  in  this  paper, we get that the only maximal subgroups of [22XS[122X
  that  contain  elements  of  order [22X65[122X occur in the rows 9–14 and 23–25; they
  have  the  isomorphism  types  [22XS_6(4)  =  Sp(6,4) ≅ O_7(4) = Ω(7,4)[122X and [22X(5 ×
  O_6^-(4)).2  =  (5 × Ω^-(6,4)).2[122X, respectively, and for each of these, there
  are three conjugacy classes of subgroups in [22XS[122X, which are conjugate under the
  triality graph automorphism of [22XS[122X.[133X
  
  [33X[0;0YWe  start  with  the natural matrix representation of [22XS[122X. For convenience, we
  compute an isomorphic permutation group on [22X5525[122X points.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xq:= 4;;  n:= 8;;[127X[104X
    [4X[25Xgap>[125X [27XG:= DerivedSubgroup( SO( 1, n, q ) );;[127X[104X
    [4X[25Xgap>[125X [27Xpoints:= NormedRowVectors( GF(q)^n );;[127X[104X
    [4X[25Xgap>[125X [27Xorbs:= OrbitsDomain( G, points, OnLines );;[127X[104X
    [4X[25Xgap>[125X [27XList( orbs, Length );[127X[104X
    [4X[28X[ 5525, 16320 ][128X[104X
    [4X[25Xgap>[125X [27Xhom:= ActionHomomorphism( G, orbs[1], OnLines );;[127X[104X
    [4X[25Xgap>[125X [27XG:= Image( hom );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  group [22XS[122X contains exactly six conjugacy classes of (cyclic) subgroups of
  order  [22X65[122X;  this  follows from the fact that the centralizer of any Sylow [22X13[122X
  subgroup in [22XS[122X has the structure [22X5 × 5 × 13[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCollected( Factors( Size( G ) ) );[127X[104X
    [4X[28X[ [ 2, 24 ], [ 3, 5 ], [ 5, 4 ], [ 7, 1 ], [ 13, 1 ], [ 17, 2 ] ][128X[104X
    [4X[25Xgap>[125X [27XResetGlobalRandomNumberGenerators();[127X[104X
    [4X[25Xgap>[125X [27Xrepeat x:= Random( G );[127X[104X
    [4X[25X>[125X [27X   until Order( x ) mod 13 = 0;[127X[104X
    [4X[25Xgap>[125X [27Xx:= x^( Order( x ) / 13 );;[127X[104X
    [4X[25Xgap>[125X [27Xc:= Centralizer( G, x );;[127X[104X
    [4X[25Xgap>[125X [27XIsAbelian( c );  AbelianInvariants( c );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X[ 5, 5, 13 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  group [22XS_6(4)[122X contains exactly one class of subgroups of order [22X65[122X, since
  the  conjugacy  classes  of elements of order [22X65[122X in [22XS_6(4)[122X are algebraically
  conjugate.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "S6(4)" );;[127X[104X
    [4X[25Xgap>[125X [27Xord65:= PositionsProperty( OrdersClassRepresentatives( t ),[127X[104X
    [4X[25X>[125X [27X                              x -> x = 65 );[127X[104X
    [4X[28X[ 105, 106, 107, 108, 109, 110, 111, 112 ][128X[104X
    [4X[25Xgap>[125X [27Xord65 = ClassOrbit( t, ord65[1] );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThus there are at least three classes of order [22X65[122X elements in [22XS[122X that are [13Xnot[113X
  contained in [22XS_6(4)[122X type subgroups of [22XS[122X. So we choose such an element [22Xs[122X, and
  have to consider only overgroups of the type [22X(5 × Ω^-(6,4)).2[122X.[133X
  
  [33X[0;0YThe group [22XΩ^-(6,4) ≅ U_4(4)[122X contains exactly one class of subgroups of order
  [22X65[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "U4(4)" );;[127X[104X
    [4X[25Xgap>[125X [27Xords:= OrdersClassRepresentatives( t );;[127X[104X
    [4X[25Xgap>[125X [27Xord65:= PositionsProperty( ords, x -> x = 65 );;[127X[104X
    [4X[25Xgap>[125X [27Xord65 = ClassOrbit( t, ord65[1] );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo [22X5 × Ω^-(6,4)[122X contains exactly six such classes. Furthermore, subgroups in
  different classes are not [22XS[122X-conjugate.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsyl5:= SylowSubgroup( c, 5 );;[127X[104X
    [4X[25Xgap>[125X [27Xelms:= Filtered( Elements( syl5 ), y -> Order( y ) = 5 );;[127X[104X
    [4X[25Xgap>[125X [27Xreps:= Set( elms, SmallestGeneratorPerm );;  Length( reps );[127X[104X
    [4X[28X6[128X[104X
    [4X[25Xgap>[125X [27Xreps65:= List( reps, y -> SubgroupNC( G, [ y * x ] ) );;[127X[104X
    [4X[25Xgap>[125X [27Xpairs:= Combinations( reps65, 2 );;[127X[104X
    [4X[25Xgap>[125X [27XForAny( pairs, p -> IsConjugate( G, p[1], p[2] ) );[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  consider  only  subgroups  [22XM ≤ S[122X in the three [22XS[122X-classes of the type [22X(5 ×
  Ω^-(6,4)).2[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xcand:= List( reps, y -> Normalizer( G, SubgroupNC( G, [ y ] ) ) );;[127X[104X
    [4X[25Xgap>[125X [27Xcand:= Filtered( cand, y -> Size( y ) = 10 * Size( t ) );;[127X[104X
    [4X[25Xgap>[125X [27XLength( cand );[127X[104X
    [4X[28X3[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y(Note  that  one  of  the  members  in [22XMM(S,s)[122X is the stabilizer in [22XS[122X of the
  orthogonal  decomposition  [22X2^-  perp  6^-[122X,  the  other  two  members are not
  reducible.)[133X
  
  [33X[0;0YBy  the  above,  the  classes of subgroups of order [22X65[122X in each such [22XM[122X are in
  bijection  with  the  corresponding classes in [22XS[122X. Since [22XN_S(⟨ g ⟩) ⊆ M[122X holds
  for  any  [22Xg ∈ M[122X of order [22X65[122X, also the conjugacy classes of [13Xelements[113X of order
  [22X65[122X in [22XM[122X are in bijection with those in [22XS[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xnorms:= List( reps65, y -> Normalizer( G, y ) );;[127X[104X
    [4X[25Xgap>[125X [27XForAll( norms, y -> ForAll( cand, M -> IsSubset( M, y ) ) );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAs  a consequence, we have [22Xg^S ∩ M = g^M[122X and thus [22X1_M^S(g) = 1[122X. This implies
  statement (a).[133X
  
  [33X[0;0YIn   order   to   show   statement (b),   we   want   to  use  the  function
  [10XUpperBoundFixedPointRatios[110X  introduced in Section [14X11.3-3[114X. For that, we first
  compute  the  conjugacy classes of the three class representatives [22XM[122X. (Since
  the  groups  have  elementary  abelian  Sylow  [22X5[122X subgroups of the order [22X5^4[122X,
  computing   all   conjugacy   classes   appears  to  be  faster  than  using
  [10XClassesOfPrimeOrder[110X.) Then we compute an upper bound for [22Xσ(S,s)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsyl5:= SylowSubgroup( cand[1], 5 );;[127X[104X
    [4X[25Xgap>[125X [27XSize( syl5 );  IsElementaryAbelian( syl5 );[127X[104X
    [4X[28X625[128X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XUpperBoundFixedPointRatios( G, List( cand, ConjugacyClasses ), false );[127X[104X
    [4X[28X[ 34817/1645056, false ][128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[13XRemark:[113X[133X
  
  [33X[0;0YComputing  the  exact  value [22Xσ(S,s)[122X in the above setup would require to test
  the  [22XS[122X-conjugacy  of  certain  order  [22X5[122X  elements in [22XM[122X. With the current [5XGAP[105X
  implementation, some of the relevant tests need several hours of CPU time.[133X
  
  [33X[0;0YAn  alternative  approach would be to compute the permutation action of [22XS[122X on
  the  cosets  of  [22XM[122X,  of  degree  [22X6580224[122X,  and  to count the fixed points of
  conjugacy  class representatives of prime order. The currently available [5XGAP[105X
  library  methods  are  not sufficient for computing this in reasonable time.
  [21XAd-hoc code[121X for this special case works, but it seemed to be not appropriate
  to include it here.[133X
  
  [33X[0;0YIn   the  proof  of  statement (c),  again  we  consult  the  result  matrix
  in [Kle87].  For  [22XS.3[122X,  the maximal subgroups are in the rows [22X4[122X, [22X15[122X, [22X22[122X, [22X26[122X,
  and  [22X61[122X.  Only  row [22X26[122X yields subgroups that contain elements [22Xs[122X of order [22X65[122X,
  they have the isomorphism type [22X(5 × GU(3,4)).2 ≅ (5^2 × U_3(4)).2[122X. Note that
  the  conjugacy  classes  of the members in [22XMM(S,s)[122X are permuted by the outer
  automorphism of order [22X3[122X, so none of the subgroups in [22XMM(S,s)[122X extends to [22XS.3[122X.
  By [BGK08,  Lemma 2.4 (2)],  if there is a maximal subgroup of [22XS.3[122X besides [22XS[122X
  that  contains  [22Xs[122X  then  this  subgroup  is  the  normalizer  in  [22XS.3[122X of the
  intersection  of  the  three members of [22XMM(S,s)[122X, i. e., [22Xs[122X is contained in at
  most one such subgroup.[133X
  
  [33X[0;0YFor  [22XS.2_1[122X,  only  the  rows  [22X9[122X  and  [22X23[122X  yield maximal subgroups containing
  elements of order [22X65[122X, and since we had chosen [22Xs[122X in such a way that row [22X9[122X was
  excluded  already  for  the simple group, only extensions of the elements in
  [22XMM(S,s)[122X  can  appear.  Exactly  one of these three subgroups of [22XS[122X extends to
  [22XS.2_1[122X,  so  again we get just one maximal subgroup of [22XS.2_1[122X, besides [22XS[122X, that
  contains [22Xs[122X.[133X
  
  [33X[0;0YAll  subgroups  in  [22XMM(S,s)[122X  extend  to  [22XS.2_2[122X,  see [Kle87]. We compute the
  extensions  of  the  above  subgroups  [22XM[122X  of [22XS[122X to [22XS.2_2[122X, by constructing the
  action  of  the field automorphism in the permutation representation we used
  for  [22XS[122X.  In  other  words, we compute the projective action of the Frobenius
  map.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfrob:= PermList( List( orbs[1], v -> Position( orbs[1],[127X[104X
    [4X[25X>[125X [27X             List( v, x -> x^2 ) ) ) );;[127X[104X
    [4X[25Xgap>[125X [27XG2:= ClosureGroupDefault( G, frob );;[127X[104X
    [4X[25Xgap>[125X [27Xcand2:= List( cand, M -> Normalizer( G2, M ) );;[127X[104X
    [4X[25Xgap>[125X [27Xccl:= List( cand2,[127X[104X
    [4X[25X>[125X [27X               M2 -> PcConjugacyClassReps( SylowSubgroup( M2, 2 ) ) );;[127X[104X
    [4X[25Xgap>[125X [27XList( ccl, l -> Number( l, x -> Order( x ) = 2 and not x in G ) );[127X[104X
    [4X[28X[ 0, 0, 0 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo in each case, the extension of [22XM[122X to its normalizer in [22XS.2_2[122X is non-split.
  This implies [22Xσ^'(S.2_2,s) = 0[122X.[133X
  
  [33X[0;0YFinally,  in  the  extension of [22XS[122X by the product of a graph automorphism and
  the  field automorphism, exactly that member of [22XMM(S,s)[122X is invariant that is
  invariant under the graph automorphism, hence statement (c) holds.[133X
  
  [33X[0;0YIt is again time to clean the workspace.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCleanWorkspace();[127X[104X
  [4X[32X[104X
  
  
  [1X11.5-15 [33X[0;0Y[22X∗[122X[101X[1X [22XO_9(3)[122X[101X[1X[133X[101X
  
  [33X[0;0YThe  group  [22XS = O_9(3) = Ω_9(3)[122X is the first member in the series dealt with
  in [BGK08,  Proposition 5.7],  and  serves  as an example to illustrate this
  statement.[133X
  
  [8X(a)[108X
        [33X[0;6YFor  [22Xs  ∈  S[122X  of  the type [22X1 perp 8^-[122X (i. e., [22Xs[122X decomposes the natural
        [22X9[122X-dimensional  module  for [22XS[122X into an orthogonal sum of two irreducible
        modules  of  the dimensions [22X1[122X and [22X8[122X, respectively) and of order [22X(3^4 +
        1)/2  =  41[122X,  [22XMM(S,s)[122X consists of one group of the type [22XO_8^-(3).2_1 =
        PGO^-(8,3)[122X.[133X
  
  [8X(b)[108X
        [33X[0;6Y[22Xσ(S,s) = 1/3[122X.[133X
  
  [8X(c)[108X
        [33X[0;6YThe uniform spread of [22XS[122X is at least three, with [22Xs[122X of order [22X41[122X.[133X
  
  [33X[0;0YBy [MSW94], the only maximal subgroup of [22XS[122X that contains [22Xs[122X is the stabilizer
  [22XM[122X  of  the  orthogonal decomposition. The group [22X2 × O_8^-(3).2_1 = GO^-(8,3)[122X
  embeds  naturally  into [22XSO(9,3)[122X, its intersection with [22XS[122X is [22XPGO^-(8,3)[122X. This
  proves statement (a).[133X
  
  [33X[0;0YThe group [22XM[122X is the stabilizer of a [22X1[122X-space, it has index [22X3240[122X in [22XS[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:= SO( 9, 3 );;[127X[104X
    [4X[25Xgap>[125X [27Xg:= DerivedSubgroup( g );;[127X[104X
    [4X[25Xgap>[125X [27XSize( g );[127X[104X
    [4X[28X65784756654489600[128X[104X
    [4X[25Xgap>[125X [27Xorbs:= OrbitsDomain( g, NormedRowVectors( GF(3)^9 ), OnLines );;[127X[104X
    [4X[25Xgap>[125X [27XList( orbs, Length ) / 41;[127X[104X
    [4X[28X[ 3240/41, 81, 80 ][128X[104X
    [4X[25Xgap>[125X [27XSize( SO( 9, 3 ) ) / Size( GO( -1, 8, 3 ) );[127X[104X
    [4X[28X3240[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo  we  compute  the  unique  transitive permutation character of [22XS[122X that has
  degree [22X3240[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "O9(3)" );;[127X[104X
    [4X[25Xgap>[125X [27Xpi:= PermChars( t, rec( torso:= [ 3240 ] ) );[127X[104X
    [4X[28X[ Character( CharacterTable( "O9(3)" ),[128X[104X
    [4X[28X  [ 3240, 1080, 380, 132, 48, 324, 378, 351, 0, 0, 54, 27, 54, 27, 0, [128X[104X
    [4X[28X      118, 0, 36, 46, 18, 12, 2, 8, 45, 0, 108, 108, 135, 126, 0, 0, [128X[104X
    [4X[28X      56, 0, 0, 36, 47, 38, 27, 39, 36, 24, 12, 18, 18, 15, 24, 2, [128X[104X
    [4X[28X      18, 15, 9, 0, 0, 0, 2, 0, 18, 11, 3, 9, 6, 6, 9, 6, 3, 6, 3, 0, [128X[104X
    [4X[28X      6, 16, 0, 4, 6, 2, 45, 36, 0, 0, 0, 0, 0, 0, 0, 9, 9, 6, 3, 0, [128X[104X
    [4X[28X      0, 15, 13, 0, 5, 7, 36, 0, 10, 0, 10, 19, 6, 15, 0, 0, 0, 0, [128X[104X
    [4X[28X      12, 3, 10, 0, 3, 3, 7, 0, 6, 6, 2, 8, 0, 4, 0, 2, 0, 1, 3, 0, [128X[104X
    [4X[28X      0, 3, 0, 3, 2, 2, 3, 3, 6, 2, 2, 9, 6, 3, 0, 0, 18, 9, 0, 0, [128X[104X
    [4X[28X      12, 0, 0, 8, 0, 6, 9, 5, 0, 0, 0, 0, 0, 0, 0, 0, 3, 3, 3, 2, 1, [128X[104X
    [4X[28X      3, 3, 1, 0, 0, 4, 1, 0, 0, 1, 0, 3, 3, 1, 1, 2, 2, 0, 0, 1, 3, [128X[104X
    [4X[28X      4, 0, 1, 2, 0, 0, 1, 0, 4, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, [128X[104X
    [4X[28X      1, 1, 1, 1, 0, 0, 1, 1, 1, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27Xspos:= Position( OrdersClassRepresentatives( t ), 41 );[127X[104X
    [4X[28X208[128X[104X
    [4X[25Xgap>[125X [27Xapprox:= ApproxP( pi, spos );;[127X[104X
    [4X[25Xgap>[125X [27XMaximum( approx );[127X[104X
    [4X[28X1/3[128X[104X
    [4X[25Xgap>[125X [27XPositionsProperty( approx, x -> x = 1/3 );[127X[104X
    [4X[28X[ 2 ][128X[104X
    [4X[25Xgap>[125X [27XSizesConjugacyClasses( t )[2];[127X[104X
    [4X[28X3321[128X[104X
    [4X[25Xgap>[125X [27XOrdersClassRepresentatives( t )[2];[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  see  that  [22XP( S, s ) = σ( S, s ) = 1/3[122X holds, and that [22Xσ( g, s )[122X attains
  this  maximum  only for [22Xg[122X in one class of involutions in [22XS[122X; let us call this
  class [10X2A[110X. (This class consists of the negatives of a class of [13Xreflections[113X in
  [22XGO(9,3)[122X.) This shows statement (b).[133X
  
  [33X[0;0YIn order to show that the uniform spread of [22XS[122X is at least three, it suffices
  to  show that for each triple of [10X2A[110X elements, there is an element [22Xs[122X of order
  [22X41[122X in [22XS[122X that generates [22XS[122X with each element of the triple.[133X
  
  [33X[0;0YWe  work  with the primitive permutation representation of [22XS[122X on [22X3240[122X points.
  In  this  representation, [22Xs[122X fixes exactly one point, and by statement (a), [22Xs[122X
  generates  [22XS[122X  with [22Xx ∈ S[122X if and only if [22Xx[122X moves this point. Since the number
  of fixed points of each [10X2A[110X involution in [22XS[122X is exactly one third of the moved
  points  of  [22XS[122X,  it  suffices  to  show  that  we  cannot  choose  three such
  involutions  with  mutually  disjoint  fixed  point  sets. And this is shown
  particularly  easily  because  it  will  turn  out  that already for any two
  different [10X2A[110X involutions, the sets of fixed points of are never disjoint.[133X
  
  [33X[0;0YFirst  we  compute  a  [10X2A[110X element, which is determined as an involution with
  exactly [22X1080[122X fixed points.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:= Action( g, orbs[1], OnLines );;[127X[104X
    [4X[25Xgap>[125X [27Xrepeat[127X[104X
    [4X[25X>[125X [27X     repeat x:= Random( g ); ord:= Order( x ); until ord mod 2 = 0;[127X[104X
    [4X[25X>[125X [27X     y:= x^(ord/2);[127X[104X
    [4X[25X>[125X [27Xuntil NrMovedPoints( y ) = 3240 - 1080;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YNext we compute the sets of fixed points of the elements in the class [10X2A[110X, by
  forming the [22XS[122X-orbit of the set of fixed points of the chosen [10X2A[110X element.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfp:= Difference( MovedPoints( g ), MovedPoints( y ) );;[127X[104X
    [4X[25Xgap>[125X [27Xorb:= Orbit( g, fp, OnSets );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YFinally,  we  show  that  for  any  pair of [10X2A[110X elements, their sets of fixed
  points  intersect  nontrivially.  (Of  course  we  can  fix  one  of the two
  elements.) This proves statement (c).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XForAny( orb, l -> IsEmpty( Intersection( l, fp ) ) );[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  
  [1X11.5-16 [33X[0;0Y[22XO_10^-(3)[122X[101X[1X[133X[101X
  
  [33X[0;0YWe show that the group [22XS = O_10^-(3) = PΩ^-(10,3)[122X satisfies the following.[133X
  
  [8X(a)[108X
        [33X[0;6YFor  [22Xs  ∈ S[122X irreducible of order [22X(3^5 + 1)/4 = 61[122X, [22XMM(S,s)[122X consists of
        one subgroup of the type [22XSU(5,3) ≅ U_5(3)[122X.[133X
  
  [8X(b)[108X
        [33X[0;6Y[22Xσ(S,s) = 1/1066[122X.[133X
  
  [33X[0;0YBy [Ber00],  the  maximal subgroups of [22XS[122X containing [22Xs[122X are of extension field
  type,   and  by [KL90,  Prop. 4.3.18  and 4.3.20],  these  groups  have  the
  structure  [22XSU(5,3) = U_5(3)[122X (which lift to [22X2 × U_5(3) < GU(5,3)[122X in [22XΩ^-(10,3)
  =  2.S[122X)  or  [22XΩ(5,9).2[122X, but the order of the latter group is not divisible by
  [22X|s|[122X.  Furthermore,  by [BGK08,  Lemma 2.12 (b)],  [22Xs[122X is contained in only one
  member of the former class.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSize( GO(5,9) ) / 61;[127X[104X
    [4X[28X6886425600/61[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[13XWhen  the  first  version  of  these computations was written, the character
  tables  of  both  [22XS[122X and [22XU_5(3)[122X were not contained in the [5XGAP[105X Character Table
  Library,  so  we  worked with the groups. Meanwhile the character tables are
  available, thus we can show also a character theoretic solution.)[113X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "O10-(3)" );  s:= CharacterTable( "U5(3)" );[127X[104X
    [4X[28XCharacterTable( "O10-(3)" )[128X[104X
    [4X[28XCharacterTable( "U5(3)" )[128X[104X
    [4X[25Xgap>[125X [27XSigmaFromMaxes( t, "61A", [ s ], [ 1 ] );[127X[104X
    [4X[28X1/1066[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[13X(Now follow the computations with groups.)[113X[133X
  
  [33X[0;0YThe  first step is the construction of the embedding of [22XM = SU(5,3)[122X into the
  matrix  group  [22X2.S[122X,  that  is, we write the matrix generators of [22XM[122X as linear
  mappings  on  the  natural module for [22X2.S[122X, and then conjugate them such that
  the  result  matrices  respect the bilinear form of [22X2.S[122X. For convenience, we
  choose  a  basis for the field extension [22XFF_9/FF_3[122X such that the [22XFF_3[122X-linear
  mapping  given by the invariant form of [22XM[122X is invariant under the [22XFF_3[122X-linear
  mappings given by the generators of [22XM[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm:= SU(5,3);;[127X[104X
    [4X[25Xgap>[125X [27Xso:= SO(-1,10,3);;[127X[104X
    [4X[25Xgap>[125X [27Xomega:= DerivedSubgroup( so );;[127X[104X
    [4X[25Xgap>[125X [27Xom:= InvariantBilinearForm( so ).matrix;;[127X[104X
    [4X[25Xgap>[125X [27XDisplay( om );[127X[104X
    [4X[28X . 1 . . . . . . . .[128X[104X
    [4X[28X 1 . . . . . . . . .[128X[104X
    [4X[28X . . 1 . . . . . . .[128X[104X
    [4X[28X . . . 2 . . . . . .[128X[104X
    [4X[28X . . . . 2 . . . . .[128X[104X
    [4X[28X . . . . . 2 . . . .[128X[104X
    [4X[28X . . . . . . 2 . . .[128X[104X
    [4X[28X . . . . . . . 2 . .[128X[104X
    [4X[28X . . . . . . . . 2 .[128X[104X
    [4X[28X . . . . . . . . . 2[128X[104X
    [4X[25Xgap>[125X [27Xb:= Basis( GF(9), [ Z(3)^0, Z(3^2)^2 ] );[127X[104X
    [4X[28XBasis( GF(3^2), [ Z(3)^0, Z(3^2)^2 ] )[128X[104X
    [4X[25Xgap>[125X [27Xblow:= List( GeneratorsOfGroup( m ), x -> BlownUpMat( b, x ) );;[127X[104X
    [4X[25Xgap>[125X [27Xform:= BlownUpMat( b, InvariantSesquilinearForm( m ).matrix );;[127X[104X
    [4X[25Xgap>[125X [27XForAll( blow, x -> x * form * TransposedMat( x ) = form );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XDisplay( form );[127X[104X
    [4X[28X . . . . . . . . 1 .[128X[104X
    [4X[28X . . . . . . . . . 1[128X[104X
    [4X[28X . . . . . . 1 . . .[128X[104X
    [4X[28X . . . . . . . 1 . .[128X[104X
    [4X[28X . . . . 1 . . . . .[128X[104X
    [4X[28X . . . . . 1 . . . .[128X[104X
    [4X[28X . . 1 . . . . . . .[128X[104X
    [4X[28X . . . 1 . . . . . .[128X[104X
    [4X[28X 1 . . . . . . . . .[128X[104X
    [4X[28X . 1 . . . . . . . .[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  matrix  [10Xom[110X  of  the invariant bilinear form of [22X2.S[122X is equivalent to the
  identity  matrix  [22XI[122X.  So we compute matrices [10XT1[110X and [10XT2[110X that transform [10Xom[110X and
  [10Xform[110X, respectively, to [22X± I[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XT1:= IdentityMat( 10, GF(3) );;[127X[104X
    [4X[25Xgap>[125X [27XT1{[1..3]}{[1..3]}:= [[1,1,0],[1,-1,1],[1,-1,-1]]*Z(3)^0;;[127X[104X
    [4X[25Xgap>[125X [27Xpi:= PermutationMat( (1,10)(3,8), 10, GF(3) );;[127X[104X
    [4X[25Xgap>[125X [27Xtr:= NullMat( 10,10,GF(3) );;[127X[104X
    [4X[25Xgap>[125X [27Xtr{[1, 2]}{[1, 2]}:= [[1,1],[1,-1]]*Z(3)^0;;[127X[104X
    [4X[25Xgap>[125X [27Xtr{[3, 4]}{[3, 4]}:= [[1,1],[1,-1]]*Z(3)^0;;[127X[104X
    [4X[25Xgap>[125X [27Xtr{[7, 8]}{[7, 8]}:= [[1,1],[1,-1]]*Z(3)^0;;[127X[104X
    [4X[25Xgap>[125X [27Xtr{[9,10]}{[9,10]}:= [[1,1],[1,-1]]*Z(3)^0;;[127X[104X
    [4X[25Xgap>[125X [27Xtr{[5, 6]}{[5, 6]}:= [[1,0],[0,1]]*Z(3)^0;;[127X[104X
    [4X[25Xgap>[125X [27Xtr2:= IdentityMat( 10,GF(3) );;[127X[104X
    [4X[25Xgap>[125X [27Xtr2{[1,3]}{[1,3]}:= [[-1,1],[1,1]]*Z(3)^0;;[127X[104X
    [4X[25Xgap>[125X [27Xtr2{[7,9]}{[7,9]}:= [[-1,1],[1,1]]*Z(3)^0;;[127X[104X
    [4X[25Xgap>[125X [27XT2:= tr2 * tr * pi;;[127X[104X
    [4X[25Xgap>[125X [27XD:= T1^-1 * T2;;[127X[104X
    [4X[25Xgap>[125X [27Xtblow:= List( blow, x -> D * x * D^-1 );;[127X[104X
    [4X[25Xgap>[125X [27XIsSubset( omega, tblow );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow we switch to a permutation representation of [22XS[122X, and use the embedding of
  [22XM[122X  into  [22X2.S[122X  to  obtain  the corresponding subgroup of type [22XM[122X in [22XS[122X. Then we
  compute an upper bound for [22Xmax{ μ(g,S/M); g ∈ S^× }[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xorbs:= OrbitsDomain( omega, NormedRowVectors( GF(3)^10 ), OnLines );;[127X[104X
    [4X[25Xgap>[125X [27XList( orbs, Length );[127X[104X
    [4X[28X[ 9882, 9882, 9760 ][128X[104X
    [4X[25Xgap>[125X [27Xpermgrp:= Action( omega, orbs[3], OnLines );;[127X[104X
    [4X[25Xgap>[125X [27XM:= SubgroupNC( permgrp,[127X[104X
    [4X[25X>[125X [27X           List( tblow, x -> Permutation( x, orbs[3], OnLines ) ) );;[127X[104X
    [4X[25Xgap>[125X [27Xccl:= ClassesOfPrimeOrder( M, PrimeDivisors( Size( M ) ),[127X[104X
    [4X[25X>[125X [27X                              TrivialSubgroup( M ) );;[127X[104X
    [4X[25Xgap>[125X [27XUpperBoundFixedPointRatios( permgrp, [ ccl ], false );[127X[104X
    [4X[28X[ 1/1066, true ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  entry  [9Xtrue[109X in the second position of the result indicates that in fact
  the  [13Xexact[113X value for the maximum of [22Xμ(g,S/M)[122X has been computed. This implies
  statement (b).[133X
  
  [33X[0;0YWe clean the workspace.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCleanWorkspace();[127X[104X
  [4X[32X[104X
  
  
  [1X11.5-17 [33X[0;0Y[22XO_14^-(2)[122X[101X[1X[133X[101X
  
  [33X[0;0YWe show that the group [22XS = O_14^-(2) = Ω^-(14,2)[122X satisfies the following.[133X
  
  [8X(a)[108X
        [33X[0;6YFor  [22Xs  ∈  S[122X irreducible of order [22X2^7+1 = 129[122X, [22XMM(S,s)[122X consists of one
        subgroup [22XM[122X of the type [22XGU(7,2) ≅ 3 × U_7(2)[122X.[133X
  
  [8X(b)[108X
        [33X[0;6Y[22Xσ(S,s) = 1/2015[122X.[133X
  
  [33X[0;0YBy [Ber00],  any  maximal  subgroup  of [22XS[122X containing [22Xs[122X is of extension field
  type,  and  by [KL90, Table 3.5.F, Prop. 4.3.18], these groups have the type
  [22XGU(7,2)[122X,  and  there  is  exactly  one  class  of  subgroups  of  this type.
  Furthermore,  by [BGK08,  Lemma 2.12 (a)], [22Xs[122X is contained in only one member
  of this class.[133X
  
  [33X[0;0YWe embed [22XU_7(2)[122X into [22XS[122X, by first replacing each element in [22XFF_4[122X by the [22X2 × 2[122X
  matrix  of the induced [22XFF_2[122X-linear mapping w.r.t. a suitable basis, and then
  conjugating  the  images of the generators such that the invariant quadratic
  form of [22XS[122X is respected.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xo:= SO(-1,14,2);;[127X[104X
    [4X[25Xgap>[125X [27Xg:= SU(7,2);;[127X[104X
    [4X[25Xgap>[125X [27Xb:= Basis( GF(4) );;[127X[104X
    [4X[25Xgap>[125X [27Xblow:= List( GeneratorsOfGroup( g ), x -> BlownUpMat( b, x ) );;[127X[104X
    [4X[25Xgap>[125X [27Xform:= NullMat( 14, 14, GF(2) );;[127X[104X
    [4X[25Xgap>[125X [27Xfor i in [ 1 .. 14 ] do form[i][ 15-i ]:= Z(2); od;[127X[104X
    [4X[25Xgap>[125X [27XForAll( blow, x -> x * form * TransposedMat( x ) = form );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xpi:= PermutationMat( (1,13)(3,11)(5,9), 14, GF(2) );;[127X[104X
    [4X[25Xgap>[125X [27Xpi * form * TransposedMat( pi ) = InvariantBilinearForm( o ).matrix;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xpi2:= PermutationMat( (7,3)(8,4), 14, GF(2) );;[127X[104X
    [4X[25Xgap>[125X [27XD:= pi2 * pi;;[127X[104X
    [4X[25Xgap>[125X [27Xtblow:= List( blow, x -> D * x * D^-1 );;[127X[104X
    [4X[25Xgap>[125X [27XIsSubset( o, tblow );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that the central subgroup of order three in [22XGU(7,2)[122X consists of scalar
  matrices.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xomega:= DerivedSubgroup( o );;[127X[104X
    [4X[25Xgap>[125X [27XIsSubset( omega, tblow );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xz:= Z(4) * One( g );;[127X[104X
    [4X[25Xgap>[125X [27Xtz:= D * BlownUpMat( b, z ) * D^-1;;[127X[104X
    [4X[25Xgap>[125X [27Xtz in omega;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we  switch  to  a  permutation  representation  of  [22XS[122X,  and compute the
  conjugacy  classes  of  prime element order in the subgroup [22XM[122X. The latter is
  done in two steps, first class representatives of the simple subgroup [22XU_7(2)[122X
  of [22XM[122X are computed, and then they are multiplied with the scalars in [22XM[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xorbs:= OrbitsDomain( omega, NormedRowVectors( GF(2)^14 ), OnLines );;[127X[104X
    [4X[25Xgap>[125X [27XList( orbs, Length );[127X[104X
    [4X[28X[ 8127, 8256 ][128X[104X
    [4X[25Xgap>[125X [27Xomega:= Action( omega, orbs[1], OnLines );;[127X[104X
    [4X[25Xgap>[125X [27Xgens:= List( GeneratorsOfGroup( g ),[127X[104X
    [4X[25X>[125X [27X            x -> Permutation( D * BlownUpMat( b, x ) * D^-1, orbs[1] ) );;[127X[104X
    [4X[25Xgap>[125X [27Xg:= Group( gens );;[127X[104X
    [4X[25Xgap>[125X [27Xccl:= ClassesOfPrimeOrder( g, PrimeDivisors( Size( g ) ),[127X[104X
    [4X[25X>[125X [27X                              TrivialSubgroup( g ) );;[127X[104X
    [4X[25Xgap>[125X [27Xtz:= Permutation( tz, orbs[1] );;[127X[104X
    [4X[25Xgap>[125X [27Xprimereps:= List( ccl, Representative );;[127X[104X
    [4X[25Xgap>[125X [27XAdd( primereps, () );[127X[104X
    [4X[25Xgap>[125X [27Xreps:= Concatenation( List( primereps,[127X[104X
    [4X[25X>[125X [27X              x -> List( [ 0 .. 2 ], i -> x * tz^i ) ) );;[127X[104X
    [4X[25Xgap>[125X [27Xprimereps:= Filtered( reps, x -> IsPrimeInt( Order( x ) ) );;[127X[104X
    [4X[25Xgap>[125X [27XLength( primereps );[127X[104X
    [4X[28X48[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFinally, we apply [10XUpperBoundFixedPointRatios[110X (see Section [14X11.3-3[114X) to compute
  an upper bound for [22Xμ(g,S/M)[122X, for [22Xg ∈ S^×[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XM:= ClosureGroup( g, tz );;[127X[104X
    [4X[25Xgap>[125X [27Xbccl:= List( primereps, x -> ConjugacyClass( M, x ) );;[127X[104X
    [4X[25Xgap>[125X [27XUpperBoundFixedPointRatios( omega, [ bccl ], false );[127X[104X
    [4X[28X[ 1/2015, true ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YAlthough  some  of the classes of [22XM[122X in the list [10Xbccl[110X may be [22XS[122X-conjugate, the
  entry  [9Xtrue[109X  in the second position of the result indicates that in fact the
  [13Xexact[113X  value  for  the  maximum of [22Xμ(g,S/M)[122X, for [22Xg ∈ S^×[122X, has been computed.
  This implies statement (b).[133X
  
  [33X[0;0YWe clean the workspace.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCleanWorkspace();[127X[104X
  [4X[32X[104X
  
  
  [1X11.5-18 [33X[0;0Y[22XO_12^+(3)[122X[101X[1X[133X[101X
  
  [33X[0;0YWe show that the group [22XS = O_12^+(3) = PΩ^+(12,3)[122X satisfies the following.[133X
  
  [8X(a)[108X
        [33X[0;6Y[22XS[122X  has  a maximal subgroup [22XM[122X of the type [22XN_S(PΩ^+(6,9))[122X, which has the
        structure [22XPΩ^+(6,9).[4][122X.[133X
  
  [8X(b)[108X
        [33X[0;6Y[22Xμ(g,S/M) ≤ 2/88209[122X holds for all [22Xg ∈ S^×[122X.[133X
  
  [33X[0;0Y(This  result is used in the proof of [BGK08, Proposition 5.14], where it is
  shown  that  for  [22Xs  ∈  S[122X  of  order  [22X205[122X, [22XMM(S,s)[122X consists of one reducible
  subgroup  [22XG_8[122X  and  at  most  two extension field type subgroups of the type
  [22XN_S(PΩ^+(6,9))[122X.  By [GK00,  Proposition 3.16], [22Xμ(g,S/G_8) ≤ 19/3^5[122X holds for
  all [22Xg ∈ S^×[122X. This implies [22XP(g,s) ≤ 19/3^5 + 2 ⋅ 2/88209 = 6901/88209 < 1/3[122X.)[133X
  
  [33X[0;0YStatement (a) follows from [KL90, Prop. 4.3.14].[133X
  
  [33X[0;0YFor  statement (b),  we  embed  [22XGO^+(6,9)  ≅  Ω^+(6,9).2^2[122X into [22XSO^+(12,3) =
  2.S.2[122X,  by replacing each element in [22XFF_9[122X by the [22X2 × 2[122X matrix of the induced
  [22XFF_3[122X-linear  mapping  w.r.t. a  suitable basis [22X(b_1, b_2)[122X. We choose a basis
  with  the  property [22Xb_1 = 1[122X and [22Xb_2^2 = 1 + b_2[122X, because then the image of a
  symmetric  matrix  is again symmetric (so the image of the invariant form is
  an  invariant  form  for  the  image of the group), and apply an appropriate
  transformation to the images of the generators.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xso:= SO(+1,12,3);;[127X[104X
    [4X[25Xgap>[125X [27XDisplay( InvariantBilinearForm( so ).matrix );[127X[104X
    [4X[28X . 1 . . . . . . . . . .[128X[104X
    [4X[28X 1 . . . . . . . . . . .[128X[104X
    [4X[28X . . 1 . . . . . . . . .[128X[104X
    [4X[28X . . . 2 . . . . . . . .[128X[104X
    [4X[28X . . . . 2 . . . . . . .[128X[104X
    [4X[28X . . . . . 2 . . . . . .[128X[104X
    [4X[28X . . . . . . 2 . . . . .[128X[104X
    [4X[28X . . . . . . . 2 . . . .[128X[104X
    [4X[28X . . . . . . . . 2 . . .[128X[104X
    [4X[28X . . . . . . . . . 2 . .[128X[104X
    [4X[28X . . . . . . . . . . 2 .[128X[104X
    [4X[28X . . . . . . . . . . . 2[128X[104X
    [4X[25Xgap>[125X [27Xg:= GO(+1,6,9);;[127X[104X
    [4X[25Xgap>[125X [27XZ(9)^2 = Z(3)^0 + Z(9);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xb:= Basis( GF(9), [ Z(3)^0, Z(9) ] );[127X[104X
    [4X[28XBasis( GF(3^2), [ Z(3)^0, Z(3^2) ] )[128X[104X
    [4X[25Xgap>[125X [27Xblow:= List( GeneratorsOfGroup( g ), x -> BlownUpMat( b, x ) );;[127X[104X
    [4X[25Xgap>[125X [27Xm:= BlownUpMat( b, InvariantBilinearForm( g ).matrix );;[127X[104X
    [4X[25Xgap>[125X [27XDisplay( m );[127X[104X
    [4X[28X . . 1 . . . . . . . . .[128X[104X
    [4X[28X . . . 1 . . . . . . . .[128X[104X
    [4X[28X 1 . . . . . . . . . . .[128X[104X
    [4X[28X . 1 . . . . . . . . . .[128X[104X
    [4X[28X . . . . 2 . . . . . . .[128X[104X
    [4X[28X . . . . . 2 . . . . . .[128X[104X
    [4X[28X . . . . . . 2 . . . . .[128X[104X
    [4X[28X . . . . . . . 2 . . . .[128X[104X
    [4X[28X . . . . . . . . 2 . . .[128X[104X
    [4X[28X . . . . . . . . . 2 . .[128X[104X
    [4X[28X . . . . . . . . . . 2 .[128X[104X
    [4X[28X . . . . . . . . . . . 2[128X[104X
    [4X[25Xgap>[125X [27Xpi:= PermutationMat( (2,3), 12, GF(3) );;[127X[104X
    [4X[25Xgap>[125X [27Xtr:= IdentityMat( 12, GF(3) );;[127X[104X
    [4X[25Xgap>[125X [27Xtr{[3,4]}{[3,4]}:= [[1,-1],[1,1]]*Z(3)^0;;[127X[104X
    [4X[25Xgap>[125X [27XD:= tr * pi;;[127X[104X
    [4X[25Xgap>[125X [27XD * m * TransposedMat( D ) = InvariantBilinearForm( so ).matrix;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xtblow:= List( blow, x -> D * x * D^-1 );;[127X[104X
    [4X[25Xgap>[125X [27XIsSubset( so, tblow );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  image  of [22XGO^+(6,9)[122X under the embedding into [22XSO^+(12,3)[122X does not lie in
  [22XΩ^+(12,3)  =  2.S[122X,  so  a  factor  of  two is missing in [22XGO^+(6,9) ∩ 2.S[122X for
  getting  (the preimage [22X2.M[122X of) the required maximal subgroup [22XM[122X of [22XS[122X. Because
  of  this,  and  also  because  currently it is time consuming to compute the
  derived   subgroup   of  [22XSO^+(12,3)[122X,  we  work  with  the  upward  extension
  [22XPSO^+(12,3) = S.2[122X. Note that [22XM[122X extends to a maximal subgroup of [22XS.2[122X.[133X
  
  [33X[0;0YFirst  we  factor  out the centre of [22XSO^+(12,3)[122X, and switch to a permutation
  representation of [22XS.2[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xorbs:= OrbitsDomain( so, NormedRowVectors( GF(3)^12 ), OnLines );;[127X[104X
    [4X[25Xgap>[125X [27XList( orbs, Length );[127X[104X
    [4X[28X[ 88452, 88452, 88816 ][128X[104X
    [4X[25Xgap>[125X [27Xact:= Action( so, orbs[1], OnLines );;[127X[104X
    [4X[25Xgap>[125X [27XSetSize( act, Size( so ) / 2 );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YNext we rewrite the matrix generators for [22XGO^+(6,9)[122X accordingly, and compute
  the  normalizer  in  [22XS.2[122X  of the subgroup they generate; this is the maximal
  subgroup [22XM.2[122X we need.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xu:= SubgroupNC( act,[127X[104X
    [4X[25X>[125X [27X           List( tblow, x -> Permutation( x, orbs[1], OnLines ) ) );;[127X[104X
    [4X[25Xgap>[125X [27Xn:= Normalizer( act, u );;[127X[104X
    [4X[25Xgap>[125X [27XSize( n ) / Size( u );[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we  compute  class  representatives of prime order in [22XM.2[122X, in a smaller
  faithful  permutation  representation,  and then the desired upper bound for
  [22Xμ(g, S/M)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xnorbs:= OrbitsDomain( n, MovedPoints( n ) );;[127X[104X
    [4X[25Xgap>[125X [27XList( norbs, Length );[127X[104X
    [4X[28X[ 58968, 29484 ][128X[104X
    [4X[25Xgap>[125X [27Xhom:= ActionHomomorphism( n, norbs[2] );;[127X[104X
    [4X[25Xgap>[125X [27Xnact:= Image( hom );;[127X[104X
    [4X[25Xgap>[125X [27XSize( nact ) = Size( n );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xccl:= ClassesOfPrimeOrder( nact, PrimeDivisors( Size( nact ) ),[127X[104X
    [4X[25X>[125X [27X                              TrivialSubgroup( nact ) );;[127X[104X
    [4X[25Xgap>[125X [27XLength( ccl );[127X[104X
    [4X[28X26[128X[104X
    [4X[25Xgap>[125X [27Xpreim:= List( ccl,[127X[104X
    [4X[25X>[125X [27X       x -> PreImagesRepresentative( hom, Representative( x ) ) );;[127X[104X
    [4X[25Xgap>[125X [27Xpccl:= List( preim, x -> ConjugacyClass( n, x ) );;[127X[104X
    [4X[25Xgap>[125X [27Xfor i in [ 1 .. Length( pccl ) ] do[127X[104X
    [4X[25X>[125X [27X     SetSize( pccl[i], Size( ccl[i] ) );[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27XUpperBoundFixedPointRatios( act, [ pccl ], false );[127X[104X
    [4X[28X[ 2/88209, true ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that  we  have  computed  [22Xmax{  μ(g,S.2/M.2),  g  ∈  S.2^×  }  ≥  max{
  μ(g,S.2/M.2), g ∈ S^× } = max{ μ(g,S/M), g ∈ S^× }[122X.[133X
  
  
  [1X11.5-19 [33X[0;0Y[22X∗[122X[101X[1X [22XS_4(8)[122X[101X[1X[133X[101X
  
  [33X[0;0YWe show that the group [22XS = S_4(8) = Sp(4,8)[122X satisfies the following.[133X
  
  [8X(a)[108X
        [33X[0;6YFor   [22Xs  ∈  S[122X  irreducible  of  order  [22X65[122X,  [22XMM(S,s)[122X  consists  of  two
        nonconjugate  subgroups of the type [22XS_2(64).2 = Sp(2,64).2 ≅ L_2(64).2
        ≅ O_4^-(8).2 = Ω^-(4,8).2[122X.[133X
  
  [8X(b)[108X
        [33X[0;6Y[22Xσ(S,s) = 8/63[122X.[133X
  
  [33X[0;0YBy [Ber00],  the only maximal subgroups of [22XS[122X that contain [22Xs[122X are [22XO_4^-(8).2 =
  SO^-(4,8)[122X  or of extension field type. By [KL90, Prop. 4.3.10, 4.8.6], there
  is one class of each of these subgroups (which happen to be isomorphic).[133X
  
  [33X[0;0YThese  classes  of  subgroups  induce  different permutation characters. One
  argument  to see this is that the involutions in the outer half of extension
  field  type  subgroup [22XS_2(64).2 < S_4(8)[122X have a two-dimensional fixed space,
  whereas  the  outer  involutions in [22XSO^-(4,8)[122X have a three-dimensional fixed
  space.[133X
  
  [33X[0;0YThe  former  statement  can  be  seen  by  using a normal basis of the field
  extension  [22XFF_64/FF_8[122X,  such  that  the action of the Frobenius automorphism
  (which yields a suitable outer involution) is just a double transposition on
  the basis vectors of the natural module for [22XS[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsp:= SP(4,8);;[127X[104X
    [4X[25Xgap>[125X [27XDisplay( InvariantBilinearForm( sp ).matrix );[127X[104X
    [4X[28X . . . 1[128X[104X
    [4X[28X . . 1 .[128X[104X
    [4X[28X . 1 . .[128X[104X
    [4X[28X 1 . . .[128X[104X
    [4X[25Xgap>[125X [27Xz:= Z(64);;[127X[104X
    [4X[25Xgap>[125X [27Xf:= AsField( GF(8), GF(64) );;[127X[104X
    [4X[25Xgap>[125X [27Xrepeat[127X[104X
    [4X[25X>[125X [27X     b:= Basis( f, [ z, z^8 ] );[127X[104X
    [4X[25X>[125X [27X     z:= z * Z(64);[127X[104X
    [4X[25X>[125X [27Xuntil b <> fail;[127X[104X
    [4X[25Xgap>[125X [27Xsub:= SP(2,64);;[127X[104X
    [4X[25Xgap>[125X [27XDisplay( InvariantBilinearForm( sub ).matrix );[127X[104X
    [4X[28X . 1[128X[104X
    [4X[28X 1 .[128X[104X
    [4X[25Xgap>[125X [27Xext:= Group( List( GeneratorsOfGroup( sub ),[127X[104X
    [4X[25X>[125X [27X                      x -> BlownUpMat( b, x ) ) );;[127X[104X
    [4X[25Xgap>[125X [27Xtr:= PermutationMat( (3,4), 4, GF(2) );;[127X[104X
    [4X[25Xgap>[125X [27Xconj:= ConjugateGroup( ext, tr );;[127X[104X
    [4X[25Xgap>[125X [27XIsSubset( sp, conj );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xinv:= [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]] * Z(2);;[127X[104X
    [4X[25Xgap>[125X [27Xinv in sp;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xinv in conj;[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XLength( NullspaceMat( inv - inv^0 ) );[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  latter  statement  can  be  shown  by looking at an outer involution in
  [22XSO^-(4,8)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xso:= SO(-1,4,8);;[127X[104X
    [4X[25Xgap>[125X [27Xder:= DerivedSubgroup( so );;[127X[104X
    [4X[25Xgap>[125X [27Xx:= First( GeneratorsOfGroup( so ), x -> not x in der );;[127X[104X
    [4X[25Xgap>[125X [27Xx:= x^( Order(x)/2 );;[127X[104X
    [4X[25Xgap>[125X [27XLength( NullspaceMat( x - x^0 ) );[127X[104X
    [4X[28X3[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  character  table  of  [22XL_2(64).2[122X  is  currently not available in the [5XGAP[105X
  Character  Table  Library, so we compute the possible permutation characters
  with a combinatorial approach, and show statement (a).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCharacterTable( "L2(64).2" );[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "S4(8)" );;[127X[104X
    [4X[25Xgap>[125X [27Xdegree:= Size( t ) / ( 2 * Size( SL(2,64) ) );;[127X[104X
    [4X[25Xgap>[125X [27Xpi:= PermChars( t, rec( torso:= [ degree ] ) );[127X[104X
    [4X[28X[ Character( CharacterTable( "S4(8)" ),[128X[104X
    [4X[28X  [ 2016, 0, 256, 32, 0, 36, 0, 8, 1, 0, 4, 0, 0, 0, 28, 28, 28, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 36, 36, 36, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, [128X[104X
    [4X[28X      4, 4, 4, 0, 0, 0, 4, 4, 4, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, [128X[104X
    [4X[28X      1, 1, 1 ] ), Character( CharacterTable( "S4(8)" ),[128X[104X
    [4X[28X  [ 2016, 256, 0, 32, 36, 0, 0, 8, 1, 4, 0, 28, 28, 28, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 36, 36, 36, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 4, 4, 4, [128X[104X
    [4X[28X      0, 0, 0, 4, 4, 4, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, [128X[104X
    [4X[28X      1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, [128X[104X
    [4X[28X      1, 1, 1 ] ) ][128X[104X
    [4X[25Xgap>[125X [27Xspos:= Position( OrdersClassRepresentatives( t ), 65 );;[127X[104X
    [4X[25Xgap>[125X [27XList( pi, x -> x[ spos ] );[127X[104X
    [4X[28X[ 1, 1 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow we compute [22Xσ(S,s)[122X, which yields statement (b).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMaximum( ApproxP( pi, spos ) );[127X[104X
    [4X[28X8/63[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe clean the workspace.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCleanWorkspace();[127X[104X
  [4X[32X[104X
  
  
  [1X11.5-20 [33X[0;0Y[22XS_6(2)[122X[101X[1X[133X[101X
  
  [33X[0;0YWe show that the group [22XS = S_6(2) = Sp(6,2)[122X satisfies the following.[133X
  
  [8X(a)[108X
        [33X[0;6Y[22Xσ(S)  =  4/7[122X,  and this value is attained exactly for [22Xσ(S,s)[122X with [22Xs[122X of
        order [22X9[122X.[133X
  
  [8X(b)[108X
        [33X[0;6YFor  [22Xs  ∈  S[122X  of order [22X9[122X, [22XMM(S,s)[122X consists of one subgroup of the type
        [22XU_4(2).2  =  Ω^-(6,2).2[122X  and  three  conjugate  subgroups  of the type
        [22XL_2(8).3 = Sp(2,8).3[122X.[133X
  
  [8X(c)[108X
        [33X[0;6YFor  [22Xs  ∈ S[122X of order [22X9[122X, and [22Xg ∈ S^×[122X, we have [22XP(g,s) < 1/3[122X, except if [22Xg[122X
        is in one of the classes [10X2A[110X (the transvection class) or [10X3A[110X.[133X
  
  [8X(d)[108X
        [33X[0;6YFor  [22Xs ∈ S[122X of order [22X15[122X, and [22Xg ∈ S^×[122X, we have [22XP(g,s) < 1/3[122X, except if [22Xg[122X
        is in one of the classes [10X2A[110X or [10X2B[110X.[133X
  
  [8X(e)[108X
        [33X[0;6Y[22XP(S)  = 11/21[122X, and this value is attained exactly for [22XP(S,s)[122X with [22Xs[122X of
        order [22X15[122X.[133X
  
  [8X(f)[108X
        [33X[0;6YFor all [22Xs^' ∈ S[122X, we have [22XP(g,s^') > 1/3[122X for [22Xg[122X in at least two classes.[133X
  
  [8X(g)[108X
        [33X[0;6YThe uniform spread of [22XS[122X is at least two, with [22Xs[122X of order [22X9[122X.[133X
  
  [33X[0;0Y(Note  that  in  this  example, the optimal choice of [22Xs[122X w.r.t. [22Xσ(S,s)[122X is not
  optimal w.r.t. [22XP(S,s)[122X.)[133X
  
  [33X[0;0YStatement (a)  follows  from  the  inspection  of  the primitive permutation
  characters, cf. Section [14X11.4-3[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "S6(2)" );;[127X[104X
    [4X[25Xgap>[125X [27XProbGenInfoSimple( t );[127X[104X
    [4X[28X[ "S6(2)", 4/7, 1, [ "9A" ], [ 4 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YAlso  statement (b)  follows  from the information provided by the character
  table of [22XS[122X (cf. [CCN+85, p. 46]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xprim:= PrimitivePermutationCharacters( t );;[127X[104X
    [4X[25Xgap>[125X [27Xord:= OrdersClassRepresentatives( t );;[127X[104X
    [4X[25Xgap>[125X [27Xspos:= Position( ord, 9 );;[127X[104X
    [4X[25Xgap>[125X [27Xfilt:= PositionsProperty( prim, x -> x[ spos ] <> 0 );[127X[104X
    [4X[28X[ 1, 8 ][128X[104X
    [4X[25Xgap>[125X [27XMaxes( t ){ filt };[127X[104X
    [4X[28X[ "U4(2).2", "L2(8).3" ][128X[104X
    [4X[25Xgap>[125X [27XList( prim{ filt }, x -> x[ spos ] );[127X[104X
    [4X[28X[ 1, 3 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we consider statement (c). For [22Xs[122X of order [22X9[122X and [22Xg[122X in one of the classes
  [10X2A[110X,  [10X3A[110X,  we observe that [22XP(g,s) = σ(g,s)[122X holds. This is because exactly one
  maximal  subgroup  of  [22XS[122X contains both [22Xs[122X and [22Xg[122X. For all other elements [22Xg[122X, we
  have even [22Xσ(g,s) < 1/3[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xprim:= PrimitivePermutationCharacters( t );;[127X[104X
    [4X[25Xgap>[125X [27Xspos9:= Position( ord, 9 );;[127X[104X
    [4X[25Xgap>[125X [27Xapprox9:= ApproxP( prim, spos9 );;[127X[104X
    [4X[25Xgap>[125X [27Xfilt9:= PositionsProperty( approx9, x -> x >= 1/3 );[127X[104X
    [4X[28X[ 2, 6 ][128X[104X
    [4X[25Xgap>[125X [27XAtlasClassNames( t ){ filt9 };[127X[104X
    [4X[28X[ "2A", "3A" ][128X[104X
    [4X[25Xgap>[125X [27Xapprox9{ filt9 };[127X[104X
    [4X[28X[ 4/7, 5/14 ][128X[104X
    [4X[25Xgap>[125X [27XList( Filtered( prim, x -> x[ spos9 ] <> 0 ), x -> x{ filt9 } );[127X[104X
    [4X[28X[ [ 16, 10 ], [ 0, 0 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YSimilarly,  statement (d)  follows.  For  [22Xs[122X  of order [22X15[122X and [22Xg[122X in one of the
  classes  [10X2A[110X, [10X2B[110X, already the degree [22X36[122X permutation character yields [22XP(g,s) ≥
  1/3[122X. And for all other elements [22Xg[122X, again we have [22Xσ(g,s) < 1/3[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xspos15:= Position( ord, 15 );;[127X[104X
    [4X[25Xgap>[125X [27Xapprox15:= ApproxP( prim, spos15 );;[127X[104X
    [4X[25Xgap>[125X [27Xfilt15:= PositionsProperty( approx15, x -> x >= 1/3 );[127X[104X
    [4X[28X[ 2, 3 ][128X[104X
    [4X[25Xgap>[125X [27XPositionsProperty( ApproxP( prim{ [ 2 ] }, spos15 ), x -> x >= 1/3 );[127X[104X
    [4X[28X[ 2, 3 ][128X[104X
    [4X[25Xgap>[125X [27XAtlasClassNames( t ){ filt15 };[127X[104X
    [4X[28X[ "2A", "2B" ][128X[104X
    [4X[25Xgap>[125X [27Xapprox15{ filt15 };[127X[104X
    [4X[28X[ 46/63, 8/21 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  the  remaining  statements, we use explicit computations with [22XS[122X, in the
  transitive  degree  [22X63[122X  permutation representation. We start with a function
  that  computes  a  transvection  in [22XS_d(2)[122X; note that the invariant bilinear
  form used for symplectic groups in [5XGAP[105X is described by a matrix with nonzero
  entries exactly in the positions [22X(i,d+1-i)[122X, for [22X1 ≤ i ≤ d[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtransvection:= function( d )[127X[104X
    [4X[25X>[125X [27X    local mat;[127X[104X
    [4X[25X>[125X [27X    mat:= IdentityMat( d, Z(2) );[127X[104X
    [4X[25X>[125X [27X    mat{ [ 1, d ] }{ [ 1, d ] }:= [ [ 0, 1 ], [ 1, 0 ] ] * Z(2);[127X[104X
    [4X[25X>[125X [27X    return mat;[127X[104X
    [4X[25X>[125X [27Xend;;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YFirst we compute, for statement (d), the exact values [22XP(g,s)[122X for [22Xg[122X in one of
  the  classes  [10X2A[110X  or [10X2B[110X, and [22Xs[122X of order [22X15[122X. Note that the classes [10X2A[110X, [10X2B[110X are
  the unique classes of the lengths [22X63[122X and [22X315[122X, respectively.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPositionsProperty( SizesConjugacyClasses( t ), x -> x in [ 63, 315 ] );[127X[104X
    [4X[28X[ 2, 3 ][128X[104X
    [4X[25Xgap>[125X [27Xd:= 6;;[127X[104X
    [4X[25Xgap>[125X [27Xmatgrp:= Sp(d,2);;[127X[104X
    [4X[25Xgap>[125X [27Xhom:= ActionHomomorphism( matgrp, NormedRowVectors( GF(2)^d ) );;[127X[104X
    [4X[25Xgap>[125X [27Xg:= Image( hom, matgrp );;[127X[104X
    [4X[25Xgap>[125X [27XResetGlobalRandomNumberGenerators();[127X[104X
    [4X[25Xgap>[125X [27Xrepeat s15:= Random( g );[127X[104X
    [4X[25X>[125X [27X   until Order( s15 ) = 15;[127X[104X
    [4X[25Xgap>[125X [27X2A:= Image( hom, transvection( d ) );;[127X[104X
    [4X[25Xgap>[125X [27XSize( ConjugacyClass( g, 2A ) );[127X[104X
    [4X[28X63[128X[104X
    [4X[25Xgap>[125X [27XIsTransitive( g, MovedPoints( g ) );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XRatioOfNongenerationTransPermGroup( g, 2A, s15 );[127X[104X
    [4X[28X11/21[128X[104X
    [4X[25Xgap>[125X [27Xrepeat 12C:= Random( g );[127X[104X
    [4X[25X>[125X [27X   until Order( 12C ) = 12 and Size( Centralizer( g, 12C ) ) = 12;[127X[104X
    [4X[25Xgap>[125X [27X2B:= 12C^6;;[127X[104X
    [4X[25Xgap>[125X [27XSize( ConjugacyClass( g, 2B ) );[127X[104X
    [4X[28X315[128X[104X
    [4X[25Xgap>[125X [27XRatioOfNongenerationTransPermGroup( g, 2B, s15 );[127X[104X
    [4X[28X8/21[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  statement (e),  we  compute  [22XP(g,  s^')[122X, for a transvection [22Xg[122X and class
  representatives  [22Xs^'[122X of [22XS[122X. It turns out that the minimum is [22X11/21[122X, and it is
  attained for exactly one [22Xs^'[122X; by the above, this element has order [22X15[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xccl:= ConjugacyClasses( g );;[127X[104X
    [4X[25Xgap>[125X [27Xreps:= List( ccl, Representative );;[127X[104X
    [4X[25Xgap>[125X [27Xnongen2A:= List( reps,[127X[104X
    [4X[25X>[125X [27X       x -> RatioOfNongenerationTransPermGroup( g, 2A, x ) );;[127X[104X
    [4X[25Xgap>[125X [27Xmin:= Minimum( nongen2A );[127X[104X
    [4X[28X11/21[128X[104X
    [4X[25Xgap>[125X [27XNumber( nongen2A, x -> x = min );[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  statement (f),  we show that for any choice of [22Xs^'[122X, at least two of the
  values [22XP(g,s^')[122X, with [22Xg[122X in the classes [10X2A[110X, [10X2B[110X, or [10X3A[110X, are larger than [22X1/3[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xnongen2B:= List( reps,[127X[104X
    [4X[25X>[125X [27X       x -> RatioOfNongenerationTransPermGroup( g, 2B, x ) );;[127X[104X
    [4X[25Xgap>[125X [27X3A:= s15^5;;[127X[104X
    [4X[25Xgap>[125X [27Xnongen3A:= List( reps,[127X[104X
    [4X[25X>[125X [27X       x -> RatioOfNongenerationTransPermGroup( g, 3A, x ) );;[127X[104X
    [4X[25Xgap>[125X [27Xbad:= List( [ 1 .. NrConjugacyClasses( t ) ],[127X[104X
    [4X[25X>[125X [27X               i -> Number( [ nongen2A, nongen2B, nongen3A ],[127X[104X
    [4X[25X>[125X [27X                            x -> x[i] > 1/3 ) );;[127X[104X
    [4X[25Xgap>[125X [27XMinimum( bad );[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFinally,  for  statement (g), we have to consider only the case that the two
  elements [22Xx[122X, [22Xy[122X are transvections.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPositionsProperty( approx9, x -> x + approx9[2] >= 1 );[127X[104X
    [4X[28X[ 2 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe use the random approach described in Section [14X11.3-3[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xrepeat s9:= Random( g );[127X[104X
    [4X[25X>[125X [27X   until Order( s9 ) = 9;[127X[104X
    [4X[25Xgap>[125X [27XRandomCheckUniformSpread( g, [ 2A, 2A ], s9, 20 );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X11.5-21 [33X[0;0Y[22XS_8(2)[122X[101X[1X[133X[101X
  
  [33X[0;0YWe show that the group [22XS = S_8(2)[122X satisfies the following.[133X
  
  [8X(a)[108X
        [33X[0;6YFor [22Xs ∈ S[122X of order [22X17[122X, [22XMM(S,s)[122X consists of one subgroup of each of the
        types  [22XO_8^-(2).2  =  Ω^-(8,2).2[122X,  [22XS_4(4).2 = Sp(4,4).2[122X, and [22XL_2(17) =
        PSL(2,17)[122X.[133X
  
  [8X(b)[108X
        [33X[0;6YFor  [22Xs ∈ S[122X of order [22X17[122X, and [22Xg ∈ S^×[122X, we have [22XP(g,s) < 1/3[122X, except if [22Xg[122X
        is a transvection.[133X
  
  [8X(c)[108X
        [33X[0;6YThe uniform spread of [22XS[122X is at least two, with [22Xs[122X of order [22X17[122X.[133X
  
  [33X[0;0YStatement (a)  follows  from  the list of maximal subgroups of [22XS[122X in [CCN+85,
  p. 123],  and  the  fact  that [22X1_H^S(s) = 1[122X holds for each [22XH ∈ MM(S,s)[122X. Note
  that [22X17[122X divides the indices of the maximal subgroups of the types [22XO_8^+(2).2[122X
  and  [22X2^7  :  S_6(2)[122X in [22XS[122X, and obviously [22X17[122X does not divide the orders of the
  remaining maximal subgroups.[133X
  
  [33X[0;0YThe permutation characters induced from the first two subgroups are uniquely
  determined  by  the  ordinary  character  tables.  The permutation character
  induced  from the last subgroup is uniquely determined if one considers also
  the  corresponding  Brauer tables; the correct class fusion is stored in the
  [5XGAP[105X Character Table Library, see [Brea].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "S8(2)" );;[127X[104X
    [4X[25Xgap>[125X [27Xpi1:= PossiblePermutationCharacters( CharacterTable( "O8-(2).2" ), t );;[127X[104X
    [4X[25Xgap>[125X [27Xpi2:= PossiblePermutationCharacters( CharacterTable( "S4(4).2" ), t );;[127X[104X
    [4X[25Xgap>[125X [27Xpi3:= [ TrivialCharacter( CharacterTable( "L2(17)" ) )^t ];;[127X[104X
    [4X[25Xgap>[125X [27Xprim:= Concatenation( pi1, pi2, pi3 );;[127X[104X
    [4X[25Xgap>[125X [27XLength( prim );[127X[104X
    [4X[28X3[128X[104X
    [4X[25Xgap>[125X [27Xspos:= Position( OrdersClassRepresentatives( t ), 17 );;[127X[104X
    [4X[25Xgap>[125X [27XList( prim, x -> x[ spos ] );[127X[104X
    [4X[28X[ 1, 1, 1 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  statement (b), we observe that [22Xσ(g,s) < 1/3[122X if [22Xg[122X is not a transvection,
  and  that  [22XP(g,s)  =  σ(g,s)[122X  for transvections [22Xg[122X because exactly one of the
  three  permutation  characters  is  nonzero  on  both  [22Xs[122X  and  the  class of
  transvections.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xapprox:= ApproxP( prim, spos );;[127X[104X
    [4X[25Xgap>[125X [27XPositionsProperty( approx, x -> x >= 1/3 );[127X[104X
    [4X[28X[ 2 ][128X[104X
    [4X[25Xgap>[125X [27XNumber( prim, pi -> pi[2] <> 0 and pi[ spos ] <> 0 );[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27Xapprox[2];[127X[104X
    [4X[28X8/15[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn statement (c), we have to consider only the case that the two elements [22Xx[122X,
  [22Xy[122X are transvections.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPositionsProperty( approx, x -> x + approx[2] >= 1 );[127X[104X
    [4X[28X[ 2 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe use the random approach described in Section [14X11.3-3[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xd:= 8;;[127X[104X
    [4X[25Xgap>[125X [27Xmatgrp:= Sp(d,2);;[127X[104X
    [4X[25Xgap>[125X [27Xhom:= ActionHomomorphism( matgrp, NormedRowVectors( GF(2)^d ) );;[127X[104X
    [4X[25Xgap>[125X [27Xx:= Image( hom, transvection( d ) );;[127X[104X
    [4X[25Xgap>[125X [27Xg:= Image( hom, matgrp );;[127X[104X
    [4X[25Xgap>[125X [27XC:= ConjugacyClass( g, x );;  Size( C );[127X[104X
    [4X[28X255[128X[104X
    [4X[25Xgap>[125X [27XResetGlobalRandomNumberGenerators();[127X[104X
    [4X[25Xgap>[125X [27Xrepeat s:= Random( g );[127X[104X
    [4X[25X>[125X [27X   until Order( s ) = 17;[127X[104X
    [4X[25Xgap>[125X [27XRandomCheckUniformSpread( g, [ x, x ], s, 20 );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X11.5-22 [33X[0;0Y[22X∗[122X[101X[1X [22XS_10(2)[122X[101X[1X[133X[101X
  
  [33X[0;0YWe show that the group [22XS = S_10(2)[122X satisfies the following.[133X
  
  [8X(a)[108X
        [33X[0;6YFor [22Xs ∈ S[122X of order [22X33[122X, [22XMM(S,s)[122X consists of one subgroup of each of the
        types [22XΩ^-(10,2).2[122X and [22XL_2(32).5 = Sp(2,32).5[122X.[133X
  
  [8X(b)[108X
        [33X[0;6YFor  [22Xs ∈ S[122X of order [22X33[122X, and [22Xg ∈ S^×[122X, we have [22XP(g,s) < 1/3[122X, except if [22Xg[122X
        is a transvection.[133X
  
  [8X(c)[108X
        [33X[0;6YThe uniform spread of [22XS[122X is at least two, with [22Xs[122X of order [22X33[122X.[133X
  
  [33X[0;0YBy [Ber00],  the  only  maximal subgroups of [22XS[122X that contain [22Xs[122X have the types
  stated  in (a),  and by [KL90, Prop. 4.3.10 and 4.8.6], there is exactly one
  class of each of these subgroups.[133X
  
  [33X[0;0YWe compute the values [22Xσ( g, s )[122X, for all [22Xg ∈ S^×[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "S10(2)" );;[127X[104X
    [4X[25Xgap>[125X [27Xpi1:= PossiblePermutationCharacters( CharacterTable( "O10-(2).2" ), t );;[127X[104X
    [4X[25Xgap>[125X [27Xpi2:= PossiblePermutationCharacters( CharacterTable( "L2(32).5" ), t );;[127X[104X
    [4X[25Xgap>[125X [27Xprim:= Concatenation( pi1, pi2 );;  Length( prim );[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27Xspos:= Position( OrdersClassRepresentatives( t ), 33 );;[127X[104X
    [4X[25Xgap>[125X [27Xapprox:= ApproxP( prim, spos );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  statement (b), we observe that [22Xσ(g,s) < 1/3[122X if [22Xg[122X is not a transvection,
  and  that [22XP(g,s) = σ(g,s)[122X for transvections [22Xg[122X because exactly one of the two
  permutation characters is nonzero on both [22Xs[122X and the class of transvections.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPositionsProperty( approx, x -> x >= 1/3 );[127X[104X
    [4X[28X[ 2 ][128X[104X
    [4X[25Xgap>[125X [27XNumber( prim, pi -> pi[2] <> 0 and pi[ spos ] <> 0 );[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27Xapprox[2];[127X[104X
    [4X[28X16/31[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn statement (c), we have to consider only the case that the two elements [22Xx[122X,
  [22Xy[122X are transvections. We use the random approach described in Section [14X11.3-3[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xd:= 10;;[127X[104X
    [4X[25Xgap>[125X [27Xmatgrp:= Sp(d,2);;[127X[104X
    [4X[25Xgap>[125X [27Xhom:= ActionHomomorphism( matgrp, NormedRowVectors( GF(2)^d ) );;[127X[104X
    [4X[25Xgap>[125X [27Xx:= Image( hom, transvection( d ) );;[127X[104X
    [4X[25Xgap>[125X [27Xg:= Image( hom, matgrp );;[127X[104X
    [4X[25Xgap>[125X [27XC:= ConjugacyClass( g, x );;  Size( C );[127X[104X
    [4X[28X1023[128X[104X
    [4X[25Xgap>[125X [27XResetGlobalRandomNumberGenerators();[127X[104X
    [4X[25Xgap>[125X [27Xrepeat s:= Random( g );[127X[104X
    [4X[25X>[125X [27X   until Order( s ) = 33;[127X[104X
    [4X[25Xgap>[125X [27XRandomCheckUniformSpread( g, [ x, x ], s, 20 );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X11.5-23 [33X[0;0Y[22XU_4(2)[122X[101X[1X[133X[101X
  
  [33X[0;0YWe  show  that  [22XS  =  U_4(2)  =  SU(4,2)  ≅  S_4(3) = PSp(4,3)[122X satisfies the
  following.[133X
  
  [8X(a)[108X
        [33X[0;6Y[22Xσ(S)  = 21/40[122X, and this value is attained exactly for [22Xσ(S,s)[122X with [22Xs[122X of
        order [22X12[122X.[133X
  
  [8X(b)[108X
        [33X[0;6YFor  [22Xs  ∈  S[122X  of order [22X9[122X, [22XMM(S,s)[122X consists of two groups, of the types
        [22X3^1+2_+ : 2A_4 = GU(3,2)[122X and [22X3^3 : S_4[122X, respectively.[133X
  
  [8X(c)[108X
        [33X[0;6Y[22XP(S)  =  2/5[122X,  and this value is attained exactly for [22XP(S,s)[122X with [22Xs[122X of
        order [22X9[122X.[133X
  
  [8X(d)[108X
        [33X[0;6YThe uniform spread of [22XS[122X is at least three, with [22Xs[122X of order [22X9[122X.[133X
  
  [8X(e)[108X
        [33X[0;6Y[22Xσ^'(Aut(S),s) = 7/20[122X.[133X
  
  [33X[0;0Y(Note  that  in  this  example, the optimal choice of [22Xs[122X w.r.t. [22Xσ(S,s)[122X is not
  optimal w.r.t. [22XP(S,s)[122X.)[133X
  
  [33X[0;0YStatement (a)   follows   from   inspection  of  the  primitive  permutation
  characters, cf. Section [14X11.4-3[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "U4(2)" );;[127X[104X
    [4X[25Xgap>[125X [27XProbGenInfoSimple( t );[127X[104X
    [4X[28X[ "U4(2)", 21/40, 1, [ "12A" ], [ 2 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YStatement (b)  can be read off from the permutation characters, and the fact
  that  the only classes of maximal subgroups that contain elements of order [22X9[122X
  consist  of  groups of the structures [22X3^1+2_+:2A_4[122X and [22X3^3:S_4[122X, see [CCN+85,
  p. 26].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XOrdersClassRepresentatives( t );[127X[104X
    [4X[28X[ 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 6, 6, 6, 6, 6, 6, 9, 9, 12, 12 ][128X[104X
    [4X[25Xgap>[125X [27Xprim:= PrimitivePermutationCharacters( t );[127X[104X
    [4X[28X[ Character( CharacterTable( "U4(2)" ),[128X[104X
    [4X[28X  [ 27, 3, 7, 0, 0, 9, 0, 3, 1, 2, 0, 0, 3, 3, 0, 1, 0, 0, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "U4(2)" ),[128X[104X
    [4X[28X  [ 36, 12, 8, 0, 0, 6, 3, 0, 2, 1, 0, 0, 0, 0, 3, 2, 0, 0, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "U4(2)" ),[128X[104X
    [4X[28X  [ 40, 8, 0, 13, 13, 4, 4, 4, 0, 0, 5, 5, 2, 2, 2, 0, 1, 1, 1, 1 ] ),[128X[104X
    [4X[28X  Character( CharacterTable( "U4(2)" ),[128X[104X
    [4X[28X  [ 40, 16, 4, 4, 4, 1, 7, 0, 2, 0, 4, 4, 1, 1, 1, 1, 1, 1, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "U4(2)" ),[128X[104X
    [4X[28X  [ 45, 13, 5, 9, 9, 6, 3, 1, 1, 0, 1, 1, 4, 4, 1, 2, 0, 0, 1, 1 ] ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  statement (c),  we  use  a  primitive  permutation representation on [22X40[122X
  points that occurs in the natural action of [22XSU(4,2)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:= SU(4,2);;[127X[104X
    [4X[25Xgap>[125X [27Xorbs:= OrbitsDomain( g, NormedRowVectors( GF(4)^4 ), OnLines );;[127X[104X
    [4X[25Xgap>[125X [27XList( orbs, Length );[127X[104X
    [4X[28X[ 45, 40 ][128X[104X
    [4X[25Xgap>[125X [27Xg:= Action( g, orbs[2], OnLines );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YFirst  we  show that for [22Xs[122X of order [22X9[122X, [22XP(S,s) = 2/5[122X holds. For that, we have
  to  consider only [22XP(g,s)[122X, with [22Xg[122X in one of the classes [10X2A[110X (of length [22X45[122X) and
  [10X3A[110X  (of length [22X40[122X); since the class [10X3B[110X contains the inverses of the elements
  in the class [10X3A[110X, we need not test it.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xspos:= Position( OrdersClassRepresentatives( t ), 9 );[127X[104X
    [4X[28X17[128X[104X
    [4X[25Xgap>[125X [27Xapprox:= ApproxP( prim, spos );[127X[104X
    [4X[28X[ 0, 3/5, 1/10, 17/40, 17/40, 1/8, 11/40, 1/10, 1/20, 0, 9/40, 9/40, [128X[104X
    [4X[28X  3/40, 3/40, 3/40, 1/40, 1/20, 1/20, 1/40, 1/40 ][128X[104X
    [4X[25Xgap>[125X [27Xbadpos:= PositionsProperty( approx, x -> x >= 2/5 );[127X[104X
    [4X[28X[ 2, 4, 5 ][128X[104X
    [4X[25Xgap>[125X [27XPowerMap( t, 2 )[4];[127X[104X
    [4X[28X5[128X[104X
    [4X[25Xgap>[125X [27XOrdersClassRepresentatives( t );[127X[104X
    [4X[28X[ 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 6, 6, 6, 6, 6, 6, 9, 9, 12, 12 ][128X[104X
    [4X[25Xgap>[125X [27XSizesConjugacyClasses( t );[127X[104X
    [4X[28X[ 1, 45, 270, 40, 40, 240, 480, 540, 3240, 5184, 360, 360, 720, 720, [128X[104X
    [4X[28X  1440, 2160, 2880, 2880, 2160, 2160 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YA  representative  [22Xg[122X of a class of length [22X40[122X can be found as the third power
  of any order [22X9[122X element.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPowerMap( t, 3 )[ spos ];[127X[104X
    [4X[28X4[128X[104X
    [4X[25Xgap>[125X [27XResetGlobalRandomNumberGenerators();[127X[104X
    [4X[25Xgap>[125X [27Xrepeat s:= Random( g );[127X[104X
    [4X[25X>[125X [27X   until Order( s ) = 9;[127X[104X
    [4X[25Xgap>[125X [27XSize( ConjugacyClass( g, s^3 ) );[127X[104X
    [4X[28X40[128X[104X
    [4X[25Xgap>[125X [27Xprop:= RatioOfNongenerationTransPermGroup( g, s^3, s );[127X[104X
    [4X[28X13/40[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNext we examine [22Xg[122X in the class [10X2A[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xrepeat x:= Random( g ); until Order( x ) = 12;[127X[104X
    [4X[25Xgap>[125X [27XSize( ConjugacyClass( g, x^6 ) );[127X[104X
    [4X[28X45[128X[104X
    [4X[25Xgap>[125X [27Xprop:= RatioOfNongenerationTransPermGroup( g, x^6, s );[127X[104X
    [4X[28X2/5[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFinally,  we compute that for [22Xs[122X of order different from [22X9[122X and [22Xg[122X in the class
  [10X2A[110X, [22XP(g,s)[122X is larger than [22X2/5[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xccl:= List( ConjugacyClasses( g ), Representative );;[127X[104X
    [4X[25Xgap>[125X [27XSortParallel( List( ccl, Order ), ccl );[127X[104X
    [4X[25Xgap>[125X [27XList( ccl, Order );[127X[104X
    [4X[28X[ 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 6, 6, 6, 6, 6, 6, 9, 9, 12, 12 ][128X[104X
    [4X[25Xgap>[125X [27Xprop:= List( ccl, r -> RatioOfNongenerationTransPermGroup( g, x^6, r ) );[127X[104X
    [4X[28X[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 5/9, 1, 1, 1, 1, 1, 1, 2/5, 2/5, 7/15, [128X[104X
    [4X[28X  7/15 ][128X[104X
    [4X[25Xgap>[125X [27XMinimum( prop );[127X[104X
    [4X[28X2/5[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  order to show statement (d), we have to consider triples [22X(x_1, x_2, x_3)[122X
  with  [22Xx_i[122X  of  prime  order  and  [22X∑_i=1^3  P(x_i,s)  ≥ 1[122X. This means that it
  suffices to check [22Xx[122X in the class [10X2A[110X, [22Xy[122X in [10X2A[110X[22X∪[122X[10X3A[110X, and [22Xz[122X in [10X2A[110X[22X∪[122X[10X3A[110X[22X∪[122X[10X3D[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xapprox[2]:= 2/5;;[127X[104X
    [4X[25Xgap>[125X [27Xapprox[4]:= 13/40;;[127X[104X
    [4X[25Xgap>[125X [27Xprimeord:= PositionsProperty( OrdersClassRepresentatives( t ),[127X[104X
    [4X[25X>[125X [27X                                 IsPrimeInt );[127X[104X
    [4X[28X[ 2, 3, 4, 5, 6, 7, 10 ][128X[104X
    [4X[25Xgap>[125X [27XRemoveSet( primeord, 5 );[127X[104X
    [4X[25Xgap>[125X [27Xprimeord;[127X[104X
    [4X[28X[ 2, 3, 4, 6, 7, 10 ][128X[104X
    [4X[25Xgap>[125X [27Xapprox{ primeord };[127X[104X
    [4X[28X[ 2/5, 1/10, 13/40, 1/8, 11/40, 0 ][128X[104X
    [4X[25Xgap>[125X [27XAtlasClassNames( t ){ primeord };[127X[104X
    [4X[28X[ "2A", "2B", "3A", "3C", "3D", "5A" ][128X[104X
    [4X[25Xgap>[125X [27Xtriples:= Filtered( UnorderedTuples( primeord, 3 ),[127X[104X
    [4X[25X>[125X [27X                 t -> Sum( approx{ t } ) >= 1 );[127X[104X
    [4X[28X[ [ 2, 2, 2 ], [ 2, 2, 4 ], [ 2, 2, 7 ], [ 2, 4, 4 ], [ 2, 4, 7 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe use the random approach described in Section [14X11.3-3[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xrepeat 6E:= Random( g );[127X[104X
    [4X[25X>[125X [27X   until Order( 6E ) = 6 and Size( Centralizer( g, 6E ) ) = 18;[127X[104X
    [4X[25Xgap>[125X [27X2A:= 6E^3;;[127X[104X
    [4X[25Xgap>[125X [27X3A:= s^3;;[127X[104X
    [4X[25Xgap>[125X [27X3D:= 6E^2;;[127X[104X
    [4X[25Xgap>[125X [27XRandomCheckUniformSpread( g, [ 2A, 2A, 2A ], s, 50 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XRandomCheckUniformSpread( g, [ 2A, 2A, 3A ], s, 50 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XRandomCheckUniformSpread( g, [ 3D, 2A, 2A ], s, 50 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XRandomCheckUniformSpread( g, [ 2A, 3A, 3A ], s, 50 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XRandomCheckUniformSpread( g, [ 3D, 3A, 2A ], s, 50 );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YStatement (e)    can    be   proved   using   [10XProbGenInfoAlmostSimple[110X,   cf.
  Section [14X11.4-4[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "U4(2)" );;[127X[104X
    [4X[25Xgap>[125X [27Xt2:= CharacterTable( "U4(2).2" );;[127X[104X
    [4X[25Xgap>[125X [27Xspos:= PositionsProperty( OrdersClassRepresentatives( t ), x -> x = 9 );;[127X[104X
    [4X[25Xgap>[125X [27XProbGenInfoAlmostSimple( t, t2, spos );[127X[104X
    [4X[28X[ "U4(2).2", 7/20, [ "9AB" ], [ 2 ] ][128X[104X
  [4X[32X[104X
  
  
  [1X11.5-24 [33X[0;0Y[22XU_4(3)[122X[101X[1X[133X[101X
  
  [33X[0;0YWe show that [22XS = U_4(3) = PSU(4,3)[122X satisfies the following.[133X
  
  [8X(a)[108X
        [33X[0;6Y[22Xσ(S) = 53/153[122X, and this value is attained exactly for [22Xσ(S,s)[122X with [22Xs[122X of
        order [22X7[122X.[133X
  
  [8X(b)[108X
        [33X[0;6YFor  [22Xs  ∈ S[122X of order [22X7[122X, [22XMM(S,s)[122X consists of two nonconjugate groups of
        the  type  [22XL_3(4)[122X,  one  group  of  the type [22XU_3(3)[122X, and four pairwise
        nonconjugate groups of the type [22XA_7[122X.[133X
  
  [8X(c)[108X
        [33X[0;6Y[22XP(S) = 43/135[122X, and this value is attained exactly for [22XP(S,s)[122X with [22Xs[122X of
        order [22X7[122X.[133X
  
  [8X(d)[108X
        [33X[0;6YThe uniform spread of [22XS[122X is at least three, with [22Xs[122X of order [22X7[122X.[133X
  
  [8X(e)[108X
        [33X[0;6YThe preimage of [22Xs[122X in the matrix group [22XSU(4,3) ≅ 4.U_4(3)[122X has order [22X28[122X,
        the preimages of the groups in [22XMM(S,s)[122X have the structures [22X4_2.L_3(4)[122X,
        [22X4 × U_3(3) ≅ GU(3,3)[122X, and [22X4.A_7[122X (the latter being a central product of
        a cyclic group of order four and [22X2.A_7[122X).[133X
  
  [8X(f)[108X
        [33X[0;6Y[22XP^'(S.2_1,s)  = 13/27[122X, [22Xσ^'(S.2_2) = 1/3[122X, and [22Xσ^'(S.2_3) = 31/162[122X, with
        [22Xs[122X of order [22X7[122X in each case.[133X
  
  [33X[0;0YStatement (a)   follows   from   inspection  of  the  primitive  permutation
  characters, cf. Section [14X11.4-3[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "U4(3)" );;[127X[104X
    [4X[25Xgap>[125X [27XProbGenInfoSimple( t );[127X[104X
    [4X[28X[ "U4(3)", 53/135, 2, [ "7A" ], [ 7 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YStatement (b)  can be read off from the permutation characters, and the fact
  that  the only classes of maximal subgroups that contain elements of order [22X7[122X
  consist of groups of the structures as claimed, see [CCN+85, p. 52].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xprim:= PrimitivePermutationCharacters( t );;[127X[104X
    [4X[25Xgap>[125X [27Xspos:= Position( OrdersClassRepresentatives( t ), 7 );[127X[104X
    [4X[28X13[128X[104X
    [4X[25Xgap>[125X [27XList( Filtered( prim, x -> x[ spos ] <> 0 ), l -> l{ [ 1, spos ] } );[127X[104X
    [4X[28X[ [ 162, 1 ], [ 162, 1 ], [ 540, 1 ], [ 1296, 1 ], [ 1296, 1 ], [128X[104X
    [4X[28X  [ 1296, 1 ], [ 1296, 1 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  order to show statement (c) (which then implies statement (d)), we use a
  permutation  representation  on  [22X112[122X  points.  It corresponds to an orbit of
  one-dimensional subspaces in the natural module of [22XΩ^-(6,3) ≅ S[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmatgrp:= DerivedSubgroup( SO( -1, 6, 3 ) );;[127X[104X
    [4X[25Xgap>[125X [27Xorbs:= OrbitsDomain( matgrp, NormedRowVectors( GF(3)^6 ), OnLines );;[127X[104X
    [4X[25Xgap>[125X [27XList( orbs, Length );[127X[104X
    [4X[28X[ 126, 126, 112 ][128X[104X
    [4X[25Xgap>[125X [27XG:= Action( matgrp, orbs[3], OnLines );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YIt is sufficient to compute [22XP(g,s)[122X, for involutions [22Xg ∈ S[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xapprox:= ApproxP( prim, spos );[127X[104X
    [4X[28X[ 0, 53/135, 1/10, 1/24, 1/24, 7/45, 4/45, 1/27, 1/36, 1/90, 1/216, [128X[104X
    [4X[28X  1/216, 7/405, 7/405, 1/270, 0, 0, 0, 0, 1/270 ][128X[104X
    [4X[25Xgap>[125X [27XFiltered( approx, x -> x >= 43/135 );[127X[104X
    [4X[28X[ 53/135 ][128X[104X
    [4X[25Xgap>[125X [27XOrdersClassRepresentatives( t );[127X[104X
    [4X[28X[ 1, 2, 3, 3, 3, 3, 4, 4, 5, 6, 6, 6, 7, 7, 8, 9, 9, 9, 9, 12 ][128X[104X
    [4X[25Xgap>[125X [27XResetGlobalRandomNumberGenerators();[127X[104X
    [4X[25Xgap>[125X [27Xrepeat g:= Random( G ); until Order(g) = 2;[127X[104X
    [4X[25Xgap>[125X [27Xrepeat s:= Random( G );[127X[104X
    [4X[25X>[125X [27X   until Order(s) = 7;[127X[104X
    [4X[25Xgap>[125X [27Xbad:= RatioOfNongenerationTransPermGroup( G, g, s );[127X[104X
    [4X[28X43/135[128X[104X
    [4X[25Xgap>[125X [27Xbad < 1/3;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YStatement (e)  can  be  shown  easily  with  character-theoretic methods, as
  follows.  Since  [22XSU(4,3)[122X is a Schur cover of [22XS[122X and the groups in [22XMM(S,s)[122X are
  simple, only very few possibilities have to be checked. The Schur multiplier
  of  [22XU_3(3)[122X  is  trivial  (see,  e. g.,  [CCN+85, p. 14]), so the preimage in
  [22XSU(4,3)[122X  is  a  direct  product of [22XU_3(3)[122X and the centre of [22XSU(4,3)[122X. Neither
  [22XL_3(4)[122X  nor  its  double cover [22X2.L_3(4)[122X can be a subgroup of [22XSU(4,3)[122X, so the
  preimage  of  [22XL_3(4)[122X  must  be  a Schur cover of [22XL_3(4)[122X, i. e., it must have
  either  the  type  [22X4_1.L_3(4)[122X  or [22X4_2.L_3(4)[122X (see [CCN+85, p. 23]); only the
  type [22X4_2.L_3(4)[122X turns out to be possible.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27X4t:= CharacterTable( "4.U4(3)" );;[127X[104X
    [4X[25Xgap>[125X [27XLength( PossibleClassFusions( CharacterTable( "L3(4)" ), 4t ) );[127X[104X
    [4X[28X0[128X[104X
    [4X[25Xgap>[125X [27XLength( PossibleClassFusions( CharacterTable( "2.L3(4)" ), 4t ) );[127X[104X
    [4X[28X0[128X[104X
    [4X[25Xgap>[125X [27XLength( PossibleClassFusions( CharacterTable( "4_1.L3(4)" ), 4t ) );[127X[104X
    [4X[28X0[128X[104X
    [4X[25Xgap>[125X [27XLength( PossibleClassFusions( CharacterTable( "4_2.L3(4)" ), 4t ) );[127X[104X
    [4X[28X4[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAs  for  the  preimage  of the [22XA_7[122X type subgroups, we first observe that the
  double  cover  of  [22XA_7[122X cannot be a subgroup of the double cover of [22XS[122X, so the
  preimage  of  [22XA_7[122X in the double cover of [22XU_4(3)[122X is a direct product [22X2 × A_7[122X.
  The  group  [22XSU(4,3)[122X  does  not contain [22XA_7[122X type subgroups, thus the [22XA_7[122X type
  subgroups  in  [22X2.U_4(3)[122X lift to double covers of [22XA_7[122X in [22XSU(4,3)[122X. This proves
  the claimed structure.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27X2t:= CharacterTable( "2.U4(3)" );;[127X[104X
    [4X[25Xgap>[125X [27XLength( PossibleClassFusions( CharacterTable( "2.A7" ), 2t ) );[127X[104X
    [4X[28X0[128X[104X
    [4X[25Xgap>[125X [27XLength( PossibleClassFusions( CharacterTable( "A7" ), 4t ) );[127X[104X
    [4X[28X0[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  statement (f),  we  consider automorphic extensions of [22XS[122X. The bound for
  [22XS.2_3[122X  has  been  computed in Section [14X11.4-4[114X. That for [22XS.2_2[122X can be computed
  form the fact that the classes of maximal subgroups of [22XS.2_2[122X containing [22Xs[122X of
  order  [22X7[122X are [22XS[122X, one class of [22XU_3(3).2[122X type subgroups, and two classes of [22XS_7[122X
  type  subgroups  which  induce  the same permutation character (see [CCN+85,
  p. 52]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt2:= CharacterTable( "U4(3).2_2" );;[127X[104X
    [4X[25Xgap>[125X [27Xpi1:= PossiblePermutationCharacters( CharacterTable( "U3(3).2" ), t2 );[127X[104X
    [4X[28X[ Character( CharacterTable( "U4(3).2_2" ),[128X[104X
    [4X[28X  [ 540, 12, 54, 0, 0, 9, 8, 0, 0, 6, 0, 0, 1, 2, 0, 0, 0, 2, 0, 24, [128X[104X
    [4X[28X      4, 0, 0, 0, 0, 0, 0, 3, 2, 0, 4, 0, 0, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27Xpi2:= PossiblePermutationCharacters( CharacterTable( "A7.2" ), t2 );[127X[104X
    [4X[28X[ Character( CharacterTable( "U4(3).2_2" ),[128X[104X
    [4X[28X  [ 1296, 48, 0, 27, 0, 9, 0, 4, 1, 0, 3, 0, 1, 0, 0, 0, 0, 0, 216, [128X[104X
    [4X[28X      24, 0, 4, 0, 0, 0, 9, 0, 3, 0, 1, 0, 1, 0, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27Xprim:= Concatenation( pi1, pi2, pi2 );;[127X[104X
    [4X[25Xgap>[125X [27Xouter:= Difference([127X[104X
    [4X[25X>[125X [27X     PositionsProperty( OrdersClassRepresentatives( t2 ), IsPrimeInt ),[127X[104X
    [4X[25X>[125X [27X     ClassPositionsOfDerivedSubgroup( t2 ) );;[127X[104X
    [4X[25Xgap>[125X [27Xspos:= Position( OrdersClassRepresentatives( t2 ), 7 );;[127X[104X
    [4X[25Xgap>[125X [27XMaximum( ApproxP( prim, spos ){ outer } );[127X[104X
    [4X[28X1/3[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFinally,  Section [14X11.4-4[114X  shows that the character tables are not sufficient
  for  what  we  need, so we compute the exact proportion of nongeneration for
  [22XU_4(3).2_1 ≅ SO^-(6,3)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmatgrp:= SO( -1, 6, 3 );[127X[104X
    [4X[28XSO(-1,6,3)[128X[104X
    [4X[25Xgap>[125X [27Xorbs:= OrbitsDomain( matgrp, NormedRowVectors( GF(3)^6 ), OnLines );;[127X[104X
    [4X[25Xgap>[125X [27XList( orbs, Length );[127X[104X
    [4X[28X[ 126, 126, 112 ][128X[104X
    [4X[25Xgap>[125X [27XG:= Action( matgrp, orbs[3], OnLines );;[127X[104X
    [4X[25Xgap>[125X [27Xrepeat s:= Random( G );[127X[104X
    [4X[25X>[125X [27X   until Order( s ) = 7;[127X[104X
    [4X[25Xgap>[125X [27Xrepeat[127X[104X
    [4X[25X>[125X [27X     repeat 2B:= Random( G ); until Order( 2B ) mod 2 = 0;[127X[104X
    [4X[25X>[125X [27X     2B:= 2B^( Order( 2B ) / 2 );[127X[104X
    [4X[25X>[125X [27X     c:= Centralizer( G, 2B );[127X[104X
    [4X[25X>[125X [27X   until Size( c ) = 12096;[127X[104X
    [4X[25Xgap>[125X [27XRatioOfNongenerationTransPermGroup( G, 2B, s );[127X[104X
    [4X[28X13/27[128X[104X
    [4X[25Xgap>[125X [27Xrepeat[127X[104X
    [4X[25X>[125X [27X     repeat 2C:= Random( G ); until Order( 2C ) mod 2 = 0;[127X[104X
    [4X[25X>[125X [27X     2C:= 2C^( Order( 2C ) / 2 );[127X[104X
    [4X[25X>[125X [27X     c:= Centralizer( G, 2C );[127X[104X
    [4X[25X>[125X [27X   until Size( c ) = 1440;[127X[104X
    [4X[25Xgap>[125X [27XRatioOfNongenerationTransPermGroup( G, 2C, s );[127X[104X
    [4X[28X0[128X[104X
  [4X[32X[104X
  
  
  [1X11.5-25 [33X[0;0Y[22XU_6(3)[122X[101X[1X[133X[101X
  
  [33X[0;0YWe show that [22XS = U_6(3) = PSU(6,3)[122X satisfies the following.[133X
  
  [8X(a)[108X
        [33X[0;6YFor  [22Xs  ∈  S[122X  of  the type [22X1 perp 5[122X (i. e., the preimage of [22Xs[122X in [22X2.S =
        SU(6,3)[122X  decomposes  the  natural [22X6[122X-dimensional module for [22X2.S[122X into an
        orthogonal  sum  of two irreducible modules of the dimensions [22X1[122X and [22X5[122X,
        respectively)  and of order [22X(3^5 + 1)/2 = 122[122X, [22XMM(S,s)[122X consists of one
        group  of the type [22X2 × U_5(3)[122X, which lifts to a subgroup of the type [22X4
        × U_5(3) = GU(5,3)[122X in [22X2.S[122X. (The preimage of [22Xs[122X in [22X2.S[122X has order [22X3^5 + 1
        = 244[122X.)[133X
  
  [8X(b)[108X
        [33X[0;6Y[22Xσ(S,s) = 353/3159[122X.[133X
  
  [33X[0;0YBy [MSW94], the only maximal subgroup of [22XS[122X that contains [22Xs[122X is the stabilizer
  [22XH ≅ 2 × U_5(3)[122X of the orthogonal decomposition. This proves statement (a).[133X
  
  [33X[0;0YThe  character  table  of  [22XS[122X is currently not available in the [5XGAP[105X Character
  Table  Library.  We consider the permutation action of [22XS[122X on the orbit of the
  stabilized [22X1[122X-space. So [22XM[122X can be taken as a point stabilizer in this action.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCharacterTable( "U6(3)" );[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27Xg:= SU(6,3);;[127X[104X
    [4X[25Xgap>[125X [27Xorbs:= OrbitsDomain( g, NormedRowVectors( GF(9)^6 ), OnLines );;[127X[104X
    [4X[25Xgap>[125X [27XList( orbs, Length );[127X[104X
    [4X[28X[ 22204, 44226 ][128X[104X
    [4X[25Xgap>[125X [27Xrepeat x:= PseudoRandom( g ); until Order( x ) = 244;[127X[104X
    [4X[25Xgap>[125X [27XList( orbs, o -> Number( o, v -> OnLines( v, x ) = v ) );[127X[104X
    [4X[28X[ 0, 1 ][128X[104X
    [4X[25Xgap>[125X [27Xg:= Action( g, orbs[2], OnLines );;[127X[104X
    [4X[25Xgap>[125X [27XM:= Stabilizer( g, 1 );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThen we compute a list of elements in [22XM[122X that covers the conjugacy classes of
  prime element order, from which the numbers of fixed points and thus [22Xmax{ μ(
  S/M, g ); g ∈ M^× } = σ( S, s )[122X can be derived. This way we avoid completely
  to  check  the  [22XS[122X-conjugacy  of  elements  (class  representatives  of Sylow
  subgroups in [22XM[122X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xelms:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xfor p in PrimeDivisors( Size( M ) ) do[127X[104X
    [4X[25X>[125X [27X     syl:= SylowSubgroup( M, p );[127X[104X
    [4X[25X>[125X [27X     Append( elms, Filtered( PcConjugacyClassReps( syl ),[127X[104X
    [4X[25X>[125X [27X                             r -> Order( r ) = p ) );[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27X1 - Minimum( List( elms, NrMovedPoints ) ) / Length( orbs[2] );[127X[104X
    [4X[28X353/3159[128X[104X
  [4X[32X[104X
  
  
  [1X11.5-26 [33X[0;0Y[22XU_8(2)[122X[101X[1X[133X[101X
  
  [33X[0;0YWe show that [22XS = U_8(2) = SU(8,2)[122X satisfies the following.[133X
  
  [8X(a)[108X
        [33X[0;6YFor  [22Xs  ∈  S[122X  of  the  type  [22X1 perp 7[122X (i. e., [22Xs[122X decomposes the natural
        [22X8[122X-dimensional  module  for [22XS[122X into an orthogonal sum of two irreducible
        modules  of the dimensions [22X1[122X and [22X7[122X, respectively) and of order [22X2^7 + 1
        = 129[122X, [22XMM(S,s)[122X consists of one group of the type [22X3 × U_7(2) = GU(7,2)[122X.[133X
  
  [8X(b)[108X
        [33X[0;6Y[22Xσ(S,s) = 2753/10880[122X.[133X
  
  [33X[0;0YBy [MSW94], the only maximal subgroup of [22XS[122X that contains [22Xs[122X is the stabilizer
  [22XM ≅ GU(7,2)[122X of the orthogonal decomposition. This proves statement (a).[133X
  
  [33X[0;0YThe  character  table  of  [22XS[122X is currently not available in the [5XGAP[105X Character
  Table  Library.  We  proceed exactly as in Section [14X11.5-25[114X in order to prove
  statement (b).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCharacterTable( "U8(2)" );[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27Xg:= SU(8,2);;[127X[104X
    [4X[25Xgap>[125X [27Xorbs:= OrbitsDomain( g, NormedRowVectors( GF(4)^8 ), OnLines );;[127X[104X
    [4X[25Xgap>[125X [27XList( orbs, Length );[127X[104X
    [4X[28X[ 10965, 10880 ][128X[104X
    [4X[25Xgap>[125X [27Xrepeat x:= PseudoRandom( g ); until Order( x ) = 129;[127X[104X
    [4X[25Xgap>[125X [27XList( orbs, o -> Number( o, v -> OnLines( v, x ) = v ) );[127X[104X
    [4X[28X[ 0, 1 ][128X[104X
    [4X[25Xgap>[125X [27Xg:= Action( g, orbs[2], OnLines );;[127X[104X
    [4X[25Xgap>[125X [27XM:= Stabilizer( g, 1 );;[127X[104X
    [4X[25Xgap>[125X [27Xelms:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xfor p in PrimeDivisors( Size( M ) ) do[127X[104X
    [4X[25X>[125X [27X     syl:= SylowSubgroup( M, p );[127X[104X
    [4X[25X>[125X [27X     Append( elms, Filtered( PcConjugacyClassReps( syl ),[127X[104X
    [4X[25X>[125X [27X                             r -> Order( r ) = p ) );[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27XLength( elms );[127X[104X
    [4X[28X611[128X[104X
    [4X[25Xgap>[125X [27X1 - Minimum( List( elms, NrMovedPoints ) ) / Length( orbs[2] );[127X[104X
    [4X[28X2753/10880[128X[104X
  [4X[32X[104X
  
