  
  [1X70 [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
  [21XTheory of Groups of Finite Order[121X, see [Bur55]. Therefore a table of marks is
  sometimes called a [13XBurnside matrix[113X.[133X
  
  [33X[0;0YThe  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 [22XA[122X and [22XB[122X the [22X(A, B)[122X-entry is the number of fixed points of [22XB[122X in the
  transitive  action  of  [22XG[122X  on  the  cosets  of [22XA[122X in [22XG[122X. So the table of marks
  characterizes the set of all permutation representations of [22XG[122X.[133X
  
  [33X[0;0YMoreover,  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 [22XB[122X contained in a subgroup [22XA[122X can be derived.[133X
  
  [33X[0;0YA  table  of  marks  of a given group [22XG[122X can be constructed from the subgroup
  lattice  of  [22XG[122X  (see [14X70.3[114X).  For  several  groups, the table of marks can be
  restored from the [5XGAP[105X library of tables of marks (see [14X70.13[114X).[133X
  
  [33X[0;0YGiven  the  table  of  marks  of [22XG[122X, one can display it (see [14X70.4[114X) and derive
  information  about  [22XG[122X  and its Burnside ring from it (see [14X70.7[114X, [14X70.8[114X, [14X70.9[114X).
  Moreover,  tables  of  marks in [5XGAP[105X provide an easy access to the classes of
  subgroups of their underlying groups (see [14X70.10[114X).[133X
  
  
  [1X70.1 [33X[0;0YMore about Tables of Marks[133X[101X
  
  [33X[0;0YLet [22XG[122X be a finite group with [22Xn[122X conjugacy classes of subgroups [22XC_1, C_2, ...,
  C_n[122X  and  representatives  [22XH_i  ∈ C_i[122X, [22X1 ≤ i ≤ n[122X. The [13Xtable of marks[113X of [22XG[122X is
  defined  to be the [22Xn × n[122X matrix [22XM = (m_ij)[122X where the [13Xmark[113X [22Xm_ij[122X is the number
  of  fixed  points of the subgroup [22XH_j[122X in the action of [22XG[122X on the right cosets
  of [22XH_i[122X in [22XG[122X.[133X
  
  [33X[0;0YSince  [22XH_j[122X  can  only  have  fixed  points  if  it  is  contained in a point
  stabilizer  the  matrix  [22XM[122X is lower triangular if the classes [22XC_i[122X are sorted
  according  to  the  condition that if [22XH_i[122X is contained in a conjugate of [22XH_j[122X
  then [22Xi ≤ j[122X.[133X
  
  [33X[0;0YMoreover,  the diagonal entries [22Xm_ii[122X are nonzero since [22Xm_ii[122X equals the index
  of  [22XH_i[122X  in its normalizer in [22XG[122X. Hence [22XM[122X is invertible. Since any transitive
  action of [22XG[122X is equivalent to an action on the cosets of a subgroup of [22XG[122X, one
  sees  that  the  table  of  marks  completely  characterizes  the set of all
  permutation representations of [22XG[122X.[133X
  
  [33X[0;0YThe  marks  [22Xm_ij[122X  have further meanings. If [22XH_1[122X is the trivial subgroup of [22XG[122X
  then each mark [22Xm_i1[122X in the first column of [22XM[122X is equal to the index of [22XH_i[122X in
  [22XG[122X  since  the trivial subgroup fixes all cosets of [22XH_i[122X. If [22XH_n = G[122X then each
  [22Xm_nj[122X  in  the last row of [22XM[122X is equal to [22X1[122X since there is only one coset of [22XG[122X
  in  [22XG[122X.  In  general,  [22Xm_ij[122X equals the number of conjugates of [22XH_i[122X containing
  [22XH_j[122X,  multiplied  by  the index of [22XH_i[122X in its normalizer in [22XG[122X. Moreover, the
  number  [22Xc_ij[122X  of conjugates of [22XH_j[122X which are contained in [22XH_i[122X can be derived
  from the marks [22Xm_ij[122X via the formula[133X
  
  
  [24X[33X[0;6Yc_ij = ( m_ij m_j1 ) / ( m_i1 m_jj )[133X[124X
  
  [33X[0;0Y.[133X
  
  [33X[0;0YBoth  the  marks  [22Xm_ij[122X  and the numbers of subgroups [22Xc_ij[122X are needed for the
  functions described in this chapter.[133X
  
  [33X[0;0YA  brief  survey  of  properties  of  tables  of  marks and a description of
  algorithms for the interactive construction of tables of marks using [5XGAP[105X can
  be found in [Pfe97].[133X
  
  
  [1X70.2 [33X[0;0YTable of Marks Objects in GAP[133X[101X
  
  [33X[0;0YA  table  of  marks  of  a  group  [22XG[122X  in  [5XGAP[105X is represented by an immutable
  (see [14X12.6[114X) object [3Xtom[103X in the category [2XIsTableOfMarks[102X ([14X70.6-2[114X), with defining
  attributes  [2XSubsTom[102X  ([14X70.7-1[114X)  and  [2XMarksTom[102X  ([14X70.7-1[114X). These two attributes
  encode  the matrix of marks in a compressed form. The [2XSubsTom[102X ([14X70.7-1[114X) value
  of  [3Xtom[103X  is  a  list  where  for each conjugacy class of subgroups the class
  numbers  of its subgroups are stored. These are exactly the positions in the
  corresponding  row  of  the  matrix of marks which have nonzero entries. The
  marks themselves are stored via the [2XMarksTom[102X ([14X70.7-1[114X) value of [3Xtom[103X, which is
  a  list  that  contains  for  each entry in [10XSubsTom( [3Xtom[103X[10X )[110X the corresponding
  nonzero value of the table of marks.[133X
  
  [33X[0;0YIt  is  possible to create table of marks objects that do not store a group,
  moreover  one  can  create  a  table  of marks object from a matrix of marks
  (see [2XTableOfMarks[102X  ([14X70.3-1[114X)).  So it may happen that a table of marks object
  in  [5XGAP[105X  is  in  fact [13Xnot[113X the table of marks of a group. To some extent, the
  consistency  of  a  table of marks object can be checked (see [14X70.9[114X), but [5XGAP[105X
  knows no general way to prove or disprove that a given matrix of nonnegative
  integers  is  the  matrix of marks for a group. Many functions for tables of
  marks  work  well  without access to the group –this is one of the arguments
  why   tables   of   marks  are  so  useful–,  but  for  example  normalizers
  (see [2XNormalizerTom[102X  ([14X70.9-4[114X))  and derived subgroups (see [2XDerivedSubgroupTom[102X
  ([14X70.9-2[114X))  of subgroups are in general not uniquely determined by the matrix
  of marks.[133X
  
  [33X[0;0Y[5XGAP[105X  tables of marks are assumed to be in lower triangular form, that is, if
  a  subgroup  from  the  conjugacy  class  corresponding  to  the [22Xi[122X-th row is
  contained  in a subgroup from the class corresponding to the [22Xj[122X-th row j then
  [22Xi ≤ j[122X.[133X
  
  [33X[0;0YThe  [2XMarksTom[102X  ([14X70.7-1[114X)  information  can be computed from the values of the
  attributes  [2XNrSubsTom[102X ([14X70.7-2[114X), [2XLengthsTom[102X ([14X70.7-3[114X), [2XOrdersTom[102X ([14X70.7-2[114X), and
  [2XSubsTom[102X ([14X70.7-1[114X). [2XNrSubsTom[102X ([14X70.7-2[114X) stores a list containing for each entry
  in  the  [2XSubsTom[102X  ([14X70.7-1[114X) value the corresponding number of conjugates that
  are  contained in a subgroup, [2XLengthsTom[102X ([14X70.7-3[114X) a list containing for each
  conjugacy  class  of  subgroups  its  length,  and [2XOrdersTom[102X ([14X70.7-2[114X) a list
  containing for each class of subgroups their order. So the [2XMarksTom[102X ([14X70.7-1[114X)
  value  of [3Xtom[103X may be missing provided that the values of [2XNrSubsTom[102X ([14X70.7-2[114X),
  [2XLengthsTom[102X ([14X70.7-3[114X), and [2XOrdersTom[102X ([14X70.7-2[114X) are stored in [3Xtom[103X.[133X
  
  [33X[0;0YAdditional  information  about a table of marks is needed by some functions.
  The class numbers of normalizers in [22XG[122X and the number of the derived subgroup
  of  [22XG[122X can be stored via appropriate attributes (see [2XNormalizersTom[102X ([14X70.9-4[114X),
  [2XDerivedSubgroupTom[102X ([14X70.9-2[114X)).[133X
  
  [33X[0;0YIf  [3Xtom[103X  stores its group [22XG[122X and a bijection from the rows and columns of the
  matrix  of  marks  of  [3Xtom[103X  to  the  classes  of subgroups of [22XG[122X then clearly
  normalizers,  derived  subgroup  etc. can be computed from this information.
  But  in  general a table of marks need not have access to [22XG[122X, for example [3Xtom[103X
  might have been constructed from a generic table of marks (see [14X70.12[114X), or as
  table   of   marks   of   a  factor  group  from  a  given  table  of  marks
  (see [2XFactorGroupTom[102X  ([14X70.9-11[114X)).  Access  to  the group [22XG[122X is provided by the
  attribute  [2XUnderlyingGroup[102X  ([14X70.7-7[114X)  if  this  value  is set. Access to the
  relevant  information  about conjugacy classes of subgroups of [22XG[122X –compatible
  with  the  ordering of rows and columns of the marks in [3Xtom[103X– is signalled by
  the filter [2XIsTableOfMarksWithGens[102X ([14X70.10-3[114X).[133X
  
  [33X[0;0YSeveral  examples  in  this  chapter require the [5XGAP[105X package [5XTomLib[105X (the [5XGAP[105X
  Library of Tables of Marks) to be available. If it is not yet loaded then we
  load it now.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLoadPackage( "tomlib" );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X70.3 [33X[0;0YConstructing Tables of Marks[133X[101X
  
  [1X70.3-1 TableOfMarks[101X
  
  [33X[1;0Y[29X[2XTableOfMarks[102X( [3XG[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XTableOfMarks[102X( [3Xstring[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XTableOfMarks[102X( [3Xmatrix[103X ) [32X attribute[133X
  
  [33X[0;0YIn the first form, [3XG[103X must be a finite group, and [2XTableOfMarks[102X constructs the
  table of marks of [3XG[103X. This computation requires the knowledge of the complete
  subgroup  lattice  of  [3XG[103X (see [2XLatticeSubgroups[102X ([14X39.20-1[114X)). If the lattice is
  not  yet  stored  then it will be constructed. This may take a while if [3XG[103X is
  large. The result has the [2XIsTableOfMarksWithGens[102X ([14X70.10-3[114X) value [9Xtrue[109X.[133X
  
  [33X[0;0YIn the second form, [3Xstring[103X must be a string, and [2XTableOfMarks[102X gets the table
  of  marks  with name [3Xstring[103X from the [5XGAP[105X library (see [14X70.13[114X). If no table of
  marks with this name is contained in the library then [9Xfail[109X is returned.[133X
  
  [33X[0;0YIn  the  third  form, [3Xmatrix[103X must be a matrix or a list of rows describing a
  lower  triangular  matrix  where the part above the diagonal is omitted. For
  such  an  argument  [3Xmatrix[103X,  [2XTableOfMarks[102X  returns  a  table of marks object
  (see [14X70.2[114X)  for  which  [3Xmatrix[103X  is  the matrix of marks. Note that not every
  matrix  (containing  only  nonnegative  integers and having lower triangular
  shape)  describes  a  table  of  marks  of a group. Necessary conditions are
  checked   with  [2XIsInternallyConsistent[102X  ([14X70.9-1[114X)  (see [14X70.9[114X),  and  [9Xfail[109X  is
  returned  if  [3Xmatrix[103X  is  proved  not  to describe a matrix of marks; but if
  [2XTableOfMarks[102X  returns  a table of marks object created from a matrix then it
  may  still happen that this object does not describe the table of marks of a
  group.[133X
  
  [33X[0;0YFor  an  overview  of  operations  for  table  of  marks  objects,  see  the
  introduction to Chapter [14X70[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtom:= TableOfMarks( AlternatingGroup( 5 ) );[127X[104X
    [4X[28XTableOfMarks( Alt( [ 1 .. 5 ] ) )[128X[104X
    [4X[25Xgap>[125X [27XTableOfMarks( "J5" );[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27Xa5:= TableOfMarks( "A5" );[127X[104X
    [4X[28XTableOfMarks( "A5" )[128X[104X
    [4X[25Xgap>[125X [27Xmat:=[127X[104X
    [4X[25X>[125X [27X[ [ 60, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 30, 2, 0, 0, 0, 0, 0, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X  [ 20, 0, 2, 0, 0, 0, 0, 0, 0 ], [ 15, 3, 0, 3, 0, 0, 0, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X  [ 12, 0, 0, 0, 2, 0, 0, 0, 0 ], [ 10, 2, 1, 0, 0, 1, 0, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X  [ 6, 2, 0, 0, 1, 0, 1, 0, 0 ], [ 5, 1, 2, 1, 0, 0, 0, 1, 0 ],[127X[104X
    [4X[25X>[125X [27X  [ 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ];;[127X[104X
    [4X[25Xgap>[125X [27XTableOfMarks( mat );[127X[104X
    [4X[28XTableOfMarks( <9 classes> )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe following [2XTableOfMarks[102X methods for a group are installed.[133X
  
  [30X    [33X[0;6YIf  the  group  is known to be cyclic then [2XTableOfMarks[102X constructs the
        table  of  marks  essentially without the group, instead the knowledge
        about the structure of cyclic groups is used.[133X
  
  [30X    [33X[0;6YIf  the  lattice  of  subgroups  is  already  stored in the group then
        [2XTableOfMarks[102X   computes   the   table   of   marks  from  the  lattice
        (see [2XTableOfMarksByLattice[102X ([14X70.3-2[114X)).[133X
  
  [30X    [33X[0;6YIf  the  group  is  known  to  be solvable then [2XTableOfMarks[102X takes the
        lattice  of  subgroups  (see [2XLatticeSubgroups[102X  ([14X39.20-1[114X)) of the group
        –which means that the lattice is computed if it is not yet stored– and
        then  computes  the  table  of  marks  from  it.  This  method is also
        accessible via the global function [2XTableOfMarksByLattice[102X ([14X70.3-2[114X).[133X
  
  [30X    [33X[0;6YIf  the  group  doesn't know its lattice of subgroups or its conjugacy
        classes of subgroups then the table of marks and the conjugacy classes
        of  subgroups  are  computed  at the same time by the cyclic extension
        method.  Only  the  table  of  marks  is  stored because the conjugacy
        classes  of  subgroups  or the lattice of subgroups can be easily read
        off (see [2XLatticeSubgroupsByTom[102X ([14X70.3-3[114X)).[133X
  
  [33X[0;0YConversely,  the  lattice  of subgroups of a group with known table of marks
  can   be   computed   using   the   table   of   marks,   via  the  function
  [2XLatticeSubgroupsByTom[102X  ([14X70.3-3[114X).  This  is  also  installed  as a method for
  [2XLatticeSubgroups[102X ([14X39.20-1[114X).[133X
  
  [1X70.3-2 TableOfMarksByLattice[101X
  
  [33X[1;0Y[29X[2XTableOfMarksByLattice[102X( [3XG[103X ) [32X function[133X
  
  [33X[0;0Y[2XTableOfMarksByLattice[102X  computes  the  table of marks of the group [3XG[103X from the
  lattice  of  subgroups  of  [3XG[103X. This lattice is computed via [2XLatticeSubgroups[102X
  ([14X39.20-1[114X)  if  it is not yet stored in [3XG[103X. The function [2XTableOfMarksByLattice[102X
  is  installed  as a method for [2XTableOfMarks[102X ([14X70.3-1[114X) for solvable groups and
  groups  with  stored subgroup lattice, and is available as a global variable
  only in order to provide explicit access to this method.[133X
  
  [1X70.3-3 LatticeSubgroupsByTom[101X
  
  [33X[1;0Y[29X[2XLatticeSubgroupsByTom[102X( [3XG[103X ) [32X function[133X
  
  [33X[0;0Y[2XLatticeSubgroupsByTom[102X  computes the lattice of subgroups of [3XG[103X from the table
  of marks of [3XG[103X, using [2XRepresentativeTom[102X ([14X70.10-4[114X).[133X
  
  
  [1X70.4 [33X[0;0YPrinting Tables of Marks[133X[101X
  
  [1X70.4-1 ViewObj[101X
  
  [33X[1;0Y[29X[2XViewObj[102X( [3Xtom[103X ) [32X method[133X
  
  [33X[0;0YThe  default  [2XViewObj[102X  ([14X6.3-5[114X)  method for tables of marks prints the string
  [10X"TableOfMarks"[110X,  followed  by  –if  known–  the  identifier  (see [2XIdentifier[102X
  ([14X70.7-9[114X))  or  the  group  of  the  table  of marks enclosed in brackets; if
  neither  group  nor  identifier  are known then just the number of conjugacy
  classes of subgroups is printed instead.[133X
  
  [1X70.4-2 PrintObj[101X
  
  [33X[1;0Y[29X[2XPrintObj[102X( [3Xtom[103X ) [32X method[133X
  
  [33X[0;0YThe  default  [2XPrintObj[102X  ([14X6.3-5[114X)  method for tables of marks does the same as
  [2XViewObj[102X  ([14X6.3-5[114X), except that [2XPrintObj[102X ([14X6.3-5[114X) is used for the group instead
  of [2XViewObj[102X ([14X6.3-5[114X).[133X
  
  [1X70.4-3 Display[101X
  
  [33X[1;0Y[29X[2XDisplay[102X( [3Xtom[103X[, [3Xarec[103X] ) [32X method[133X
  
  [33X[0;0YThe  default  [2XDisplay[102X  ([14X6.3-6[114X)  method  for  a table of marks [3Xtom[103X produces a
  formatted  output  of  the marks in [3Xtom[103X. Each line of output begins with the
  number  of  the corresponding class of subgroups. This number is repeated if
  the  output spreads over several pages. The number of columns printed at one
  time depends on the actual line length, which can be accessed and changed by
  the function [2XSizeScreen[102X ([14X6.12-1[114X).[133X
  
  [33X[0;0YAn  interactive  variant  of  [2XDisplay[102X ([14X6.3-6[114X) is the [2XBrowse[102X ([14XBrowse: Browse[114X)
  method  for  tables of marks that is provided by the [5XGAP[105X package [5XBrowse[105X, see
  [2XBrowse[102X ([14XBrowse: Browse for tables of marks[114X).[133X
  
  [33X[0;0YThe  optional  second argument [3Xarec[103X of [2XDisplay[102X ([14X6.3-6[114X) can be used to change
  the  default  style  for displaying a table of marks. [3Xarec[103X must be a record,
  its relevant components are the following.[133X
  
  [8X[10Xclasses[110X[8X[108X
        [33X[0;6Ya  list  of  class  numbers to select only the rows and columns of the
        matrix that correspond to this list for printing,[133X
  
  [8X[10Xform[110X[8X[108X
        [33X[0;6Yone  of the strings [10X"subgroups"[110X, [10X"supergroups"[110X; in the former case, at
        position [22X(i,j)[122X of the matrix the number of conjugates of [22XH_j[122X contained
        in  [22XH_i[122X  is  printed,  and  in  the latter case, at position [22X(i,j)[122X the
        number of conjugates of [22XH_i[122X which contain [22XH_j[122X is printed.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtom:= TableOfMarks( "A5" );;[127X[104X
    [4X[25Xgap>[125X [27XDisplay( tom );[127X[104X
    [4X[28X1:  60[128X[104X
    [4X[28X2:  30 2[128X[104X
    [4X[28X3:  20 . 2[128X[104X
    [4X[28X4:  15 3 . 3[128X[104X
    [4X[28X5:  12 . . . 2[128X[104X
    [4X[28X6:  10 2 1 . . 1[128X[104X
    [4X[28X7:   6 2 . . 1 . 1[128X[104X
    [4X[28X8:   5 1 2 1 . . . 1[128X[104X
    [4X[28X9:   1 1 1 1 1 1 1 1 1[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XDisplay( tom, rec( classes:= [ 1, 2, 3, 4, 8 ] ) );[127X[104X
    [4X[28X1:  60[128X[104X
    [4X[28X2:  30 2[128X[104X
    [4X[28X3:  20 . 2[128X[104X
    [4X[28X4:  15 3 . 3[128X[104X
    [4X[28X8:   5 1 2 1 1[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XDisplay( tom, rec( form:= "subgroups" ) );[127X[104X
    [4X[28X1:  1[128X[104X
    [4X[28X2:  1  1[128X[104X
    [4X[28X3:  1  .  1[128X[104X
    [4X[28X4:  1  3  . 1[128X[104X
    [4X[28X5:  1  .  . . 1[128X[104X
    [4X[28X6:  1  3  1 . .  1[128X[104X
    [4X[28X7:  1  5  . . 1  . 1[128X[104X
    [4X[28X8:  1  3  4 1 .  . . 1[128X[104X
    [4X[28X9:  1 15 10 5 6 10 6 5 1[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XDisplay( tom, rec( form:= "supergroups" ) );[127X[104X
    [4X[28X1:   1[128X[104X
    [4X[28X2:  15 1[128X[104X
    [4X[28X3:  10 . 1[128X[104X
    [4X[28X4:   5 1 . 1[128X[104X
    [4X[28X5:   6 . . . 1[128X[104X
    [4X[28X6:  10 2 1 . . 1[128X[104X
    [4X[28X7:   6 2 . . 1 . 1[128X[104X
    [4X[28X8:   5 1 2 1 . . . 1[128X[104X
    [4X[28X9:   1 1 1 1 1 1 1 1 1[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X70.5 [33X[0;0YSorting Tables of Marks[133X[101X
  
  [1X70.5-1 SortedTom[101X
  
  [33X[1;0Y[29X[2XSortedTom[102X( [3Xtom[103X, [3Xperm[103X ) [32X operation[133X
  
  [33X[0;0Y[2XSortedTom[102X  returns  a table of marks where the rows and columns of the table
  of marks [3Xtom[103X are reordered according to the permutation [3Xperm[103X.[133X
  
  [33X[0;0Y[13XNote[113X  that  in each table of marks in [5XGAP[105X, the matrix of marks is assumed to
  have  lower  triangular  shape  (see [14X70.2[114X). If the permutation [3Xperm[103X does [13Xnot[113X
  have this property then the functions for tables of marks might return wrong
  results when applied to the output of [2XSortedTom[102X.[133X
  
  [33X[0;0YThe  returned table of marks has only those attribute values stored that are
  known for [3Xtom[103X and listed in [2XTableOfMarksComponents[102X ([14X70.6-4[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtom:= TableOfMarksCyclic( 6 );;  Display( tom );[127X[104X
    [4X[28X1:  6[128X[104X
    [4X[28X2:  3 3[128X[104X
    [4X[28X3:  2 . 2[128X[104X
    [4X[28X4:  1 1 1 1[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xsorted:= SortedTom( tom, (2,3) );;  Display( sorted );[127X[104X
    [4X[28X1:  6[128X[104X
    [4X[28X2:  2 2[128X[104X
    [4X[28X3:  3 . 3[128X[104X
    [4X[28X4:  1 1 1 1[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xwrong:= SortedTom( tom, (1,2) );;  Display( wrong );[127X[104X
    [4X[28X1:  3[128X[104X
    [4X[28X2:  . 6[128X[104X
    [4X[28X3:  . 2 2[128X[104X
    [4X[28X4:  1 1 1 1[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X70.5-2 PermutationTom[101X
  
  [33X[1;0Y[29X[2XPermutationTom[102X( [3Xtom[103X ) [32X attribute[133X
  
  [33X[0;0YFor the table of marks [3Xtom[103X of the group [22XG[122X stored as [2XUnderlyingGroup[102X ([14X70.7-7[114X)
  value of [3Xtom[103X, [2XPermutationTom[102X is a permutation [22Xπ[122X such that the [22Xi[122X-th conjugacy
  class  of  subgroups  of  [22XG[122X belongs to the [22Xi^π[122X-th column and row of marks in
  [3Xtom[103X.[133X
  
  [33X[0;0YThis attribute value is bound only if [3Xtom[103X was obtained from another table of
  marks  by  permuting with [2XSortedTom[102X ([14X70.5-1[114X), and there is no default method
  to compute its value.[133X
  
  [33X[0;0YThe  attribute  is  necessary  because  the original and the sorted table of
  marks  have the same identifier and the same group, and information computed
  from  the  group may depend on the ordering of marks, for example the fusion
  from the ordinary character table of [22XG[122X into [3Xtom[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMarksTom( tom )[2];[127X[104X
    [4X[28X[ 3, 3 ][128X[104X
    [4X[25Xgap>[125X [27XMarksTom( sorted )[2];[127X[104X
    [4X[28X[ 2, 2 ][128X[104X
    [4X[25Xgap>[125X [27XHasPermutationTom( sorted );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XPermutationTom( sorted );[127X[104X
    [4X[28X(2,3)[128X[104X
  [4X[32X[104X
  
  
  [1X70.6 [33X[0;0YTechnical Details about Tables of Marks[133X[101X
  
  [1X70.6-1 InfoTom[101X
  
  [33X[1;0Y[29X[2XInfoTom[102X [32X info class[133X
  
  [33X[0;0Yis the info class for computations concerning tables of marks.[133X
  
  [1X70.6-2 IsTableOfMarks[101X
  
  [33X[1;0Y[29X[2XIsTableOfMarks[102X( [3Xobj[103X ) [32X Category[133X
  
  [33X[0;0YEach table of marks belongs to this category.[133X
  
  [1X70.6-3 TableOfMarksFamily[101X
  
  [33X[1;0Y[29X[2XTableOfMarksFamily[102X [32X family[133X
  
  [33X[0;0YEach table of marks belongs to this family.[133X
  
  [1X70.6-4 TableOfMarksComponents[101X
  
  [33X[1;0Y[29X[2XTableOfMarksComponents[102X [32X global variable[133X
  
  [33X[0;0YThe  list  [2XTableOfMarksComponents[102X  is  used  when a table of marks object is
  created     from    a    record    via    [2XConvertToTableOfMarks[102X    ([14X70.6-5[114X).
  [2XTableOfMarksComponents[102X  contains at position [22X2i-1[122X a name of an attribute and
  at position [22X2i[122X the corresponding attribute getter function.[133X
  
  [1X70.6-5 ConvertToTableOfMarks[101X
  
  [33X[1;0Y[29X[2XConvertToTableOfMarks[102X( [3Xrecord[103X ) [32X function[133X
  
  [33X[0;0Y[2XConvertToTableOfMarks[102X    converts    a    record    with   components   from
  [2XTableOfMarksComponents[102X  ([14X70.6-4[114X)  into  a  table  of  marks  object with the
  corresponding attributes.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xrecord:= rec( MarksTom:= [ [ 4 ], [ 2, 2 ], [ 1, 1, 1 ] ],[127X[104X
    [4X[25X>[125X [27X SubsTom:= [ [ 1 ], [ 1, 2 ], [ 1, 2, 3 ] ] );;[127X[104X
    [4X[25Xgap>[125X [27XConvertToTableOfMarks( record );;[127X[104X
    [4X[25Xgap>[125X [27Xrecord;[127X[104X
    [4X[28XTableOfMarks( <3 classes> )[128X[104X
  [4X[32X[104X
  
  
  [1X70.7 [33X[0;0YAttributes of Tables of Marks[133X[101X
  
  [1X70.7-1 MarksTom[101X
  
  [33X[1;0Y[29X[2XMarksTom[102X( [3Xtom[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XSubsTom[102X( [3Xtom[103X ) [32X attribute[133X
  
  [33X[0;0YThe  matrix  of  marks  (see [14X70.1[114X)  of the table of marks [3Xtom[103X is stored in a
  compressed  form  where zeros are omitted, using the attributes [2XMarksTom[102X and
  [2XSubsTom[102X.  If  [22XM[122X  is the square matrix of marks of [3Xtom[103X (see [2XMatTom[102X ([14X70.7-10[114X))
  then the [2XSubsTom[102X value of [3Xtom[103X is a list that contains at position [22Xi[122X the list
  of  all  positions of nonzero entries of the [22Xi[122X-th row of [22XM[122X, and the [2XMarksTom[102X
  value  of  [3Xtom[103X  is  a  list  that  contains  at  position  [22Xi[122X the list of the
  corresponding marks.[133X
  
  [33X[0;0Y[2XMarksTom[102X  and [2XSubsTom[102X are defining attributes of tables of marks (see [14X70.2[114X).
  There  is no default method for computing the [2XSubsTom[102X value, and the default
  [2XMarksTom[102X  method  needs  the  values  of  [2XNrSubsTom[102X  ([14X70.7-2[114X)  and [2XOrdersTom[102X
  ([14X70.7-2[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xa5:= TableOfMarks( "A5" );[127X[104X
    [4X[28XTableOfMarks( "A5" )[128X[104X
    [4X[25Xgap>[125X [27XMarksTom( a5 );[127X[104X
    [4X[28X[ [ 60 ], [ 30, 2 ], [ 20, 2 ], [ 15, 3, 3 ], [ 12, 2 ],[128X[104X
    [4X[28X  [ 10, 2, 1, 1 ], [ 6, 2, 1, 1 ], [ 5, 1, 2, 1, 1 ],[128X[104X
    [4X[28X  [ 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27XSubsTom( a5 );[127X[104X
    [4X[28X[ [ 1 ], [ 1, 2 ], [ 1, 3 ], [ 1, 2, 4 ], [ 1, 5 ], [ 1, 2, 3, 6 ],[128X[104X
    [4X[28X  [ 1, 2, 5, 7 ], [ 1, 2, 3, 4, 8 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9 ] ][128X[104X
  [4X[32X[104X
  
  [1X70.7-2 NrSubsTom[101X
  
  [33X[1;0Y[29X[2XNrSubsTom[102X( [3Xtom[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XOrdersTom[102X( [3Xtom[103X ) [32X attribute[133X
  
  [33X[0;0YInstead  of  storing the marks (see [2XMarksTom[102X ([14X70.7-1[114X)) of the table of marks
  [3Xtom[103X  one  can  use  a  matrix which contains at position [22X(i,j)[122X the number of
  subgroups  of  conjugacy  class  [22Xj[122X  that  are contained in one member of the
  conjugacy  class  [22Xi[122X.  These  values are stored in the [2XNrSubsTom[102X value in the
  same way as the marks in the [2XMarksTom[102X ([14X70.7-1[114X) value.[133X
  
  [33X[0;0Y[2XOrdersTom[102X  returns  a  list  that  contains  at  position  [22Xi[122X  the order of a
  representative of the [22Xi[122X-th conjugacy class of subgroups of [3Xtom[103X.[133X
  
  [33X[0;0YOne  can  compute  the  [2XNrSubsTom[102X  and  [2XOrdersTom[102X  values  from the [2XMarksTom[102X
  ([14X70.7-1[114X) value of [3Xtom[103X and vice versa.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XNrSubsTom( a5 );[127X[104X
    [4X[28X[ [ 1 ], [ 1, 1 ], [ 1, 1 ], [ 1, 3, 1 ], [ 1, 1 ], [ 1, 3, 1, 1 ],[128X[104X
    [4X[28X  [ 1, 5, 1, 1 ], [ 1, 3, 4, 1, 1 ], [ 1, 15, 10, 5, 6, 10, 6, 5, 1 ][128X[104X
    [4X[28X ][128X[104X
    [4X[25Xgap>[125X [27XOrdersTom( a5 );[127X[104X
    [4X[28X[ 1, 2, 3, 4, 5, 6, 10, 12, 60 ][128X[104X
  [4X[32X[104X
  
  [1X70.7-3 LengthsTom[101X
  
  [33X[1;0Y[29X[2XLengthsTom[102X( [3Xtom[103X ) [32X attribute[133X
  
  [33X[0;0YFor  a  table  of marks [3Xtom[103X, [2XLengthsTom[102X returns a list of the lengths of the
  conjugacy classes of subgroups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLengthsTom( a5 );[127X[104X
    [4X[28X[ 1, 15, 10, 5, 6, 10, 6, 5, 1 ][128X[104X
  [4X[32X[104X
  
  [1X70.7-4 ClassTypesTom[101X
  
  [33X[1;0Y[29X[2XClassTypesTom[102X( [3Xtom[103X ) [32X attribute[133X
  
  [33X[0;0Y[2XClassTypesTom[102X distinguishes isomorphism types of the classes of subgroups of
  the  table of marks [3Xtom[103X as far as this is possible from the [2XSubsTom[102X ([14X70.7-1[114X)
  and [2XMarksTom[102X ([14X70.7-1[114X) values of [3Xtom[103X.[133X
  
  [33X[0;0YTwo  subgroups  are  clearly  not  isomorphic if they have different orders.
  Moreover,  isomorphic subgroups must contain the same number of subgroups of
  each type.[133X
  
  [33X[0;0YEach  type  is  represented by a positive integer. [2XClassTypesTom[102X returns the
  list which contains for each class of subgroups its corresponding type.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xa6:= TableOfMarks( "A6" );;[127X[104X
    [4X[25Xgap>[125X [27XClassTypesTom( a6 );[127X[104X
    [4X[28X[ 1, 2, 3, 3, 4, 5, 6, 6, 7, 7, 8, 9, 10, 11, 11, 12, 13, 13, 14, 15,[128X[104X
    [4X[28X  15, 16 ][128X[104X
  [4X[32X[104X
  
  [1X70.7-5 ClassNamesTom[101X
  
  [33X[1;0Y[29X[2XClassNamesTom[102X( [3Xtom[103X ) [32X attribute[133X
  
  [33X[0;0Y[2XClassNamesTom[102X   constructs  generic  names  for  the  conjugacy  classes  of
  subgroups of the table of marks [3Xtom[103X. In general, the generic name of a class
  of non-cyclic subgroups consists of three parts and has the form [10X"([110X[3Xo[103X[10X)_{[110X[3Xt[103X[10X}[110X[3Xl[103X[10X"[110X,
  where   [3Xo[103X  indicates  the  order  of  the  subgroup,  [3Xt[103X  is  a  number  that
  distinguishes  different  types  of  subgroups of the same order, and [3Xl[103X is a
  letter  that  distinguishes classes of subgroups of the same type and order.
  The  type  of  a  subgroup  is determined by the numbers of its subgroups of
  other  types  (see [2XClassTypesTom[102X  ([14X70.7-4[114X)).  This  is  slightly weaker than
  isomorphism.[133X
  
  [33X[0;0YThe  letter is omitted if there is only one class of subgroups of that order
  and  type, and the type is omitted if there is only one class of that order.
  Moreover,  the  braces [10X{}[110X around the type are omitted if the type number has
  only one digit.[133X
  
  [33X[0;0YFor  classes  of  cyclic  subgroups, the parentheses round the order and the
  type  are  omitted.  Hence  the  most general form of their generic names is
  [10X"[3Xo[103X[10X,[3Xl[103X[10X"[110X.  Again,  the  letter  is omitted if there is only one class of cyclic
  subgroups of that order.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XClassNamesTom( a6 );[127X[104X
    [4X[28X[ "1", "2", "3a", "3b", "5", "4", "(4)_2a", "(4)_2b", "(6)a", "(6)b",[128X[104X
    [4X[28X  "(8)", "(9)", "(10)", "(12)a", "(12)b", "(18)", "(24)a", "(24)b",[128X[104X
    [4X[28X  "(36)", "(60)a", "(60)b", "(360)" ][128X[104X
  [4X[32X[104X
  
  [1X70.7-6 FusionsTom[101X
  
  [33X[1;0Y[29X[2XFusionsTom[102X( [3Xtom[103X ) [32X attribute[133X
  
  [33X[0;0YFor  a table of marks [3Xtom[103X, [2XFusionsTom[102X is a list of fusions into other tables
  of  marks.  Each  fusion  is a list of length two, the first entry being the
  [2XIdentifier[102X  ([14X70.7-9[114X))  value  of the image table, the second entry being the
  list of images of the class positions of [3Xtom[103X in the image table.[133X
  
  [33X[0;0YThis  attribute  is  mainly  used  for  tables  of  marks in the [5XGAP[105X library
  (see [14X70.13[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfus:= FusionsTom( a6 );;[127X[104X
    [4X[25Xgap>[125X [27Xfus[1];[127X[104X
    [4X[28X[ "L3(4)",[128X[104X
    [4X[28X  [ 1, 2, 3, 3, 14, 5, 9, 7, 15, 15, 24, 26, 27, 32, 33, 50, 57, 55,[128X[104X
    [4X[28X      63, 73, 77, 90 ] ][128X[104X
  [4X[32X[104X
  
  [1X70.7-7 UnderlyingGroup[101X
  
  [33X[1;0Y[29X[2XUnderlyingGroup[102X( [3Xtom[103X ) [32X attribute[133X
  
  [33X[0;0Y[2XUnderlyingGroup[102X  is used to access an underlying group that is stored on the
  table  of  marks  [3Xtom[103X.  There  is no default method to compute an underlying
  group if it is not stored.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XUnderlyingGroup( a6 );[127X[104X
    [4X[28XGroup([ (1,2)(3,4), (1,2,4,5)(3,6) ])[128X[104X
  [4X[32X[104X
  
  [1X70.7-8 IdempotentsTom[101X
  
  [33X[1;0Y[29X[2XIdempotentsTom[102X( [3Xtom[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XIdempotentsTomInfo[102X( [3Xtom[103X ) [32X attribute[133X
  
  [33X[0;0Y[2XIdempotentsTom[102X  encodes  the  idempotents  of  the  integral  Burnside  ring
  described  by  the  table  of  marks  [3Xtom[103X.  The  return value is a list [22Xl[122X of
  positive   integers  such  that  each  row  vector  describing  a  primitive
  idempotent  has  value [22X1[122X at all positions with the same entry in [22Xl[122X, and [22X0[122X at
  all other positions.[133X
  
  [33X[0;0YAccording   to   A. Dress [Dre69]   (see  also [Pfe97]),  these  idempotents
  correspond  to the classes of perfect subgroups, and each such idempotent is
  the  characteristic  function  of  all  those subgroups that arise by cyclic
  extension  from  the corresponding perfect subgroup (see [2XCyclicExtensionsTom[102X
  ([14X70.9-7[114X)).[133X
  
  [33X[0;0Y[2XIdempotentsTomInfo[102X  returns  a  record  with  components [10Xfixpointvectors[110X and
  [10Xprimidems[110X,  both  bound to lists. The [22Xi[122X-th entry of the [10Xfixpointvectors[110X list
  is  the  [22X0-1[122X-vector  describing  the [22Xi[122X-th primitive idempotent, and the [22Xi[122X-th
  entry  of  [10Xprimidems[110X  is the decomposition of this idempotent in the rows of
  [3Xtom[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIdempotentsTom( a5 );[127X[104X
    [4X[28X[ 1, 1, 1, 1, 1, 1, 1, 1, 9 ][128X[104X
    [4X[25Xgap>[125X [27XIdempotentsTomInfo( a5 );[127X[104X
    [4X[28Xrec([128X[104X
    [4X[28X  fixpointvectors := [ [ 1, 1, 1, 1, 1, 1, 1, 1, 0 ],[128X[104X
    [4X[28X      [ 0, 0, 0, 0, 0, 0, 0, 0, 1 ] ],[128X[104X
    [4X[28X  primidems := [ [ 1, -2, -1, 0, 0, 1, 1, 1 ],[128X[104X
    [4X[28X      [ -1, 2, 1, 0, 0, -1, -1, -1, 1 ] ] )[128X[104X
  [4X[32X[104X
  
  [1X70.7-9 Identifier[101X
  
  [33X[1;0Y[29X[2XIdentifier[102X( [3Xtom[103X ) [32X attribute[133X
  
  [33X[0;0YThe  identifier of a table of marks [3Xtom[103X is a string. It is used for printing
  the  table  of  marks  (see [14X70.4[114X)  and  in  fusions  between tables of marks
  (see [2XFusionsTom[102X ([14X70.7-6[114X)).[133X
  
  [33X[0;0YIf  [3Xtom[103X  is  a  table  of  marks  from  the  [5XGAP[105X  library of tables of marks
  (see [14X70.13[114X)  then  it  has  an identifier, and if [3Xtom[103X was constructed from a
  group with [2XName[102X ([14X12.8-2[114X) then this name is chosen as [2XIdentifier[102X value. There
  is no default method to compute an identifier in all other cases.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIdentifier( a5 );[127X[104X
    [4X[28X"A5"[128X[104X
  [4X[32X[104X
  
  [1X70.7-10 MatTom[101X
  
  [33X[1;0Y[29X[2XMatTom[102X( [3Xtom[103X ) [32X attribute[133X
  
  [33X[0;0Y[2XMatTom[102X  returns  the square matrix of marks (see [14X70.1[114X) of the table of marks
  [3Xtom[103X  which  is  stored  in  a  compressed form using the attributes [2XMarksTom[102X
  ([14X70.7-1[114X)  and  [2XSubsTom[102X  ([14X70.7-1[114X) This may need substantially more space than
  the values of [2XMarksTom[102X ([14X70.7-1[114X) and [2XSubsTom[102X ([14X70.7-1[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMatTom( a5 );[127X[104X
    [4X[28X[ [ 60, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 30, 2, 0, 0, 0, 0, 0, 0, 0 ],[128X[104X
    [4X[28X  [ 20, 0, 2, 0, 0, 0, 0, 0, 0 ], [ 15, 3, 0, 3, 0, 0, 0, 0, 0 ],[128X[104X
    [4X[28X  [ 12, 0, 0, 0, 2, 0, 0, 0, 0 ], [ 10, 2, 1, 0, 0, 1, 0, 0, 0 ],[128X[104X
    [4X[28X  [ 6, 2, 0, 0, 1, 0, 1, 0, 0 ], [ 5, 1, 2, 1, 0, 0, 0, 1, 0 ],[128X[104X
    [4X[28X  [ 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ][128X[104X
  [4X[32X[104X
  
  [1X70.7-11 MoebiusTom[101X
  
  [33X[1;0Y[29X[2XMoebiusTom[102X( [3Xtom[103X ) [32X attribute[133X
  
  [33X[0;0Y[2XMoebiusTom[102X  computes  the  Möbius values both of the subgroup lattice of the
  group  [22XG[122X  with  table  of marks [3Xtom[103X and of the poset of conjugacy classes of
  subgroups  of  [22XG[122X.  It  returns  a record where the component [10Xmu[110X contains the
  Möbius  values  of  the  subgroup lattice, and the component [10Xnu[110X contains the
  Möbius values of the poset.[133X
  
  [33X[0;0YMoreover,  according  to  an  observation  of  Isaacs  et  al. (see [HIÖ89],
  [Pah93]), the values on the subgroup lattice often can be derived from those
  of the poset of conjugacy classes. These [21Xexpected values[121X are returned in the
  component  [10Xex[110X, and the list of numbers of those subgroups where the expected
  value  does not coincide with the actual value are returned in the component
  [10Xhyp[110X.  For  the  computation  of  these  values,  the position of the derived
  subgroup  of  [22XG[122X  is  needed  (see [2XDerivedSubgroupTom[102X ([14X70.9-2[114X)). If it is not
  uniquely determined then the result does not have the components [10Xex[110X and [10Xhyp[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMoebiusTom( a5 );[127X[104X
    [4X[28Xrec( ex := [ -60, 4, 2,,, -1, -1, -1, 1 ], hyp := [  ],[128X[104X
    [4X[28X  mu := [ -60, 4, 2,,, -1, -1, -1, 1 ],[128X[104X
    [4X[28X  nu := [ -1, 2, 1,,, -1, -1, -1, 1 ] )[128X[104X
    [4X[25Xgap>[125X [27Xtom:= TableOfMarks( "M12" );;[127X[104X
    [4X[25Xgap>[125X [27Xmoebius:= MoebiusTom( tom );;[127X[104X
    [4X[25Xgap>[125X [27Xmoebius.hyp;[127X[104X
    [4X[28X[ 1, 2, 4, 16, 39, 45, 105 ][128X[104X
    [4X[25Xgap>[125X [27Xmoebius.mu[1];  moebius.ex[1];[127X[104X
    [4X[28X95040[128X[104X
    [4X[28X190080[128X[104X
  [4X[32X[104X
  
  [1X70.7-12 WeightsTom[101X
  
  [33X[1;0Y[29X[2XWeightsTom[102X( [3Xtom[103X ) [32X attribute[133X
  
  [33X[0;0Y[2XWeightsTom[102X  extracts  the  [13Xweights[113X  from  the  table of marks [3Xtom[103X, i.e., the
  diagonal  entries of the matrix of marks (see [2XMarksTom[102X ([14X70.7-1[114X)), indicating
  the index of a subgroup in its normalizer.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xwt:= WeightsTom( a5 );[127X[104X
    [4X[28X[ 60, 2, 2, 3, 2, 1, 1, 1, 1 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThis  information may be used to obtain the numbers of conjugate supergroups
  from the marks.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmarks:= MarksTom( a5 );;[127X[104X
    [4X[25Xgap>[125X [27XList( [ 1 .. 9 ], x -> marks[x] / wt[x] );[127X[104X
    [4X[28X[ [ 1 ], [ 15, 1 ], [ 10, 1 ], [ 5, 1, 1 ], [ 6, 1 ], [ 10, 2, 1, 1 ],[128X[104X
    [4X[28X  [ 6, 2, 1, 1 ], [ 5, 1, 2, 1, 1 ], [ 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ][128X[104X
  [4X[32X[104X
  
  
  [1X70.8 [33X[0;0YProperties of Tables of Marks[133X[101X
  
  [33X[0;0YFor  a  table  of  marks [3Xtom[103X of a group [22XG[122X, the following properties have the
  same  meaning  as  the  corresponding  properties  for [22XG[122X. Additionally, if a
  positive  integer  [3Xsub[103X is given as the second argument then the value of the
  corresponding property for the [3Xsub[103X-th class of subgroups of [3Xtom[103X is returned.[133X
  
  [1X70.8-1 IsAbelianTom[101X
  
  [33X[1;0Y[29X[2XIsAbelianTom[102X( [3Xtom[103X[, [3Xsub[103X] ) [32X property[133X
  [33X[1;0Y[29X[2XIsCyclicTom[102X( [3Xtom[103X[, [3Xsub[103X] ) [32X property[133X
  [33X[1;0Y[29X[2XIsNilpotentTom[102X( [3Xtom[103X[, [3Xsub[103X] ) [32X property[133X
  [33X[1;0Y[29X[2XIsPerfectTom[102X( [3Xtom[103X[, [3Xsub[103X] ) [32X property[133X
  [33X[1;0Y[29X[2XIsSolvableTom[102X( [3Xtom[103X[, [3Xsub[103X] ) [32X property[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtom:= TableOfMarks( "A5" );;[127X[104X
    [4X[25Xgap>[125X [27XIsAbelianTom( tom );  IsPerfectTom( tom );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsAbelianTom( tom, 3 );  IsNilpotentTom( tom, 7 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsPerfectTom( tom, 7 );  IsSolvableTom( tom, 7 );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xfor i in [ 1 .. 6 ] do[127X[104X
    [4X[25X>[125X [27XPrint( i, ": ", IsCyclicTom(a5, i), "  " );[127X[104X
    [4X[25X>[125X [27Xod;  Print( "\n" );[127X[104X
    [4X[28X1: true  2: true  3: true  4: false  5: true  6: false[128X[104X
  [4X[32X[104X
  
  
  [1X70.9 [33X[0;0YOther Operations for Tables of Marks[133X[101X
  
  [1X70.9-1 IsInternallyConsistent[101X
  
  [33X[1;0Y[29X[2XIsInternallyConsistent[102X( [3Xtom[103X ) [32X method[133X
  
  [33X[0;0YFor  a  table  of  marks  [3Xtom[103X,  [2XIsInternallyConsistent[102X decomposes all tensor
  products  of  rows  of [3Xtom[103X. It returns [9Xtrue[109X if all decomposition numbers are
  nonnegative   integers,   and   [9Xfalse[109X  otherwise.  This  provides  a  strong
  consistency check for a table of marks.[133X
  
  [1X70.9-2 DerivedSubgroupTom[101X
  
  [33X[1;0Y[29X[2XDerivedSubgroupTom[102X( [3Xtom[103X, [3Xsub[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDerivedSubgroupsTom[102X( [3Xtom[103X ) [32X function[133X
  
  [33X[0;0YFor  a  table  of  marks  [3Xtom[103X and a positive integer [3Xsub[103X, [2XDerivedSubgroupTom[102X
  returns either a positive integer [22Xi[122X or a list [22Xl[122X of positive integers. In the
  former case, the result means that the derived subgroups of the subgroups in
  the [3Xsub[103X-th class of [3Xtom[103X lie in the [22Xi[122X-th class. In the latter case, the class
  of  the derived subgroups could not be uniquely determined, and the position
  of the class of derived subgroups is an entry of [22Xl[122X.[133X
  
  [33X[0;0YValues  computed  with  [2XDerivedSubgroupTom[102X  are  stored  using the attribute
  [2XDerivedSubgroupsTomPossible[102X ([14X70.9-3[114X).[133X
  
  [33X[0;0Y[2XDerivedSubgroupsTom[102X  is  just  the list of [2XDerivedSubgroupTom[102X values for all
  values of [3Xsub[103X.[133X
  
  [1X70.9-3 DerivedSubgroupsTomPossible[101X
  
  [33X[1;0Y[29X[2XDerivedSubgroupsTomPossible[102X( [3Xtom[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XDerivedSubgroupsTomUnique[102X( [3Xtom[103X ) [32X attribute[133X
  
  [33X[0;0YLet   [3Xtom[103X   be   a   table   of   marks.   The   value   of   the  attribute
  [2XDerivedSubgroupsTomPossible[102X  is  a list in which the value at position [22Xi[122X –if
  bound– is a positive integer or a list; the meaning of the entry is the same
  as in [2XDerivedSubgroupTom[102X ([14X70.9-2[114X).[133X
  
  [33X[0;0YIf  the  value  of  the attribute [2XDerivedSubgroupsTomUnique[102X is known for [3Xtom[103X
  then  it  is  a list of positive integers, the value at position [22Xi[122X being the
  position of the class of derived subgroups of the [22Xi[122X-th class of subgroups in
  [3Xtom[103X.[133X
  
  [33X[0;0YThe derived subgroups are in general not uniquely determined by the table of
  marks if no [2XUnderlyingGroup[102X ([14X70.7-7[114X) value is stored, so there is no default
  method   for  [2XDerivedSubgroupsTomUnique[102X.  But  in  some  cases  the  derived
  subgroups are explicitly set when the table of marks is constructed. In this
  case,   [2XDerivedSubgroupTom[102X   ([14X70.9-2[114X)   does   not   set   values   in   the
  [2XDerivedSubgroupsTomPossible[102X list.[133X
  
  [33X[0;0YThe  [2XDerivedSubgroupsTomUnique[102X  value  is  automatically  set  when the last
  missing  unique  value is entered in the [2XDerivedSubgroupsTomPossible[102X list by
  [2XDerivedSubgroupTom[102X ([14X70.9-2[114X).[133X
  
  [1X70.9-4 NormalizerTom[101X
  
  [33X[1;0Y[29X[2XNormalizerTom[102X( [3Xtom[103X, [3Xsub[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XNormalizersTom[102X( [3Xtom[103X ) [32X attribute[133X
  
  [33X[0;0YLet  [3Xtom[103X be the table of marks of a group [22XG[122X. [2XNormalizerTom[102X tries to find the
  conjugacy class of the normalizer [22XN[122X in [22XG[122X of a subgroup [22XU[122X in the [3Xsub[103X-th class
  of  [3Xtom[103X.  The  return  value  is  either  the list of class numbers of those
  subgroups  that  have  the  right  size  and  contain  the  subgroup and all
  subgroups  that clearly contain it as a normal subgroup, or the class number
  of  the  normalizer if it is uniquely determined by these conditions. If [3Xtom[103X
  knows  the subgroup lattice of [22XG[122X (see [2XIsTableOfMarksWithGens[102X ([14X70.10-3[114X)) then
  all  normalizers  are uniquely determined. [2XNormalizerTom[102X should never return
  an empty list.[133X
  
  [33X[0;0Y[2XNormalizersTom[102X  returns  the list of positions of the classes of normalizers
  of  subgroups  in  [3Xtom[103X.  In  addition  to the criteria for a single class of
  subgroup  used  by  [2XNormalizerTom[102X,  the  approximations  of  normalizers for
  several   classes  are  used  and  thus  [2XNormalizersTom[102X  may  return  better
  approximations than [2XNormalizerTom[102X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XNormalizerTom( a5, 4 );[127X[104X
    [4X[28X8[128X[104X
    [4X[25Xgap>[125X [27XNormalizersTom( a5 );[127X[104X
    [4X[28X[ 9, 4, 6, 8, 7, 6, 7, 8, 9 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  example  shows  that  a subgroup with class number 4 in [22XA_5[122X (which is a
  Kleinian  four  group)  is  normalized  by a subgroup in class 8. This class
  contains the subgroups of [22XA_5[122X which are isomorphic to [22XA_4[122X.[133X
  
  [1X70.9-5 ContainedTom[101X
  
  [33X[1;0Y[29X[2XContainedTom[102X( [3Xtom[103X, [3Xsub1[103X, [3Xsub2[103X ) [32X operation[133X
  
  [33X[0;0Y[2XContainedTom[102X  returns  the number of subgroups in class [3Xsub1[103X of the table of
  marks [3Xtom[103X that are contained in one fixed member of the class [3Xsub2[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XContainedTom( a5, 3, 5 );  ContainedTom( a5, 3, 8 );[127X[104X
    [4X[28X0[128X[104X
    [4X[28X4[128X[104X
  [4X[32X[104X
  
  [1X70.9-6 ContainingTom[101X
  
  [33X[1;0Y[29X[2XContainingTom[102X( [3Xtom[103X, [3Xsub1[103X, [3Xsub2[103X ) [32X operation[133X
  
  [33X[0;0Y[2XContainingTom[102X  returns the number of subgroups in class [3Xsub2[103X of the table of
  marks [3Xtom[103X that contain one fixed member of the class [3Xsub1[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XContainingTom( a5, 3, 5 );  ContainingTom( a5, 3, 8 );[127X[104X
    [4X[28X0[128X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [1X70.9-7 CyclicExtensionsTom[101X
  
  [33X[1;0Y[29X[2XCyclicExtensionsTom[102X( [3Xtom[103X, [3Xp[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XCyclicExtensionsTom[102X( [3Xtom[103X[, [3Xlist[103X] ) [32X attribute[133X
  
  [33X[0;0YAccording  to  A. Dress [Dre69],  two  columns of the table of marks [3Xtom[103X are
  equal  modulo  the  prime  [3Xp[103X  if and only if the corresponding subgroups are
  connected by a chain of normal extensions of order [3Xp[103X.[133X
  
  [33X[0;0YCalled  with  [3Xtom[103X  and  [3Xp[103X,  [2XCyclicExtensionsTom[102X  returns the classes of this
  equivalence relation.[133X
  
  [33X[0;0YIn  the  second form, [3Xlist[103X must be a list of primes, and the return value is
  the list of classes of the relation obtained by considering chains of normal
  extensions  of  prime  order where all primes are in [3Xlist[103X. The default value
  for [3Xlist[103X is the set of prime divisors of the order of the group of [3Xtom[103X.[133X
  
  [33X[0;0Y(This  information  is  [13Xnot[113X used by [2XNormalizerTom[102X ([14X70.9-4[114X) although it might
  give additional restrictions in the search of normalizers.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCyclicExtensionsTom( a5, 2 );[127X[104X
    [4X[28X[ [ 1, 2, 4 ], [ 3, 6 ], [ 5, 7 ], [ 8 ], [ 9 ] ][128X[104X
  [4X[32X[104X
  
  [1X70.9-8 DecomposedFixedPointVector[101X
  
  [33X[1;0Y[29X[2XDecomposedFixedPointVector[102X( [3Xtom[103X, [3Xfix[103X ) [32X operation[133X
  
  [33X[0;0YLet  [3Xtom[103X be the table of marks of a group [22XG[122X and let [3Xfix[103X be a vector of fixed
  point  numbers w.r.t. an action of [22XG[122X, i.e., a vector which contains for each
  class  of  subgroups  the  number  of  fixed  points under the given action.
  [2XDecomposedFixedPointVector[102X returns the decomposition of [3Xfix[103X into rows of the
  table  of  marks.  This  decomposition corresponds to a decomposition of the
  action into transitive constituents. Trailing zeros in [3Xfix[103X may be omitted.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDecomposedFixedPointVector( a5, [ 16, 4, 1, 0, 1, 1, 1 ] );[127X[104X
    [4X[28X[ 0, 0, 0, 0, 0, 1, 1 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  vector  [3Xfix[103X may be any vector of integers. The resulting decomposition,
  however, will not be integral, in general.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDecomposedFixedPointVector( a5, [ 0, 0, 0, 0, 1, 1 ] );[127X[104X
    [4X[28X[ 2/5, -1, -1/2, 0, 1/2, 1 ][128X[104X
  [4X[32X[104X
  
  [1X70.9-9 EulerianFunctionByTom[101X
  
  [33X[1;0Y[29X[2XEulerianFunctionByTom[102X( [3Xtom[103X, [3Xn[103X[, [3Xsub[103X] ) [32X operation[133X
  
  [33X[0;0YCalled  with  two  arguments,  [2XEulerianFunctionByTom[102X  computes  the Eulerian
  function  (see [2XEulerianFunction[102X  ([14X39.16-3[114X)) of the underlying group [22XG[122X of the
  table  of  marks  [3Xtom[103X, that is, the number of [3Xn[103X-tuples of elements in [22XG[122X that
  generate [22XG[122X. If the optional argument [3Xsub[103X is given then [2XEulerianFunctionByTom[102X
  computes  the  Eulerian  function  of  each  subgroup in the [3Xsub[103X-th class of
  subgroups of [3Xtom[103X.[133X
  
  [33X[0;0YFor  a  group  [22XG[122X  whose  table  of  marks is known, [2XEulerianFunctionByTom[102X is
  installed as a method for [2XEulerianFunction[102X ([14X39.16-3[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XEulerianFunctionByTom( a5, 2 );[127X[104X
    [4X[28X2280[128X[104X
    [4X[25Xgap>[125X [27XEulerianFunctionByTom( a5, 3 );[127X[104X
    [4X[28X200160[128X[104X
    [4X[25Xgap>[125X [27XEulerianFunctionByTom( a5, 2, 3 );[127X[104X
    [4X[28X8[128X[104X
  [4X[32X[104X
  
  [1X70.9-10 IntersectionsTom[101X
  
  [33X[1;0Y[29X[2XIntersectionsTom[102X( [3Xtom[103X, [3Xsub1[103X, [3Xsub2[103X ) [32X operation[133X
  
  [33X[0;0YThe  intersections of the groups in the [3Xsub1[103X-th conjugacy class of subgroups
  of  the  table of marks [3Xtom[103X with the groups in the [3Xsub2[103X-th conjugacy classes
  of  subgroups  of [3Xtom[103X are determined up to conjugacy by the decomposition of
  the tensor product of their rows of marks. [2XIntersectionsTom[102X returns a list [22Xl[122X
  that  describes  this decomposition. The [22Xi[122X-th entry in [22Xl[122X is the multiplicity
  of groups in the [22Xi[122X-th conjugacy class as an intersection.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIntersectionsTom( a5, 8, 8 );[127X[104X
    [4X[28X[ 0, 0, 1, 0, 0, 0, 0, 1 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YAny  two subgroups of class number 8 ([22XA_4[122X) of [22XA_5[122X are either equal and their
  intersection  has  again  class  number  8,  or their intersection has class
  number [22X3[122X, and is a cyclic subgroup of order 3.[133X
  
  [1X70.9-11 FactorGroupTom[101X
  
  [33X[1;0Y[29X[2XFactorGroupTom[102X( [3Xtom[103X, [3Xn[103X ) [32X operation[133X
  
  [33X[0;0YFor  a  table  of  marks  [3Xtom[103X  of  a  group [22XG[122X and the normal subgroup [22XN[122X of [22XG[122X
  corresponding  to the [3Xn[103X-th class of subgroups of [3Xtom[103X, [2XFactorGroupTom[102X returns
  the table of marks of the factor group [22XG / N[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs4:= TableOfMarks( SymmetricGroup( 4 ) );[127X[104X
    [4X[28XTableOfMarks( Sym( [ 1 .. 4 ] ) )[128X[104X
    [4X[25Xgap>[125X [27XLengthsTom( s4 );[127X[104X
    [4X[28X[ 1, 3, 6, 4, 1, 3, 3, 4, 3, 1, 1 ][128X[104X
    [4X[25Xgap>[125X [27XOrdersTom( s4 );[127X[104X
    [4X[28X[ 1, 2, 2, 3, 4, 4, 4, 6, 8, 12, 24 ][128X[104X
    [4X[25Xgap>[125X [27Xs3:= FactorGroupTom( s4, 5 );[127X[104X
    [4X[28XTableOfMarks( Group([ f1, f2 ]) )[128X[104X
    [4X[25Xgap>[125X [27XDisplay( s3 );[127X[104X
    [4X[28X1:  6[128X[104X
    [4X[28X2:  3 1[128X[104X
    [4X[28X3:  2 . 2[128X[104X
    [4X[28X4:  1 1 1 1[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X70.9-12 MaximalSubgroupsTom[101X
  
  [33X[1;0Y[29X[2XMaximalSubgroupsTom[102X( [3Xtom[103X[, [3Xsub[103X] ) [32X attribute[133X
  
  [33X[0;0YCalled  with  a  table  of  marks [3Xtom[103X, [2XMaximalSubgroupsTom[102X returns a list of
  length  two,  the  first entry being the list of positions of the classes of
  maximal subgroups of the whole group of [3Xtom[103X, the second entry being the list
  of class lengths of these groups.[133X
  
  [33X[0;0YCalled  with  a  table of marks [3Xtom[103X and a position [3Xsub[103X, the same information
  for the [3Xsub[103X-th class of subgroups is returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMaximalSubgroupsTom( s4 );[127X[104X
    [4X[28X[ [ 10, 9, 8 ], [ 1, 3, 4 ] ][128X[104X
    [4X[25Xgap>[125X [27XMaximalSubgroupsTom( s4, 10 );[127X[104X
    [4X[28X[ [ 5, 4 ], [ 1, 4 ] ][128X[104X
  [4X[32X[104X
  
  [1X70.9-13 MinimalSupergroupsTom[101X
  
  [33X[1;0Y[29X[2XMinimalSupergroupsTom[102X( [3Xtom[103X, [3Xsub[103X ) [32X operation[133X
  
  [33X[0;0YFor  a  table  of  marks [3Xtom[103X, [2XMinimalSupergroupsTom[102X returns a list of length
  two,  the  first entry being the list of positions of the classes containing
  the  minimal  supergroups  of the groups in the [3Xsub[103X-th class of subgroups of
  [3Xtom[103X, the second entry being the list of class lengths of these groups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMinimalSupergroupsTom( s4, 5 );[127X[104X
    [4X[28X[ [ 9, 10 ], [ 3, 1 ] ][128X[104X
  [4X[32X[104X
  
  
  [1X70.10 [33X[0;0YAccessing Subgroups via Tables of Marks[133X[101X
  
  [33X[0;0YLet [3Xtom[103X be the table of marks of the group [22XG[122X, and assume that [3Xtom[103X has access
  to  [22XG[122X via the [2XUnderlyingGroup[102X ([14X70.7-7[114X) value. Then it makes sense to use [3Xtom[103X
  and  its  ordering  of  conjugacy  classes  of  subgroups  of  [22XG[122X for storing
  information  for  constructing representatives of these classes. The group [22XG[122X
  is  in  general not sufficient for this, [3Xtom[103X needs more information; this is
  available  if  and only if the [2XIsTableOfMarksWithGens[102X ([14X70.10-3[114X) value of [3Xtom[103X
  is  [9Xtrue[109X.  In  this  case,  [2XRepresentativeTom[102X ([14X70.10-4[114X) can be used to get a
  subgroup of the [22Xi[122X-th class, for all [22Xi[122X.[133X
  
  [33X[0;0Y[5XGAP[105X  provides  two  different  possibilities  to  store  generators  of  the
  representatives  of  classes  of  subgroups. The first is implemented by the
  attribute  [2XGeneratorsSubgroupsTom[102X  ([14X70.10-1[114X), which uses explicit generators
  of  the  subgroups.  The second, more general, possibility is implemented by
  the   attribute   [2XStraightLineProgramsTom[102X   ([14X70.10-2[114X),   which  encodes  the
  generators  as  straight  line  programs  (see [14X37.8[114X)  that  evaluate  to the
  generators  in question when applied to [13Xstandard generators[113X of [22XG[122X. This means
  that on the one hand, standard generators of [22XG[122X must be known in order to use
  [2XStraightLineProgramsTom[102X  ([14X70.10-2[114X).  On  the  other  hand, the straight line
  programs  allow one to compute easily generators not only of a subgroup [22XU[122X of
  [22XG[122X but also generators of the image of [22XU[122X in any representation of [22XG[122X, provided
  that   one   knows  standard  generators  of  the  image  of  [22XG[122X  under  this
  representation.  See  the  manual  of  the package [5XTomLib[105X for details and an
  example.[133X
  
  [1X70.10-1 GeneratorsSubgroupsTom[101X
  
  [33X[1;0Y[29X[2XGeneratorsSubgroupsTom[102X( [3Xtom[103X ) [32X attribute[133X
  
  [33X[0;0YLet  [3Xtom[103X  be  a  table  of marks with [2XIsTableOfMarksWithGens[102X ([14X70.10-3[114X) value
  [9Xtrue[109X.  Then  [2XGeneratorsSubgroupsTom[102X  returns a list of length two, the first
  entry  being  a  list  [22Xl[122X  of elements of the group stored as [2XUnderlyingGroup[102X
  ([14X70.7-7[114X)  value  of  [3Xtom[103X,  the  second  entry  being a list that contains at
  position  [22Xi[122X  a list of positions in [22Xl[122X of generators of a representative of a
  subgroup in class [22Xi[122X.[133X
  
  [33X[0;0YThe  [2XGeneratorsSubgroupsTom[102X value is known for all tables of marks that have
  been computed with [2XTableOfMarks[102X ([14X70.3-1[114X) from a group, and there is a method
  to  compute  the  value  for  a table of marks that admits [2XRepresentativeTom[102X
  ([14X70.10-4[114X).[133X
  
  [1X70.10-2 StraightLineProgramsTom[101X
  
  [33X[1;0Y[29X[2XStraightLineProgramsTom[102X( [3Xtom[103X ) [32X attribute[133X
  
  [33X[0;0YFor  a  table of marks [3Xtom[103X with [2XIsTableOfMarksWithGens[102X ([14X70.10-3[114X) value [9Xtrue[109X,
  [2XStraightLineProgramsTom[102X  returns a list that contains at position [22Xi[122X either a
  list  of  straight  line  programs  or  a  straight line program (see [14X37.8[114X),
  encoding  the  generators of a representative of the [22Xi[122X-th conjugacy class of
  subgroups  of [10XUnderlyingGroup( [3Xtom[103X[10X )[110X; in the former case, each straight line
  program  returns  a  generator,  in the latter case, the program returns the
  list of generators.[133X
  
  [33X[0;0YThere is no default method to compute the [2XStraightLineProgramsTom[102X value of a
  table of marks if they are not yet stored. The value is known for all tables
  of marks that belong to the [5XGAP[105X library of tables of marks (see [14X70.13[114X).[133X
  
  [1X70.10-3 IsTableOfMarksWithGens[101X
  
  [33X[1;0Y[29X[2XIsTableOfMarksWithGens[102X( [3Xtom[103X ) [32X filter[133X
  
  [33X[0;0YThis  filter  shall  express  the  union  of  the filters [10XIsTableOfMarks and
  HasStraightLineProgramsTom[110X and [10XIsTableOfMarks and HasGeneratorsSubgroupsTom[110X.
  If a table of marks [3Xtom[103X has this filter set then [3Xtom[103X can be asked to compute
  information  that is in general not uniquely determined by a table of marks,
  for  example  the positions of derived subgroups or normalizers of subgroups
  (see [2XDerivedSubgroupTom[102X ([14X70.9-2[114X), [2XNormalizerTom[102X ([14X70.9-4[114X)).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xa5:= TableOfMarks( "A5" );;  IsTableOfMarksWithGens( a5 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XHasGeneratorsSubgroupsTom( a5 );  HasStraightLineProgramsTom( a5 );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xalt5:= TableOfMarks( AlternatingGroup( 5 ) );;[127X[104X
    [4X[25Xgap>[125X [27XIsTableOfMarksWithGens( alt5 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XHasGeneratorsSubgroupsTom(alt5); HasStraightLineProgramsTom(alt5);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xprogs:= StraightLineProgramsTom( a5 );;[127X[104X
    [4X[25Xgap>[125X [27XOrdersTom( a5 );[127X[104X
    [4X[28X[ 1, 2, 3, 4, 5, 6, 10, 12, 60 ][128X[104X
    [4X[25Xgap>[125X [27XIsCyclicTom( a5, 4 );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XLength( progs[4] );[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27Xprogs[4][1];[127X[104X
    [4X[28X<straight line program>[128X[104X
    [4X[25Xgap>[125X [27X# first generator of an el. ab group of order 4:[127X[104X
    [4X[25Xgap>[125X [27XDisplay( progs[4][1] );[127X[104X
    [4X[28X# input:[128X[104X
    [4X[28Xr:= [ g1, g2 ];[128X[104X
    [4X[28X# program:[128X[104X
    [4X[28Xr[3]:= r[2]*r[1];[128X[104X
    [4X[28Xr[4]:= r[3]*r[2]^-1*r[1]*r[3]*r[2]^-1*r[1]*r[2];[128X[104X
    [4X[28X# return value:[128X[104X
    [4X[28Xr[4][128X[104X
    [4X[25Xgap>[125X [27Xx:= [ [ Z(2)^0, 0*Z(2) ], [ Z(2^2), Z(2)^0 ] ];;[127X[104X
    [4X[25Xgap>[125X [27Xy:= [ [ Z(2^2), Z(2)^0 ], [ 0*Z(2), Z(2^2)^2 ] ];;[127X[104X
    [4X[25Xgap>[125X [27Xres1:= ResultOfStraightLineProgram( progs[4][1], [ x, y ] );[127X[104X
    [4X[28X[ [ Z(2)^0, 0*Z(2) ], [ Z(2^2)^2, Z(2)^0 ] ][128X[104X
    [4X[25Xgap>[125X [27Xres2:= ResultOfStraightLineProgram( progs[4][2], [ x, y ] );[127X[104X
    [4X[28X[ [ Z(2)^0, 0*Z(2) ], [ Z(2^2), Z(2)^0 ] ][128X[104X
    [4X[25Xgap>[125X [27Xw:= y*x;;[127X[104X
    [4X[25Xgap>[125X [27Xres1 = w*y^-1*x*w*y^-1*x*y;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xsubgrp:= Group( res1, res2 );;  Size( subgrp );  IsCyclic( subgrp );[127X[104X
    [4X[28X4[128X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X70.10-4 RepresentativeTom[101X
  
  [33X[1;0Y[29X[2XRepresentativeTom[102X( [3Xtom[103X, [3Xsub[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XRepresentativeTomByGenerators[102X( [3Xtom[103X, [3Xsub[103X, [3Xgens[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XRepresentativeTomByGeneratorsNC[102X( [3Xtom[103X, [3Xsub[103X, [3Xgens[103X ) [32X operation[133X
  
  [33X[0;0YLet  [3Xtom[103X  be  a  table  of marks with [2XIsTableOfMarksWithGens[102X ([14X70.10-3[114X) value
  [9Xtrue[109X, and [3Xsub[103X a positive integer. [2XRepresentativeTom[102X returns a representative
  of the [3Xsub[103X-th conjugacy class of subgroups of [3Xtom[103X.[133X
  
  [33X[0;0YIf  the  attribute  [2XStraightLineProgramsTom[102X  ([14X70.10-2[114X)  is  set  in [3Xtom[103X then
  methods     for    the    operations    [2XRepresentativeTomByGenerators[102X    and
  [2XRepresentativeTomByGeneratorsNC[102X are available, which return a representative
  of  the  [3Xsub[103X-th  conjugacy  class  of subgroups of [3Xtom[103X, as a subgroup of the
  group  generated by [3Xgens[103X. This means that the standard generators of [3Xtom[103X are
  replaced by [3Xgens[103X.[133X
  
  [33X[0;0Y[2XRepresentativeTomByGenerators[102X checks whether mapping the standard generators
  of  [3Xtom[103X  to  [3Xgens[103X  extends  to a group isomorphism, and returns [9Xfail[109X if not.
  [2XRepresentativeTomByGeneratorsNC[102X       omits       all       checks.       So
  [2XRepresentativeTomByGenerators[102X is thought mainly for debugging purposes; note
  that  when  several  representatives  are  constructed,  it  is  cheaper  to
  construct  (and  check) the isomorphism once, and to map the groups returned
  by    [2XRepresentativeTom[102X    under   this   isomorphism.   The   idea   behind
  [2XRepresentativeTomByGeneratorsNC[102X,  however, is to avoid the overhead of using
  isomorphisms  when  [3Xgens[103X  are  known  to be standard generators. In order to
  proceed  like  this,  the  attribute  [2XStraightLineProgramsTom[102X  ([14X70.10-2[114X)  is
  needed.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XRepresentativeTom( a5, 4 );[127X[104X
    [4X[28XGroup([ (2,3)(4,5), (2,4)(3,5) ])[128X[104X
  [4X[32X[104X
  
  
  [1X70.11 [33X[0;0YThe Interface between Tables of Marks and Character Tables[133X[101X
  
  [33X[0;0YThe  following  examples  require  the  [5XGAP[105X  Character  Table  Library to be
  available. If it is not yet loaded then we load it now.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLoadPackage( "ctbllib" );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X70.11-1 FusionCharTableTom[101X
  
  [33X[1;0Y[29X[2XFusionCharTableTom[102X( [3Xtbl[103X, [3Xtom[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XPossibleFusionsCharTableTom[102X( [3Xtbl[103X, [3Xtom[103X[, [3Xoptions[103X] ) [32X operation[133X
  
  [33X[0;0YLet  [3Xtbl[103X  be  the  ordinary character table of the group [22XG[122X, say, and [3Xtom[103X the
  table of marks of [22XG[122X. [2XFusionCharTableTom[102X determines the fusion of the classes
  of  elements  from [3Xtbl[103X to the classes of cyclic subgroups on [3Xtom[103X, that is, a
  list  that  contains  at  position  [22Xi[122X  the  position  of the class of cyclic
  subgroups  in [3Xtom[103X that are generated by elements in the [22Xi[122X-th conjugacy class
  of elements in [3Xtbl[103X.[133X
  
  [33X[0;0YThree cases are handled differently.[133X
  
  [31X1[131X   [33X[0;6YThe  fusion  is explicitly stored on [3Xtbl[103X. Then nothing has to be done.
        This  happens only if both [3Xtbl[103X and [3Xtom[103X are tables from the [5XGAP[105X library
        (see [14X70.13[114X and the manual of the [5XGAP[105X Character Table Library).[133X
  
  [31X2[131X   [33X[0;6YThe  [2XUnderlyingGroup[102X  ([14X70.7-7[114X)  values  of  [3Xtbl[103X  and [3Xtom[103X are known and
        equal. Then the group is used to compute the fusion.[133X
  
  [31X3[131X   [33X[0;6YThere  is neither fusion nor group information available. In this case
        only  necessary  conditions  can  be  checked,  and  if  they  are not
        sufficient  to  determine the fusion uniquely then [9Xfail[109X is returned by
        [2XFusionCharTableTom[102X.[133X
  
  [33X[0;0Y[2XPossibleFusionsCharTableTom[102X  computes  the list of possible fusions from [3Xtbl[103X
  to   [3Xtom[103X,   according  to  the  criteria  that  have  been  checked.  So  if
  [2XFusionCharTableTom[102X  returns  a  unique  fusion  then  the  list  returned by
  [2XPossibleFusionsCharTableTom[102X  for  the  same  arguments contains exactly this
  fusion,  and if [2XFusionCharTableTom[102X returns [9Xfail[109X then the length of this list
  is different from [22X1[122X.[133X
  
  [33X[0;0YThe  optional  argument [3Xoptions[103X must be a record that may have the following
  components.[133X
  
  [8X[10Xfusionmap[110X[8X[108X
        [33X[0;6Ya parametrized map which is an approximation of the desired map,[133X
  
  [8X[10Xquick[110X[8X[108X
        [33X[0;6Ya  Boolean;  if [9Xtrue[109X then as soon as only one possibility remains this
        possibility is returned immediately; the default value is [9Xfalse[109X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xa5c:= CharacterTable( "A5" );;[127X[104X
    [4X[25Xgap>[125X [27Xfus:= FusionCharTableTom( a5c, a5 );[127X[104X
    [4X[28X[ 1, 2, 3, 5, 5 ][128X[104X
  [4X[32X[104X
  
  [1X70.11-2 PermCharsTom[101X
  
  [33X[1;0Y[29X[2XPermCharsTom[102X( [3Xfus[103X, [3Xtom[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XPermCharsTom[102X( [3Xtbl[103X, [3Xtom[103X ) [32X operation[133X
  
  [33X[0;0Y[2XPermCharsTom[102X  returns the list of transitive permutation characters from the
  table  of  marks [3Xtom[103X. In the first form, [3Xfus[103X must be the fusion map from the
  ordinary  character table of the group of [3Xtom[103X to [3Xtom[103X (see [2XFusionCharTableTom[102X
  ([14X70.11-1[114X)). In the second form, [3Xtbl[103X must be the character table of the group
  of  which  [3Xtom[103X  is  the  table  of  marks. If the fusion map is not uniquely
  determined (see [2XFusionCharTableTom[102X ([14X70.11-1[114X)) then [9Xfail[109X is returned.[133X
  
  [33X[0;0YIf  the  fusion  map  [3Xfus[103X  is  given  as first argument then each transitive
  permutation  character  is  represented by its values list. If the character
  table  [3Xtbl[103X  is  given  then  the  permutation  characters are class function
  objects (see Chapter [14X72[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPermCharsTom( a5c, a5 );[127X[104X
    [4X[28X[ Character( CharacterTable( "A5" ), [ 60, 0, 0, 0, 0 ] ),[128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 30, 2, 0, 0, 0 ] ),[128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 20, 0, 2, 0, 0 ] ),[128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 15, 3, 0, 0, 0 ] ),[128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 12, 0, 0, 2, 2 ] ),[128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 10, 2, 1, 0, 0 ] ),[128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 6, 2, 0, 1, 1 ] ),[128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 5, 1, 2, 0, 0 ] ),[128X[104X
    [4X[28X  Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XPermCharsTom( fus, a5 )[1];[127X[104X
    [4X[28X[ 60, 0, 0, 0, 0 ][128X[104X
  [4X[32X[104X
  
  
  [1X70.12 [33X[0;0YGeneric Construction of Tables of Marks[133X[101X
  
  [33X[0;0YThe following three operations construct a table of marks only from the data
  given, i.e., without underlying group.[133X
  
  [1X70.12-1 TableOfMarksCyclic[101X
  
  [33X[1;0Y[29X[2XTableOfMarksCyclic[102X( [3Xn[103X ) [32X operation[133X
  
  [33X[0;0Y[2XTableOfMarksCyclic[102X  returns  the table of marks of the cyclic group of order
  [3Xn[103X.[133X
  
  [33X[0;0YA  cyclic  group  of  order [3Xn[103X has as its subgroups for each divisor [22Xd[122X of [3Xn[103X a
  cyclic subgroup of order [22Xd[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDisplay( TableOfMarksCyclic( 6 ) );[127X[104X
    [4X[28X1:  6[128X[104X
    [4X[28X2:  3 3[128X[104X
    [4X[28X3:  2 . 2[128X[104X
    [4X[28X4:  1 1 1 1[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X70.12-2 TableOfMarksDihedral[101X
  
  [33X[1;0Y[29X[2XTableOfMarksDihedral[102X( [3Xn[103X ) [32X operation[133X
  
  [33X[0;0Y[2XTableOfMarksDihedral[102X  returns  the  table  of marks of the dihedral group of
  order [3Xm[103X.[133X
  
  [33X[0;0YFor each divisor [22Xd[122X of [3Xm[103X, a dihedral group of order [22Xm = 2n[122X contains subgroups
  of  order  [22Xd[122X according to the following rule. If [22Xd[122X is odd and divides [22Xn[122X then
  there  is  only  one  cyclic subgroup of order [22Xd[122X. If [22Xd[122X is even and divides [22Xn[122X
  then  there  are  a  cyclic  subgroup of order [22Xd[122X and two classes of dihedral
  subgroups  of  order  [22Xd[122X  (which  are cyclic, too, in the case [22Xd = 2[122X, see the
  example  below).  Otherwise (i.e., if [22Xd[122X does not divide [22Xn[122X) there is just one
  class of dihedral subgroups of order [22Xd[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDisplay( TableOfMarksDihedral( 12 ) );[127X[104X
    [4X[28X 1:  12[128X[104X
    [4X[28X 2:   6 6[128X[104X
    [4X[28X 3:   6 . 2[128X[104X
    [4X[28X 4:   6 . . 2[128X[104X
    [4X[28X 5:   4 . . . 4[128X[104X
    [4X[28X 6:   3 3 1 1 . 1[128X[104X
    [4X[28X 7:   2 2 . . 2 . 2[128X[104X
    [4X[28X 8:   2 . 2 . 2 . . 2[128X[104X
    [4X[28X 9:   2 . . 2 2 . . . 2[128X[104X
    [4X[28X10:   1 1 1 1 1 1 1 1 1 1[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X70.12-3 TableOfMarksFrobenius[101X
  
  [33X[1;0Y[29X[2XTableOfMarksFrobenius[102X( [3Xp[103X, [3Xq[103X ) [32X operation[133X
  
  [33X[0;0Y[2XTableOfMarksFrobenius[102X  computes  the  table of marks of a Frobenius group of
  order [22Xp q[122X, where [22Xp[122X is a prime and [22Xq[122X divides [22Xp-1[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDisplay( TableOfMarksFrobenius( 5, 4 ) );[127X[104X
    [4X[28X1:  20[128X[104X
    [4X[28X2:  10 2[128X[104X
    [4X[28X3:   5 1 1[128X[104X
    [4X[28X4:   4 . . 4[128X[104X
    [4X[28X5:   2 2 . 2 2[128X[104X
    [4X[28X6:   1 1 1 1 1 1[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X70.13 [33X[0;0YThe Library of Tables of Marks[133X[101X
  
  [33X[0;0YThe [5XGAP[105X package [5XTomLib[105X provides access to several hundred tables of marks of
  almost  simple  groups  and  their  maximal  subgroups.  If  this package is
  installed   then   the  tables  from  this  database  can  be  accessed  via
  [2XTableOfMarks[102X  ([14X70.3-1[114X)  with  argument  a  string. If also the [5XGAP[105X Character
  Table  Library is installed and contains the ordinary character table of the
  group for which one wants to fetch the table of marks then one can also call
  [2XTableOfMarks[102X ([14X70.3-1[114X) with argument the character table.[133X
  
  [33X[0;0YA  list  of  all  names  of  tables of marks that are provided by the [5XTomLib[105X
  package can be obtained via [2XAllLibTomNames[102X ([14XTomLib: AllLibTomNames[114X).[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
  
