  
  [1X1 [33X[0;0YThe GAP Table of Marks Library[133X[101X
  
  
  [1X1.1 [33X[0;0YTables Of Marks[133X[101X
  
  [33X[0;0YThe  concept  of  a  [13XTable of Marks[113X was introduced by W.Burnside in his book
  ``Theory  of Groups of Finite Order'' [Bur55]. Therefore a table of marks is
  sometimes  called  a [13XBurnside matrix[113X. The table of marks of a finite group [22XG[122X
  is  a matrix whose rows and columns are labelled by the conjugacy classes of
  subgroups  of  [22XG[122X and where for two subgroups [22XH[122X and [22XK[122X the [22X(H, K)[122X–entry is the
  number of fixed points of [22XK[122X in the transitive action of [22XG[122X on the cosets of [22XH[122X
  in  [22XG[122X.  So  the  table  of  marks  characterizes  the set of all permutation
  representations  of  [22XG[122X.  Moreover,  the  table  of  marks  gives  a  compact
  description  of  the  subgroup lattice of [22XG[122X, since from the numbers of fixed
  points  the  numbers of conjugates of a subgroup [22XK[122X contained in a subgroup [22XH[122X
  can  be derived. For small groups the table of marks of [22XG[122X can be constructed
  directly  in  GAP  by  first  computing  the  entire  subgroup lattice of [22XG[122X.
  However, for larger groups this method is unfeasible. The GAP Table of Marks
  library  provides access to several hundred table of marks and their maximal
  subgroups.[133X
  
  
  [1X1.2 [33X[0;0YInstalling The Table of Marks Library[133X[101X
  
  [33X[0;0YDownload  the  archives in your preferred format. Unpack the archives inside
  the pkg dirctory of your GAP installation. Load the package[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLoadPackage("tomlib");[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X1.3 [33X[0;0YContents[133X[101X
  
  [33X[0;0YTomLib  contains several hundred tables of marks. For a complete list of the
  contents of the library do the following.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xnames:=AllLibTomNames();;[127X[104X
    [4X[25Xgap>[125X [27X"A5" in names;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe current version of the tomlib contains the tables of marks of the groups
  listed  below  as  well as the tables of many of their maximal subgroups and
  automorphism groups. The Alternating groups [22XA_n[122X[133X
  
  [30X    [33X[0;6Yfor [22Xn = 5, 6, 7, 8, 9, 10, 11, 12, 13[122X.[133X
  
  [33X[0;0YThe Symmetric groups [22XS_n[122X[133X
  
  [30X    [33X[0;6Yfor [22Xn = 4, 5, 6, 7, 8, 9, 10, 11, 12, 13[122X.[133X
  
  [33X[0;0YThe Linear groups [22XL_2(n)[122X for[133X
  
  [30X    [33X[0;6Y[22Xn  = 7, 8, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47,
        49, 53[122X[133X
  
  [30X    [33X[0;6Y[22Xn  =  59,  61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109,
        113, 121, 125 .[122X[133X
  
  [33X[0;0Yalong with[133X
  
  [30X    [33X[0;6Y[22XL_3(4), L_3(3), L_3(5), L_3(7), L_3(9)[122X[133X
  
  [30X    [33X[0;6Y[22XL_4(3), L_3(8), L_3(11)[122X.[133X
  
  [33X[0;0YThe Unitary groups[133X
  
  [30X    [33X[0;6Y[22XU_3(3), U_4(3), U_3(5), U_3(4), U_3(11), U_3(7), U_3(8)[122X[133X
  
  [30X    [33X[0;6Y[22XU_3(9), U_4(2), U_5(2)[122X[133X
  
  [33X[0;0YThe Sporadic Groups[133X
  
  [30X    [33X[0;6Y[22XCo_3, HS, McL, He, J_1, J_2, J_3, M_11, M_12, M_22, M_23, M_24[122X[133X
  
  [33X[0;0YThe  names  given  to each subgroup are consistent with those used in Robert
  Wilson's  atlas  [WWT+] For example if you wish to access the table of marks
  of  the  maximal  subgroup  [22X"5:4  ×  A5"[122X  of  the  Higman-Sims  group do the
  following:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XTableOfMarks("5:4xA5");[127X[104X
    [4X[28XTableOfMarks( "5:4xA5" )[128X[104X
  [4X[32X[104X
  
  
  [1X1.4 [33X[0;0YAdministrative Functions[133X[101X
  
  [33X[0;0YHere  we  document  some  of  the  administrative facilities for the the [5XGAP[105X
  library of tables of marks.[133X
  
  [1X1.4-1 LIBTOMKNOWN[101X
  
  [33X[1;0Y[29X[2XLIBTOMKNOWN[102X[32X global variable[133X
  
  [33X[0;0Y[10XLIBTOMKNOWN[110X  is  a record that controls the loading of data files of the [5XGAP[105X
  library of tables of marks. It has the following components.[133X
  
  [8X[10XACTUAL[110X[8X [108X
        [33X[0;6Ythe   name   of   the   file   to  be  read  at  the  moment  (set  by
        [10XSetActualLibFileName[110X),[133X
  
  [8X[10XLOADSTATUS[110X[8X [108X
        [33X[0;6Ya  record whose components are names of files in the library of tables
        of  marks,  with  value  a  list whose first entry is one of [10X"loaded"[110X,
        [10X"unloaded"[110X,  [10X"userloaded"[110X  and  whose  second entry is an integer that
        controls  when  the  corresponding tables of marks can be removed from
        [5XGAP[105X,[133X
  
  [8X[10XMAX[110X[8X [108X
        [33X[0;6Y[5XGAP[105X  remembers at most [10XMAX[110X of the previously loaded library files (the
        default value is [22X10[122X),[133X
  
  [8X[10XUNLOAD[110X[8X [108X
        [33X[0;6Yis  it  allowed  to  remove previously loaded library files (is set to
        [9Xtrue[109X by default),[133X
  
  [8X[10XSTDGEN[110X[8X [108X
        [33X[0;6Ya  list  describing standard generators of almost simple groups in the
        table of marks library.[133X
  
  [33X[0;0YAdditionally  the  names  of the files already loaded occur as components of
  [10XLIBTOMKNOWN[110X; the corresponding values are given by the data of the files.[133X
  
  [1X1.4-2 IsLibTomRep[101X
  
  [33X[1;0Y[29X[2XIsLibTomRep[102X( [3Xobj[103X ) [32X Representation[133X
  
  [33X[0;0YLibrary  tables of marks have their own representation. [10XIsLibTomRep[110X tests if
  [3Xobj[103X is a library representation.[133X
  
  [1X1.4-3 TableOfMarksFromLibrary[101X
  
  [33X[1;0Y[29X[2XTableOfMarksFromLibrary[102X( [3Xstring[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ythe table of marks with name [3Xstring[103X.[133X
  
  [1X1.4-4 ConvertToLibTom[101X
  
  [33X[1;0Y[29X[2XConvertToLibTom[102X( [3Xrecord[103X ) [32X function[133X
  
  [33X[0;0Y[10XConvertToLibTom[110X     converts     a     record     with    components    from
  [10XTableOfMarksComponents[110X  into  a  library  table  of  marks  object  with the
  corresponding attribute values set.[133X
  
  [1X1.4-5 SetActualLibFileName[101X
  
  [33X[1;0Y[29X[2XSetActualLibFileName[102X( [3Xfilename[103X ) [32X function[133X
  
  [33X[0;0YSets the file name for [3Xfilename[103X.[133X
  
  [1X1.4-6 LIBTOM[101X
  
  [33X[1;0Y[29X[2XLIBTOM[102X( [3Xarg[103X ) [32X function[133X
  [33X[1;0Y[29X[2XAFLT[102X( [3Xsource[103X, [3Xdestination[103X, [3Xfusion[103X ) [32X function[133X
  [33X[1;0Y[29X[2XACLT[102X( [3Xidentifier[103X, [3Xcomponent[103X, [3Xvalue[103X ) [32X function[133X
  
  [33X[0;0YThe  library  format  of  a library table of marks is a call to the function
  [10XLIBTOM[110X, with the arguments sorted as in [10XTableOfMarksComponents[110X .[133X
  
  [33X[0;0Y[10XAFLT[110X adds a fusion map [3Xvalue[103X from the table of marks with name [3Xsource[103X to the
  table  of  marks with name [3Xdestination[103X. The fusion map is a list of positive
  integers,  storing  at  position  [22Xi[122X the position of the class in [3Xdestination[103X
  that contains the subgroups in the [22Xi[122X-th class of [3Xsource[103X.[133X
  
  [33X[0;0Y[10XACLT[110X adds the value [3Xvalue[103X of a component with name [3Xcomponent[103X to the table of
  marks with identifier value [3Xidentifier[103X in the library of tables of marks.[133X
  
  [1X1.4-7 AllLibTomNames[101X
  
  [33X[1;0Y[29X[2XAllLibTomNames[102X(  ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya list containing all names of available library tables of marks.[133X
  
  [1X1.4-8 NamesLibTom[101X
  
  [33X[1;0Y[29X[2XNamesLibTom[102X( [3Xstring[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XNamesLibTom[102X( [3Xtom[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Yall  names  of  the library table [3Xtom[103X or of the library table with
            name [3Xstring[103X[133X
  
  [1X1.4-9 NotifiedFusionsOfLibTom[101X
  
  [33X[1;0Y[29X[2XNotifiedFusionsOfLibTom[102X( [3Xtom[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XNotifiedFusionsOfLibTom[102X( [3Xstring[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XFusionsOfLibTom[102X( [3Xtom[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XFusionsOfLibTom[102X( [3Xstring[103X ) [32X attribute[133X
  
  [33X[0;0YAre  there  any  fusions from the library table of marks [3Xtom[103X or the table of
  marks with name [3Xstring[103X into other library tables marks?[133X
  
  [33X[0;0Y[10XNotifiedFusionsOfLibTom[110X  returns  the  names  of  all  such tables of marks.
  [10XFusionsOfLibTom[110X returns the complete fusion maps. For that the corresponding
  library file has to be loaded.[133X
  
  [1X1.4-10 NotifiedFusionsToLibTom[101X
  
  [33X[1;0Y[29X[2XNotifiedFusionsToLibTom[102X( [3Xtom[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XNotifiedFusionsToLibTom[102X( [3Xstring[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XFusionsToLibTom[102X( [3Xtom[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XFusionsToLibTom[102X( [3Xstring[103X ) [32X attribute[133X
  
  [33X[0;0YAre  there any fusions from other library table of marks to [3Xtom[103X or the table
  of marks with name [3Xstring[103X.[133X
  
  [33X[0;0Y[10XNotifiedFusionsToLibTom[110X  returns  the  names  of  all  such tables of marks.
  [10XFusionsToLibTom[110X returns the complete fusion maps. For that, the correponding
  library files have to be loaded.[133X
  
  [1X1.4-11 UnloadTableOfMarksData[101X
  
  [33X[1;0Y[29X[2XUnloadTableOfMarksData[102X(  ) [32X function[133X
  
  [33X[0;0Y[10XUnloadTableOfMarksData[110X clears the cache of tables of marks. This can be used
  to free up to several hundred megabytes of space in the current [5XGAP[105X session.[133X
  
  
  [1X1.5 [33X[0;0YStandard Generators of Groups[133X[101X
  
  [33X[0;0YAn  [22Xs[122X-tuple of [13Xstandard generators[113X of a given group [22XG[122X is a vector [22X(g_1, g_2,
  ...,  g_s)[122X  of  elements [22Xg_i ∈ G[122X satisfying certain conditions (depending on
  the isomorphism type of [22XG[122X) such that[133X
  
  [31X1[131X   [33X[0;6Y[22X⟨ g_1, g_2, ..., g_s ⟩ = G[122X and[133X
  
  [31X2[131X   [33X[0;6Ythe  vector  is unique up to automorphisms of [22XG[122X, i.e., for two vectors
        [22X(g_1,  g_2, ..., g_s)[122X and [22X(h_1, h_2, ..., h_s)[122X of standard generators,
        the map [22Xg_i ↦ h_i[122X extends to an automorphism of [22XG[122X.[133X
  
  [33X[0;0YFor details about standard generators, see [Wil96].[133X
  
  [1X1.5-1 StandardGeneratorsInfo[101X
  
  [33X[1;0Y[29X[2XStandardGeneratorsInfo[102X( [3XG[103X ) [32X attribute[133X
  
  [33X[0;0YWhen  called  with  the  group  [3XG[103X,  [2XStandardGeneratorsInfo[102X returns a list of
  records  with  at  least  one of the components [10Xscript[110X and [10Xdescription[110X. Each
  such  record defines [13Xstandard generators[113X of groups isomorphic to [3XG[103X, the [22Xi[122X-th
  record  is  referred  to  as  the  [22Xi[122X-th  set of standard generators for such
  groups.  The  value  of  [10Xscript[110X  is  a  dense list of lists, each encoding a
  command that has one of the following forms.[133X
  
  [8XA [13Xdefinition[113X [22X[ i, n, k ][122X or [22X[ i, n ][122X[108X
        [33X[0;6Ymeans  to search for an element of order [22Xn[122X, and to take its [22Xk[122X-th power
        as candidate for the [22Xi[122X-th standard generator (the default for [22Xk[122X is [22X1[122X),[133X
  
  [8Xa [13Xrelation[113X [22X[ i_1, k_1, i_2, k_2, ..., i_m, k_m, n ][122X with [22Xm > 1[122X[108X
        [33X[0;6Ymeans  a  check  whether  the  element  [22Xg_{i_1}^{k_1}  g_{i_2}^{k_2} ⋯
        g_{i_m}^{k_m}[122X  has  order  [22Xn[122X;  if  [22Xg_j[122X  occurs then of course the [22Xj[122X-th
        generator must have been defined before,[133X
  
  [8Xa [13Xrelation[113X [22X[ [ i_1, i_2, ..., i_m ], [3Xslp[103X[8X, n ][122X[108X
        [33X[0;6Ymeans  a  check  whether  the  result of the straight line program [3Xslp[103X
        (see [14X'Reference:  Straight  Line  Programs'[114X) applied to the candidates
        [22Xg_{i_1},  g_{i_2},  ..., g_{i_m}[122X has order [22Xn[122X, where the candidates [22Xg_j[122X
        for the [22Xj[122X-th standard generators must have been defined before,[133X
  
  [8Xa [13Xcondition[113X [22X[ [ i_1, k_1, i_2, k_2, ..., i_m, k_m ], f, v ][122X[108X
        [33X[0;6Ymeans   a   check   whether  the  [5XGAP[105X  function  in  the  global  list
        [2XStandardGeneratorsFunctions[102X  ([14X1.5-3[114X) that is followed by the list [22Xf[122X of
        strings returns the value [22Xv[122X when it is called with [22XG[122X and [22Xg_{i_1}^{k_1}
        g_{i_2}^{k_2} ⋯ g_{i_m}^{k_m}[122X.[133X
  
  [33X[0;0YOptional components of the returned records are[133X
  
  [8X[10Xgenerators[110X[8X[108X
        [33X[0;6Ya string of names of the standard generators,[133X
  
  [8X[10Xdescription[110X[8X[108X
        [33X[0;6Ya  string describing the [10Xscript[110X information in human readable form, in
        terms of the [10Xgenerators[110X value,[133X
  
  [8X[10Xclassnames[110X[8X[108X
        [33X[0;6Ya  list  of  strings,  the  [22Xi[122X-th entry being the name of the conjugacy
        class  containing  the [22Xi[122X-th standard generator, according to the [5XAtlas[105X
        character table of the group (see [2XClassNames[102X ([14XReference: ClassNames[114X)),
        and[133X
  
  [8X[10XATLAS[110X[8X[108X
        [33X[0;6Ya boolean; [9Xtrue[109X means that the standard generators coincide with those
        defined  in  Rob Wilson's [5XAtlas[105X of Group Representations (see [WWT+]),
        and [9Xfalse[109X means that this property is not guaranteed.[133X
  
  [8X[10Xstandardization[110X[8X[108X
        [33X[0;6Ya positive integer [22Xi[122X; Whenever [10XATLAS[110X returns [9Xtrue[109X the value of [22Xi[122X means
        that the generators stored in the group are standard generators w.r.t.
        standardization  [22Xi[122X,  in  the  sense  of  Rob  Wilson's  [5XAtlas[105X of Group
        Representations.[133X
  
  [33X[0;0YThere  is  no  default  method  for  an arbitrary isomorphism type, since in
  general the definition of standard generators is not obvious.[133X
  
  [33X[0;0YThe function [2XStandardGeneratorsOfGroup[102X ([14XStandardGeneratorsOfGroup???[114X) can be
  used to find standard generators of a given group isomorphic to [3XG[103X.[133X
  
  [33X[0;0YThe  [10Xgenerators[110X  and  [10Xdescription[110X  values,  if not known, can be computed by
  [2XHumanReadableDefinition[102X ([14X1.5-2[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XStandardGeneratorsInfo( TableOfMarks( "L3(3)" ) );[127X[104X
    [4X[28X[ rec( ATLAS := true, [128X[104X
    [4X[28X    description := "|a|=2, |b|=3, |C(b)|=9, |ab|=13, |ababb|=4", [128X[104X
    [4X[28X    generators := "a, b", [128X[104X
    [4X[28X    script := [ [ 1, 2 ], [ 2, 3 ], [ [ 2, 1 ], [ "|C(",, ")|" ], 9 ], [128X[104X
    [4X[28X        [ 1, 1, 2, 1, 13 ], [ 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 4 ] ], [128X[104X
    [4X[28X    standardization := 1 ) ][128X[104X
  [4X[32X[104X
  
  [1X1.5-2 HumanReadableDefinition[101X
  
  [33X[1;0Y[29X[2XHumanReadableDefinition[102X( [3Xinfo[103X ) [32X function[133X
  [33X[1;0Y[29X[2XScriptFromString[102X( [3Xstring[103X ) [32X function[133X
  
  [33X[0;0YLet  [3Xinfo[103X  be  a  record  that  is  valid as value of [2XStandardGeneratorsInfo[102X
  ([14X1.5-1[114X).   [2XHumanReadableDefinition[102X  returns  a  string  that  describes  the
  definition  of  standard generators given by the [10Xscript[110X component of [3Xinfo[103X in
  human  readable  form.  The  names  of  the  generators  are  taken from the
  [10Xgenerators[110X   component   (default   names  [10X"a"[110X,  [10X"b"[110X  etc. are  computed  if
  necessary), and the result is stored in the [10Xdescription[110X component.[133X
  
  [33X[0;0Y[2XScriptFromString[102X  does  the  converse  of  [2XHumanReadableDefinition[102X, i.e., it
  takes  a string [3Xstring[103X as returned by [2XHumanReadableDefinition[102X, and returns a
  corresponding [10Xscript[110X list.[133X
  
  [33X[0;0YIf  [21Xcondition[121X lines occur in the script (see [2XStandardGeneratorsInfo[102X ([14X1.5-1[114X))
  then     the     functions    that    occur    must    be    contained    in
  [2XStandardGeneratorsFunctions[102X ([14X1.5-3[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xscr:= ScriptFromString( "|a|=2, |b|=3, |C(b)|=9, |ab|=13, |ababb|=4" );[127X[104X
    [4X[28X[ [ 1, 2 ], [ 2, 3 ], [ [ 2, 1 ], [ "|C(",, ")|" ], 9 ], [ 1, 1, 2, 1, 13 ], [128X[104X
    [4X[28X  [ 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 4 ] ][128X[104X
    [4X[25Xgap>[125X [27Xinfo:= rec( script:= scr );[127X[104X
    [4X[28Xrec( script := [ [ 1, 2 ], [ 2, 3 ], [ [ 2, 1 ], [ "|C(",, ")|" ], 9 ], [128X[104X
    [4X[28X      [ 1, 1, 2, 1, 13 ], [ 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 4 ] ] )[128X[104X
    [4X[25Xgap>[125X [27XHumanReadableDefinition( info );[127X[104X
    [4X[28X"|a|=2, |b|=3, |C(b)|=9, |ab|=13, |ababb|=4"[128X[104X
    [4X[25Xgap>[125X [27Xinfo;[127X[104X
    [4X[28Xrec( description := "|a|=2, |b|=3, |C(b)|=9, |ab|=13, |ababb|=4", [128X[104X
    [4X[28X  generators := "a, b", [128X[104X
    [4X[28X  script := [ [ 1, 2 ], [ 2, 3 ], [ [ 2, 1 ], [ "|C(",, ")|" ], 9 ], [128X[104X
    [4X[28X      [ 1, 1, 2, 1, 13 ], [ 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 4 ] ] )[128X[104X
  [4X[32X[104X
  
  [1X1.5-3 StandardGeneratorsFunctions[101X
  
  [33X[1;0Y[29X[2XStandardGeneratorsFunctions[102X[32X global variable[133X
  
  [33X[0;0Y[2XStandardGeneratorsFunctions[102X  is  a  list of even length. At position [22X2i-1[122X, a
  function  of two arguments is stored, which are expected to be a group and a
  group  element.  At  position [22X2i[122X a list of strings is stored such that first
  inserting  a  generator name in all holes and then forming the concatenation
  yields  a  string that describes the function at the previous position; this
  string must contain the generator enclosed in round brackets [10X([110X and [10X)[110X.[133X
  
  [33X[0;0YThis   list  is  used  by  the  functions  [2XStandardGeneratorsInfo[102X  ([14X1.5-1[114X)),
  [2XHumanReadableDefinition[102X ([14X1.5-2[114X), and [2XScriptFromString[102X ([14X1.5-2[114X). Note that the
  lists at even positions must be pairwise different.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XStandardGeneratorsFunctions{ [ 1, 2 ] };[127X[104X
    [4X[28X[ function( G, g ) ... end, [ "|C(",, ")|" ] ][128X[104X
  [4X[32X[104X
  
  [1X1.5-4 IsStandardGeneratorsOfGroup[101X
  
  [33X[1;0Y[29X[2XIsStandardGeneratorsOfGroup[102X( [3Xinfo[103X, [3XG[103X, [3Xgens[103X ) [32X function[133X
  
  [33X[0;0YLet  [3Xinfo[103X  be  a  record  that  is  valid as value of [2XStandardGeneratorsInfo[102X
  ([14X1.5-1[114X),  [3XG[103X  a  group,  and  [3Xgens[103X  a list of generators for [3XG[103X. In this case,
  [2XIsStandardGeneratorsOfGroup[102X returns [9Xtrue[109X if [3Xgens[103X satisfies the conditions of
  the [10Xscript[110X component of [3Xinfo[103X, and [9Xfalse[109X otherwise.[133X
  
  [33X[0;0YNote  that  the result [9Xtrue[109X means that [3Xgens[103X is a list of standard generators
  for  [3XG[103X  only if [3XG[103X has the isomorphism type for which [3Xinfo[103X describes standard
  generators.[133X
  
  [1X1.5-5 StandardGeneratorsOfGroup[101X
  
  [33X[1;0Y[29X[2XStandardGeneratorsOfGroup[102X( [3Xinfo[103X, [3XG[103X[, [3Xrandfunc[103X] ) [32X function[133X
  
  [33X[0;0YLet  [3Xinfo[103X  be  a  record  that  is  valid as value of [2XStandardGeneratorsInfo[102X
  ([14X1.5-1[114X),  and  [3XG[103X  a  group  of the isomorphism type for which [3Xinfo[103X describes
  standard  generators. In this case, [2XStandardGeneratorsOfGroup[102X returns a list
  of standard generators of [3XG[103X, see Section [14X1.5[114X.[133X
  
  [33X[0;0YThe optional argument [3Xrandfunc[103X must be a function that returns an element of
  [3XG[103X when called with [3XG[103X; the default is [2XPseudoRandom[102X ([14XReference: PseudoRandom[114X).[133X
  
  [33X[0;0YIn  each  call  to [2XStandardGeneratorsOfGroup[102X ([14XStandardGeneratorsOfGroup???[114X),
  the  [10Xscript[110X  component  of [3Xinfo[103X is scanned line by line. [3Xrandfunc[103X is used to
  find  an  element of the prescribed order whenever a definition line is met,
  and  for  the  relation  and condition lines in the [10Xscript[110X list, the current
  generator  candidates  are  checked;  if  a  condition is not fulfilled, all
  candidates  are  thrown  away, and the procedure starts again with the first
  line.  When  the  conditions are fulfilled after processing the last line of
  the [10Xscript[110X list, the standard generators are returned.[133X
  
  [33X[0;0YNote that if [3XG[103X has the wrong isomorphism type then [2XStandardGeneratorsOfGroup[102X
  ([14XStandardGeneratorsOfGroup???[114X)  returns a list of elements in [3XG[103X that satisfy
  the  conditions  of the [10Xscript[110X component of [3Xinfo[103X if such elements exist, and
  does  not  terminate  otherwise.  In the former case, obviously the returned
  elements need not be standard generators of [3XG[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xa5:= AlternatingGroup( 5 );[127X[104X
    [4X[28XAlt( [ 1 .. 5 ] )[128X[104X
    [4X[25Xgap>[125X [27Xinfo:= StandardGeneratorsInfo( TableOfMarks( "A5" ) )[1];[127X[104X
    [4X[28Xrec( ATLAS := true, description := "|a|=2, |b|=3, |ab|=5", [128X[104X
    [4X[28Xgenerators := "a, b", script := [ [ 1, 2 ], [ 2, 3 ], [ 1, 1, 2, 1, 5 ] ], [128X[104X
    [4X[28Xstandardization := 1 )[128X[104X
    [4X[25Xgap>[125X [27XIsStandardGeneratorsOfGroup( info, a5, [ (1,3)(2,4), (3,4,5) ] );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsStandardGeneratorsOfGroup( info, a5, [ (1,3)(2,4), (1,2,3) ] );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xs5:= SymmetricGroup( 5 );;[127X[104X
    [4X[25Xgap>[125X [27XRepresentativeAction( s5, [ (1,3)(2,4), (3,4,5) ], [127X[104X
    [4X[25X>[125X [27X       StandardGeneratorsOfGroup( info, a5 ), OnPairs ) <> fail;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X1.5-6 StandardGeneratorsInfo[101X
  
  [33X[1;0Y[29X[2XStandardGeneratorsInfo[102X( [3Xtom[103X ) [32X attribute[133X
  
  [33X[0;0YFor  a  table  of  marks  [3Xtom[103X,  a  stored  value  of  [2XStandardGeneratorsInfo[102X
  ([14XStandardGeneratorsInfo,  for  tables  of marks???[114X) equals the value of this
  attribute   for   the   underlying  group  (see [2XUnderlyingGroup[102X  ([14XReference:
  UnderlyingGroup   for   tables   of  marks[114X))  of  [3Xtom[103X,  cf. Section [14XStandard
  Generators of Groups???[114X.[133X
  
  [33X[0;0YIn  this case, the [2XGeneratorsOfGroup[102X ([14XReference: GeneratorsOfGroup[114X) value of
  the  underlying  group  [22XG[122X of [3Xtom[103X is assumed to be in fact a list of standard
  generators   for   [22XG[122X;   So   one   should   be   careful  when  setting  the
  [2XStandardGeneratorsInfo[102X value by hand.[133X
  
  [33X[0;0YThere  is no default method to compute the [2XStandardGeneratorsInfo[102X value of a
  table of marks if it is not yet stored.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xa5:=TableOfMarks("A5");[127X[104X
    [4X[25Xgap>[125X [27Xstd:= StandardGeneratorsInfo( a5 );[127X[104X
    [4X[28X[ rec( ATLAS := true, description := "|a|=2, |b|=3, |ab|=5", [128X[104X
    [4X[28X    generators := "a, b", script := [ [ 1, 2 ], [ 2, 3 ], [ 1, 1, 2, 1, 5 ] [128X[104X
    [4X[28X       ], standardization := 1 ) ][128X[104X
    [4X[25Xgap>[125X [27X# Now find standard generators of an isomorphic group.[127X[104X
    [4X[25Xgap>[125X [27Xg:= SL(2,4);;[127X[104X
    [4X[25Xgap>[125X [27Xrepeat[127X[104X
    [4X[25X>[125X [27X  x:= PseudoRandom( g );[127X[104X
    [4X[25X>[125X [27Xuntil Order( x ) = 2;[127X[104X
    [4X[25Xgap>[125X [27Xrepeat  [127X[104X
    [4X[25X>[125X [27X  y:= PseudoRandom( g );[127X[104X
    [4X[25X>[125X [27Xuntil Order( y ) = 3 and Order( x*y ) = 5;[127X[104X
    [4X[25Xgap>[125X [27X# Compute a representative w.r.t. these generators.[127X[104X
    [4X[25Xgap>[125X [27XRepresentativeTomByGenerators( a5, 4, [ x, y ] );[127X[104X
    [4X[28XGroup([ [ [ 0*Z(2), Z(2)^0 ], [ Z(2)^0, 0*Z(2) ] ],[128X[104X
    [4X[28X  [ [ Z(2^2), Z(2^2)^2 ], [ Z(2^2)^2, Z(2^2) ] ] ])[128X[104X
    [4X[25Xgap>[125X [27X# Show that the new generators are really good.[127X[104X
    [4X[25Xgap>[125X [27Xgrp:= UnderlyingGroup( a5 );;[127X[104X
    [4X[25Xgap>[125X [27Xiso:= GroupGeneralMappingByImages( grp, g,[127X[104X
    [4X[25X>[125X [27X             GeneratorsOfGroup( grp ), [ x, y ] );;[127X[104X
    [4X[25Xgap>[125X [27XIsGroupHomomorphism( iso );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsBijective( iso );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
