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70 Tables of Marks
70.1 More about Tables of Marks
70.2 Table of Marks Objects in GAP
70.3 Constructing Tables of Marks
70.3-1 TableOfMarks
70.3-2 TableOfMarksByLattice
70.3-3 LatticeSubgroupsByTom
70.4 Printing Tables of Marks
70.4-1 ViewObj
70.4-2 PrintObj
70.4-3 Display
70.5 Sorting Tables of Marks
70.5-1 SortedTom
70.5-2 PermutationTom
70.6 Technical Details about Tables of Marks
70.6-1 InfoTom
70.6-2 IsTableOfMarks
70.6-3 TableOfMarksFamily
70.6-4 TableOfMarksComponents
70.6-5 ConvertToTableOfMarks
70.7 Attributes of Tables of Marks
70.7-1 MarksTom
70.7-2 NrSubsTom
70.7-3 LengthsTom
70.7-4 ClassTypesTom
70.7-5 ClassNamesTom
70.7-6 FusionsTom
70.7-7 UnderlyingGroup
70.7-8 IdempotentsTom
70.7-9 Identifier
70.7-10 MatTom
70.7-11 MoebiusTom
70.7-12 WeightsTom
70.8 Properties of Tables of Marks
70.8-1 IsAbelianTom
70.9 Other Operations for Tables of Marks
70.9-1 IsInternallyConsistent
70.9-2 DerivedSubgroupTom
70.9-3 DerivedSubgroupsTomPossible
70.9-4 NormalizerTom
70.9-5 ContainedTom
70.9-6 ContainingTom
70.9-7 CyclicExtensionsTom
70.9-8 DecomposedFixedPointVector
70.9-9 EulerianFunctionByTom
70.9-10 IntersectionsTom
70.9-11 FactorGroupTom
70.9-12 MaximalSubgroupsTom
70.9-13 MinimalSupergroupsTom
70.10 Accessing Subgroups via Tables of Marks
70.10-1 GeneratorsSubgroupsTom
70.10-2 StraightLineProgramsTom
70.10-3 IsTableOfMarksWithGens
70.10-4 RepresentativeTom
70.11 The Interface between Tables of Marks and Character Tables
70.11-1 FusionCharTableTom
70.11-2 PermCharsTom
70.12 Generic Construction of Tables of Marks
70.12-1 TableOfMarksCyclic
70.12-2 TableOfMarksDihedral
70.12-3 TableOfMarksFrobenius
70.13 The Library of Tables of Marks

### 70 Tables of Marks

The concept of a table of marks was introduced by W. Burnside in his book "Theory of Groups of Finite Order", see [Bur55]. Therefore a table of marks is sometimes called a Burnside matrix.

The table of marks of a finite group \(G\) is a matrix whose rows and columns are labelled by the conjugacy classes of subgroups of \(G\) and where for two subgroups \(A\) and \(B\) the \((A, B)\)-entry is the number of fixed points of \(B\) in the transitive action of \(G\) on the cosets of \(A\) in \(G\). So the table of marks characterizes the set of all permutation representations of \(G\).

Moreover, the table of marks gives a compact description of the subgroup lattice of \(G\), since from the numbers of fixed points the numbers of conjugates of a subgroup \(B\) contained in a subgroup \(A\) can be derived.

A table of marks of a given group \(G\) can be constructed from the subgroup lattice of \(G\) (see 70.3). For several groups, the table of marks can be restored from the GAP library of tables of marks (see 70.13).

Given the table of marks of \(G\), one can display it (see 70.4) and derive information about \(G\) and its Burnside ring from it (see 70.7, 70.8, 70.9). Moreover, tables of marks in GAP provide an easy access to the classes of subgroups of their underlying groups (see 70.10).

#### 70.1 More about Tables of Marks

Let \(G\) be a finite group with \(n\) conjugacy classes of subgroups \(C_1, C_2, \ldots, C_n\) and representatives \(H_i \in C_i\), \(1 \leq i \leq n\). The table of marks of \(G\) is defined to be the \(n \times n\) matrix \(M = (m_{ij})\) where the mark \(m_{ij}\) is the number of fixed points of the subgroup \(H_j\) in the action of \(G\) on the right cosets of \(H_i\) in \(G\).

Since \(H_j\) can only have fixed points if it is contained in a point stabilizer the matrix \(M\) is lower triangular if the classes \(C_i\) are sorted according to the condition that if \(H_i\) is contained in a conjugate of \(H_j\) then \(i \leq j\).

Moreover, the diagonal entries \(m_{ii}\) are nonzero since \(m_{ii}\) equals the index of \(H_i\) in its normalizer in \(G\). Hence \(M\) is invertible. Since any transitive action of \(G\) is equivalent to an action on the cosets of a subgroup of \(G\), one sees that the table of marks completely characterizes the set of all permutation representations of \(G\).

The marks \(m_{ij}\) have further meanings. If \(H_1\) is the trivial subgroup of \(G\) then each mark \(m_{i1}\) in the first column of \(M\) is equal to the index of \(H_i\) in \(G\) since the trivial subgroup fixes all cosets of \(H_i\). If \(H_n = G\) then each \(m_{nj}\) in the last row of \(M\) is equal to \(1\) since there is only one coset of \(G\) in \(G\). In general, \(m_{ij}\) equals the number of conjugates of \(H_i\) containing \(H_j\), multiplied by the index of \(H_i\) in its normalizer in \(G\). Moreover, the number \(c_{ij}\) of conjugates of \(H_j\) which are contained in \(H_i\) can be derived from the marks \(m_{ij}\) via the formula

\[ c_{ij} = ( m_{ij} m_{j1} ) / ( m_{i1} m_{jj} ) \]

.

Both the marks \(m_{ij}\) and the numbers of subgroups \(c_{ij}\) are needed for the functions described in this chapter.

A brief survey of properties of tables of marks and a description of algorithms for the interactive construction of tables of marks using GAP can be found in [Pfe97].

#### 70.2 Table of Marks Objects in GAP

A table of marks of a group \(G\) in GAP is represented by an immutable (see 12.6) object tom in the category `IsTableOfMarks` (70.6-2), with defining attributes `SubsTom` (70.7-1) and `MarksTom` (70.7-1). These two attributes encode the matrix of marks in a compressed form. The `SubsTom` (70.7-1) value of tom 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 `MarksTom` (70.7-1) value of tom, which is a list that contains for each entry in `SubsTom( tom )` the corresponding nonzero value of the table of marks.

It 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 `TableOfMarks` (70.3-1)). So it may happen that a table of marks object in GAP is in fact not the table of marks of a group. To some extent, the consistency of a table of marks object can be checked (see 70.9), but GAP 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 `NormalizerTom` (70.9-4)) and derived subgroups (see `DerivedSubgroupTom` (70.9-2)) of subgroups are in general not uniquely determined by the matrix of marks.

GAP tables of marks are assumed to be in lower triangular form, that is, if a subgroup from the conjugacy class corresponding to the \(i\)-th row is contained in a subgroup from the class corresponding to the \(j\)-th row j then \(i \leq j\).

The `MarksTom` (70.7-1) information can be computed from the values of the attributes `NrSubsTom` (70.7-2), `LengthsTom` (70.7-3), `OrdersTom` (70.7-2), and `SubsTom` (70.7-1). `NrSubsTom` (70.7-2) stores a list containing for each entry in the `SubsTom` (70.7-1) value the corresponding number of conjugates that are contained in a subgroup, `LengthsTom` (70.7-3) a list containing for each conjugacy class of subgroups its length, and `OrdersTom` (70.7-2) a list containing for each class of subgroups their order. So the `MarksTom` (70.7-1) value of tom may be missing provided that the values of `NrSubsTom` (70.7-2), `LengthsTom` (70.7-3), and `OrdersTom` (70.7-2) are stored in tom.

Additional information about a table of marks is needed by some functions. The class numbers of normalizers in \(G\) and the number of the derived subgroup of \(G\) can be stored via appropriate attributes (see `NormalizersTom` (70.9-4), `DerivedSubgroupTom` (70.9-2)).

If tom stores its group \(G\) and a bijection from the rows and columns of the matrix of marks of tom to the classes of subgroups of \(G\) then clearly normalizers, derived subgroup etc. can be computed from this information. But in general a table of marks need not have access to \(G\), for example tom might have been constructed from a generic table of marks (see 70.12), or as table of marks of a factor group from a given table of marks (see `FactorGroupTom` (70.9-11)). Access to the group \(G\) is provided by the attribute `UnderlyingGroup` (70.7-7) if this value is set. Access to the relevant information about conjugacy classes of subgroups of \(G\) –compatible with the ordering of rows and columns of the marks in tom– is signalled by the filter `IsTableOfMarksWithGens` (70.10-3).

Several examples in this chapter require the GAP package TomLib (the GAP Library of Tables of Marks) to be available. If it is not yet loaded then we load it now.

    gap> LoadPackage( "tomlib" );
    true

#### 70.3 Constructing Tables of Marks

##### 70.3-1 TableOfMarks

`‣ TableOfMarks`( G ) | ( attribute )

`‣ TableOfMarks`( string ) | ( attribute )

`‣ TableOfMarks`( matrix ) | ( attribute )

In the first form, G must be a finite group, and `TableOfMarks` constructs the table of marks of G. This computation requires the knowledge of the complete subgroup lattice of G (see `LatticeSubgroups` (39.20-1)). If the lattice is not yet stored then it will be constructed. This may take a while if G is large. The result has the `IsTableOfMarksWithGens` (70.10-3) value `true`.

In the second form, string must be a string, and `TableOfMarks` gets the table of marks with name string from the GAP library (see 70.13). If no table of marks with this name is contained in the library then `fail` is returned.

In the third form, matrix 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 matrix, `TableOfMarks` returns a table of marks object (see 70.2) for which matrix 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 `IsInternallyConsistent` (70.9-1) (see 70.9), and `fail` is returned if matrix is proved not to describe a matrix of marks; but if `TableOfMarks` 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.

For an overview of operations for table of marks objects, see the introduction to Chapter 70.

    gap> tom:= TableOfMarks( AlternatingGroup( 5 ) );
    TableOfMarks( Alt( [ 1 .. 5 ] ) )
    gap> TableOfMarks( "J5" );
    fail
    gap> a5:= TableOfMarks( "A5" );
    TableOfMarks( "A5" )
    gap> mat:=
    > [ [ 60, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 30, 2, 0, 0, 0, 0, 0, 0, 0 ],
    >   [ 20, 0, 2, 0, 0, 0, 0, 0, 0 ], [ 15, 3, 0, 3, 0, 0, 0, 0, 0 ],
    >   [ 12, 0, 0, 0, 2, 0, 0, 0, 0 ], [ 10, 2, 1, 0, 0, 1, 0, 0, 0 ],
    >   [ 6, 2, 0, 0, 1, 0, 1, 0, 0 ], [ 5, 1, 2, 1, 0, 0, 0, 1, 0 ],
    >   [ 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ];;
    gap> TableOfMarks( mat );
    TableOfMarks( <9 classes> )

The following `TableOfMarks` methods for a group are installed.

- If the group is known to be cyclic then `TableOfMarks` constructs the table of marks essentially without the group, instead the knowledge about the structure of cyclic groups is used.

- If the lattice of subgroups is already stored in the group then `TableOfMarks` computes the table of marks from the lattice (see `TableOfMarksByLattice` (70.3-2)).

- If the group is known to be solvable then `TableOfMarks` takes the lattice of subgroups (see `LatticeSubgroups` (39.20-1)) 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 `TableOfMarksByLattice` (70.3-2).

- If 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 `LatticeSubgroupsByTom` (70.3-3)).

Conversely, the lattice of subgroups of a group with known table of marks can be computed using the table of marks, via the function `LatticeSubgroupsByTom` (70.3-3). This is also installed as a method for `LatticeSubgroups` (39.20-1).

##### 70.3-2 TableOfMarksByLattice

`‣ TableOfMarksByLattice`( G ) | ( function )

`TableOfMarksByLattice` computes the table of marks of the group G from the lattice of subgroups of G. This lattice is computed via `LatticeSubgroups` (39.20-1) if it is not yet stored in G. The function `TableOfMarksByLattice` is installed as a method for `TableOfMarks` (70.3-1) 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.

##### 70.3-3 LatticeSubgroupsByTom

`‣ LatticeSubgroupsByTom`( G ) | ( function )

`LatticeSubgroupsByTom` computes the lattice of subgroups of G from the table of marks of G, using `RepresentativeTom` (70.10-4).

#### 70.4 Printing Tables of Marks

##### 70.4-1 ViewObj

`‣ ViewObj`( tom ) | ( method )

The default `ViewObj` (6.3-5) method for tables of marks prints the string `"TableOfMarks"`, followed by –if known– the identifier (see `Identifier` (70.7-9)) 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.

##### 70.4-2 PrintObj

`‣ PrintObj`( tom ) | ( method )

The default `PrintObj` (6.3-5) method for tables of marks does the same as `ViewObj` (6.3-5), except that `PrintObj` (6.3-5) is used for the group instead of `ViewObj` (6.3-5).

##### 70.4-3 Display

`‣ Display`( tom[, arec] ) | ( method )

The default `Display` (6.3-6) method for a table of marks tom produces a formatted output of the marks in tom. 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 `SizeScreen` (6.12-1).

An interactive variant of `Display` (6.3-6) is the `Browse` (Browse: Browse) method for tables of marks that is provided by the GAP package Browse, see `Browse` (Browse: Browse (for tables of marks)).

The optional second argument arec of `Display` (6.3-6) can be used to change the default style for displaying a table of marks. arec must be a record, its relevant components are the following.

`classes`

  a list of class numbers to select only the rows and columns of the matrix that correspond to this list for printing,

`form`

  one of the strings `"subgroups"`, `"supergroups"`; in the former case, at position \((i,j)\) of the matrix the number of conjugates of \(H_j\) contained in \(H_i\) is printed, and in the latter case, at position \((i,j)\) the number of conjugates of \(H_i\) which contain \(H_j\) is printed.

    gap> tom:= TableOfMarks( "A5" );;
    gap> Display( tom );
    1:  60
    2:  30 2
    3:  20 . 2
    4:  15 3 . 3
    5:  12 . . . 2
    6:  10 2 1 . . 1
    7:   6 2 . . 1 . 1
    8:   5 1 2 1 . . . 1
    9:   1 1 1 1 1 1 1 1 1

    gap> Display( tom, rec( classes:= [ 1, 2, 3, 4, 8 ] ) );
    1:  60
    2:  30 2
    3:  20 . 2
    4:  15 3 . 3
    8:   5 1 2 1 1

    gap> Display( tom, rec( form:= "subgroups" ) );
    1:  1
    2:  1  1
    3:  1  .  1
    4:  1  3  . 1
    5:  1  .  . . 1
    6:  1  3  1 . .  1
    7:  1  5  . . 1  . 1
    8:  1  3  4 1 .  . . 1
    9:  1 15 10 5 6 10 6 5 1

    gap> Display( tom, rec( form:= "supergroups" ) );
    1:   1
    2:  15 1
    3:  10 . 1
    4:   5 1 . 1
    5:   6 . . . 1
    6:  10 2 1 . . 1
    7:   6 2 . . 1 . 1
    8:   5 1 2 1 . . . 1
    9:   1 1 1 1 1 1 1 1 1

#### 70.5 Sorting Tables of Marks

##### 70.5-1 SortedTom

`‣ SortedTom`( tom, perm ) | ( operation )

`SortedTom` returns a table of marks where the rows and columns of the table of marks tom are reordered according to the permutation perm.

Note that in each table of marks in GAP, the matrix of marks is assumed to have lower triangular shape (see 70.2). If the permutation perm does not have this property then the functions for tables of marks might return wrong results when applied to the output of `SortedTom`.

The returned table of marks has only those attribute values stored that are known for tom and listed in `TableOfMarksComponents` (70.6-4).

    gap> tom:= TableOfMarksCyclic( 6 );;  Display( tom );
    1:  6
    2:  3 3
    3:  2 . 2
    4:  1 1 1 1

    gap> sorted:= SortedTom( tom, (2,3) );;  Display( sorted );
    1:  6
    2:  2 2
    3:  3 . 3
    4:  1 1 1 1

    gap> wrong:= SortedTom( tom, (1,2) );;  Display( wrong );
    1:  3
    2:  . 6
    3:  . 2 2
    4:  1 1 1 1

##### 70.5-2 PermutationTom

`‣ PermutationTom`( tom ) | ( attribute )

For the table of marks tom of the group \(G\) stored as `UnderlyingGroup` (70.7-7) value of tom, `PermutationTom` is a permutation \(\pi\) such that the \(i\)-th conjugacy class of subgroups of \(G\) belongs to the \(i^\pi\)-th column and row of marks in tom.

This attribute value is bound only if tom was obtained from another table of marks by permuting with `SortedTom` (70.5-1), and there is no default method to compute its value.

The 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 \(G\) into tom.

    gap> MarksTom( tom )[2];
    [ 3, 3 ]
    gap> MarksTom( sorted )[2];
    [ 2, 2 ]
    gap> HasPermutationTom( sorted );
    true
    gap> PermutationTom( sorted );
    (2,3)

#### 70.6 Technical Details about Tables of Marks

##### 70.6-1 InfoTom

`‣ InfoTom` | ( info class )

is the info class for computations concerning tables of marks.

##### 70.6-2 IsTableOfMarks

`‣ IsTableOfMarks`( obj ) | ( category )

Each table of marks belongs to this category.

##### 70.6-3 TableOfMarksFamily

`‣ TableOfMarksFamily` | ( family )

Each table of marks belongs to this family.

##### 70.6-4 TableOfMarksComponents

`‣ TableOfMarksComponents` | ( global variable )

The list `TableOfMarksComponents` is used when a table of marks object is created from a record via `ConvertToTableOfMarks` (70.6-5). `TableOfMarksComponents` contains at position \(2i-1\) a name of an attribute and at position \(2i\) the corresponding attribute getter function.

##### 70.6-5 ConvertToTableOfMarks

`‣ ConvertToTableOfMarks`( record ) | ( function )

`ConvertToTableOfMarks` converts a record with components from `TableOfMarksComponents` (70.6-4) into a table of marks object with the corresponding attributes.

    gap> record:= rec( MarksTom:= [ [ 4 ], [ 2, 2 ], [ 1, 1, 1 ] ],
    >  SubsTom:= [ [ 1 ], [ 1, 2 ], [ 1, 2, 3 ] ] );;
    gap> ConvertToTableOfMarks( record );;
    gap> record;
    TableOfMarks( <3 classes> )

#### 70.7 Attributes of Tables of Marks

##### 70.7-1 MarksTom

`‣ MarksTom`( tom ) | ( attribute )

`‣ SubsTom`( tom ) | ( attribute )

The matrix of marks (see 70.1) of the table of marks tom is stored in a compressed form where zeros are omitted, using the attributes `MarksTom` and `SubsTom`. If \(M\) is the square matrix of marks of tom (see `MatTom` (70.7-10)) then the `SubsTom` value of tom is a list that contains at position \(i\) the list of all positions of nonzero entries of the \(i\)-th row of \(M\), and the `MarksTom` value of tom is a list that contains at position \(i\) the list of the corresponding marks.

`MarksTom` and `SubsTom` are defining attributes of tables of marks (see 70.2). There is no default method for computing the `SubsTom` value, and the default `MarksTom` method needs the values of `NrSubsTom` (70.7-2) and `OrdersTom` (70.7-2).

    gap> a5:= TableOfMarks( "A5" );
    TableOfMarks( "A5" )
    gap> MarksTom( a5 );
    [ [ 60 ], [ 30, 2 ], [ 20, 2 ], [ 15, 3, 3 ], [ 12, 2 ],
      [ 10, 2, 1, 1 ], [ 6, 2, 1, 1 ], [ 5, 1, 2, 1, 1 ],
      [ 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ]
    gap> SubsTom( a5 );
    [ [ 1 ], [ 1, 2 ], [ 1, 3 ], [ 1, 2, 4 ], [ 1, 5 ], [ 1, 2, 3, 6 ],
      [ 1, 2, 5, 7 ], [ 1, 2, 3, 4, 8 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9 ] ]

##### 70.7-2 NrSubsTom

`‣ NrSubsTom`( tom ) | ( attribute )

`‣ OrdersTom`( tom ) | ( attribute )

Instead of storing the marks (see `MarksTom` (70.7-1)) of the table of marks tom one can use a matrix which contains at position \((i,j)\) the number of subgroups of conjugacy class \(j\) that are contained in one member of the conjugacy class \(i\). These values are stored in the `NrSubsTom` value in the same way as the marks in the `MarksTom` (70.7-1) value.

`OrdersTom` returns a list that contains at position \(i\) the order of a representative of the \(i\)-th conjugacy class of subgroups of tom.

One can compute the `NrSubsTom` and `OrdersTom` values from the `MarksTom` (70.7-1) value of tom and vice versa.

    gap> NrSubsTom( a5 );
    [ [ 1 ], [ 1, 1 ], [ 1, 1 ], [ 1, 3, 1 ], [ 1, 1 ], [ 1, 3, 1, 1 ],
      [ 1, 5, 1, 1 ], [ 1, 3, 4, 1, 1 ], [ 1, 15, 10, 5, 6, 10, 6, 5, 1 ]
     ]
    gap> OrdersTom( a5 );
    [ 1, 2, 3, 4, 5, 6, 10, 12, 60 ]

##### 70.7-3 LengthsTom

`‣ LengthsTom`( tom ) | ( attribute )

For a table of marks tom, `LengthsTom` returns a list of the lengths of the conjugacy classes of subgroups.

    gap> LengthsTom( a5 );
    [ 1, 15, 10, 5, 6, 10, 6, 5, 1 ]

##### 70.7-4 ClassTypesTom

`‣ ClassTypesTom`( tom ) | ( attribute )

`ClassTypesTom` distinguishes isomorphism types of the classes of subgroups of the table of marks tom as far as this is possible from the `SubsTom` (70.7-1) and `MarksTom` (70.7-1) values of tom.

Two subgroups are clearly not isomorphic if they have different orders. Moreover, isomorphic subgroups must contain the same number of subgroups of each type.

Each type is represented by a positive integer. `ClassTypesTom` returns the list which contains for each class of subgroups its corresponding type.

    gap> a6:= TableOfMarks( "A6" );;
    gap> ClassTypesTom( a6 );
    [ 1, 2, 3, 3, 4, 5, 6, 6, 7, 7, 8, 9, 10, 11, 11, 12, 13, 13, 14, 15,
      15, 16 ]

##### 70.7-5 ClassNamesTom

`‣ ClassNamesTom`( tom ) | ( attribute )

`ClassNamesTom` constructs generic names for the conjugacy classes of subgroups of the table of marks tom. In general, the generic name of a class of non-cyclic subgroups consists of three parts and has the form `"(`o`)_{`t`}`l`"`, where o indicates the order of the subgroup, t is a number that distinguishes different types of subgroups of the same order, and l 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 `ClassTypesTom` (70.7-4)). This is slightly weaker than isomorphism.

The 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 `{}` around the type are omitted if the type number has only one digit.

For classes of cyclic subgroups, the parentheses round the order and the type are omitted. Hence the most general form of their generic names is `"o,l"`. Again, the letter is omitted if there is only one class of cyclic subgroups of that order.

    gap> ClassNamesTom( a6 );
    [ "1", "2", "3a", "3b", "5", "4", "(4)_2a", "(4)_2b", "(6)a", "(6)b",
      "(8)", "(9)", "(10)", "(12)a", "(12)b", "(18)", "(24)a", "(24)b",
      "(36)", "(60)a", "(60)b", "(360)" ]

##### 70.7-6 FusionsTom

`‣ FusionsTom`( tom ) | ( attribute )

For a table of marks tom, `FusionsTom` is a list of fusions into other tables of marks. Each fusion is a list of length two, the first entry being the `Identifier` (70.7-9)) value of the image table, the second entry being the list of images of the class positions of tom in the image table.

This attribute is mainly used for tables of marks in the GAP library (see 70.13).

    gap> fus:= FusionsTom( a6 );;
    gap> fus[1];
    [ "L3(4)",
      [ 1, 2, 3, 3, 14, 5, 9, 7, 15, 15, 24, 26, 27, 32, 33, 50, 57, 55,
          63, 73, 77, 90 ] ]

##### 70.7-7 UnderlyingGroup

`‣ UnderlyingGroup`( tom ) | ( attribute )

`UnderlyingGroup` is used to access an underlying group that is stored on the table of marks tom. There is no default method to compute an underlying group if it is not stored.

    gap> UnderlyingGroup( a6 );
    Group([ (1,2)(3,4), (1,2,4,5)(3,6) ])

##### 70.7-8 IdempotentsTom

`‣ IdempotentsTom`( tom ) | ( attribute )

`‣ IdempotentsTomInfo`( tom ) | ( attribute )

`IdempotentsTom` encodes the idempotents of the integral Burnside ring described by the table of marks tom. The return value is a list \(l\) of positive integers such that each row vector describing a primitive idempotent has value \(1\) at all positions with the same entry in \(l\), and \(0\) at all other positions.

According 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 `CyclicExtensionsTom` (70.9-7)).

`IdempotentsTomInfo` returns a record with components `fixpointvectors` and `primidems`, both bound to lists. The \(i\)-th entry of the `fixpointvectors` list is the \(0-1\)-vector describing the \(i\)-th primitive idempotent, and the \(i\)-th entry of `primidems` is the decomposition of this idempotent in the rows of tom.

    gap> IdempotentsTom( a5 );
    [ 1, 1, 1, 1, 1, 1, 1, 1, 9 ]
    gap> IdempotentsTomInfo( a5 );
    rec(
      fixpointvectors := [ [ 1, 1, 1, 1, 1, 1, 1, 1, 0 ],
          [ 0, 0, 0, 0, 0, 0, 0, 0, 1 ] ],
      primidems := [ [ 1, -2, -1, 0, 0, 1, 1, 1 ],
          [ -1, 2, 1, 0, 0, -1, -1, -1, 1 ] ] )

##### 70.7-9 Identifier

`‣ Identifier`( tom ) | ( attribute )

The identifier of a table of marks tom is a string. It is used for printing the table of marks (see 70.4) and in fusions between tables of marks (see `FusionsTom` (70.7-6)).

If tom is a table of marks from the GAP library of tables of marks (see 70.13) then it has an identifier, and if tom was constructed from a group with `Name` (12.8-2) then this name is chosen as `Identifier` value. There is no default method to compute an identifier in all other cases.

    gap> Identifier( a5 );
    "A5"

##### 70.7-10 MatTom

`‣ MatTom`( tom ) | ( attribute )

`MatTom` returns the square matrix of marks (see 70.1) of the table of marks tom which is stored in a compressed form using the attributes `MarksTom` (70.7-1) and `SubsTom` (70.7-1) This may need substantially more space than the values of `MarksTom` (70.7-1) and `SubsTom` (70.7-1).

    gap> MatTom( a5 );
    [ [ 60, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 30, 2, 0, 0, 0, 0, 0, 0, 0 ],
      [ 20, 0, 2, 0, 0, 0, 0, 0, 0 ], [ 15, 3, 0, 3, 0, 0, 0, 0, 0 ],
      [ 12, 0, 0, 0, 2, 0, 0, 0, 0 ], [ 10, 2, 1, 0, 0, 1, 0, 0, 0 ],
      [ 6, 2, 0, 0, 1, 0, 1, 0, 0 ], [ 5, 1, 2, 1, 0, 0, 0, 1, 0 ],
      [ 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ]

##### 70.7-11 MoebiusTom

`‣ MoebiusTom`( tom ) | ( attribute )

`MoebiusTom` computes the Möbius values both of the subgroup lattice of the group \(G\) with table of marks tom and of the poset of conjugacy classes of subgroups of \(G\). It returns a record where the component `mu` contains the Möbius values of the subgroup lattice, and the component `nu` contains the Möbius values of the poset.

Moreover, 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 "expected values" are returned in the component `ex`, and the list of numbers of those subgroups where the expected value does not coincide with the actual value are returned in the component `hyp`. For the computation of these values, the position of the derived subgroup of \(G\) is needed (see `DerivedSubgroupTom` (70.9-2)). If it is not uniquely determined then the result does not have the components `ex` and `hyp`.

    gap> MoebiusTom( a5 );
    rec( ex := [ -60, 4, 2,,, -1, -1, -1, 1 ], hyp := [  ],
      mu := [ -60, 4, 2,,, -1, -1, -1, 1 ],
      nu := [ -1, 2, 1,,, -1, -1, -1, 1 ] )
    gap> tom:= TableOfMarks( "M12" );;
    gap> moebius:= MoebiusTom( tom );;
    gap> moebius.hyp;
    [ 1, 2, 4, 16, 39, 45, 105 ]
    gap> moebius.mu[1];  moebius.ex[1];
    95040
    190080

##### 70.7-12 WeightsTom

`‣ WeightsTom`( tom ) | ( attribute )

`WeightsTom` extracts the weights from the table of marks tom, i.e., the diagonal entries of the matrix of marks (see `MarksTom` (70.7-1)), indicating the index of a subgroup in its normalizer.

    gap> wt:= WeightsTom( a5 );
    [ 60, 2, 2, 3, 2, 1, 1, 1, 1 ]

This information may be used to obtain the numbers of conjugate supergroups from the marks.

    gap> marks:= MarksTom( a5 );;
    gap> List( [ 1 .. 9 ], x -> marks[x] / wt[x] );
    [ [ 1 ], [ 15, 1 ], [ 10, 1 ], [ 5, 1, 1 ], [ 6, 1 ], [ 10, 2, 1, 1 ],
      [ 6, 2, 1, 1 ], [ 5, 1, 2, 1, 1 ], [ 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ]

#### 70.8 Properties of Tables of Marks

For a table of marks tom of a group \(G\), the following properties have the same meaning as the corresponding properties for \(G\). Additionally, if a positive integer sub is given as the second argument then the value of the corresponding property for the sub-th class of subgroups of tom is returned.

##### 70.8-1 IsAbelianTom

`‣ IsAbelianTom`( tom[, sub] ) | ( property )

`‣ IsCyclicTom`( tom[, sub] ) | ( property )

`‣ IsNilpotentTom`( tom[, sub] ) | ( property )

`‣ IsPerfectTom`( tom[, sub] ) | ( property )

`‣ IsSolvableTom`( tom[, sub] ) | ( property )

    gap> tom:= TableOfMarks( "A5" );;
    gap> IsAbelianTom( tom );  IsPerfectTom( tom );
    false
    true
    gap> IsAbelianTom( tom, 3 );  IsNilpotentTom( tom, 7 );
    true
    false
    gap> IsPerfectTom( tom, 7 );  IsSolvableTom( tom, 7 );
    false
    true
    gap> for i in [ 1 .. 6 ] do
    > Print( i, ": ", IsCyclicTom(a5, i), "  " );
    > od;  Print( "\n" );
    1: true  2: true  3: true  4: false  5: true  6: false

#### 70.9 Other Operations for Tables of Marks

##### 70.9-1 IsInternallyConsistent

`‣ IsInternallyConsistent`( tom ) | ( method )

For a table of marks tom, `IsInternallyConsistent` decomposes all tensor products of rows of tom. It returns `true` if all decomposition numbers are nonnegative integers, and `false` otherwise. This provides a strong consistency check for a table of marks.

##### 70.9-2 DerivedSubgroupTom

`‣ DerivedSubgroupTom`( tom, sub ) | ( operation )

`‣ DerivedSubgroupsTom`( tom ) | ( function )

For a table of marks tom and a positive integer sub, `DerivedSubgroupTom` returns either a positive integer \(i\) or a list \(l\) of positive integers. In the former case, the result means that the derived subgroups of the subgroups in the sub-th class of tom lie in the \(i\)-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 \(l\).

Values computed with `DerivedSubgroupTom` are stored using the attribute `DerivedSubgroupsTomPossible` (70.9-3).

`DerivedSubgroupsTom` is just the list of `DerivedSubgroupTom` values for all values of sub.

##### 70.9-3 DerivedSubgroupsTomPossible

`‣ DerivedSubgroupsTomPossible`( tom ) | ( attribute )

`‣ DerivedSubgroupsTomUnique`( tom ) | ( attribute )

Let tom be a table of marks. The value of the attribute `DerivedSubgroupsTomPossible` is a list in which the value at position \(i\) –if bound– is a positive integer or a list; the meaning of the entry is the same as in `DerivedSubgroupTom` (70.9-2).

If the value of the attribute `DerivedSubgroupsTomUnique` is known for tom then it is a list of positive integers, the value at position \(i\) being the position of the class of derived subgroups of the \(i\)-th class of subgroups in tom.

The derived subgroups are in general not uniquely determined by the table of marks if no `UnderlyingGroup` (70.7-7) value is stored, so there is no default method for `DerivedSubgroupsTomUnique`. But in some cases the derived subgroups are explicitly set when the table of marks is constructed. In this case, `DerivedSubgroupTom` (70.9-2) does not set values in the `DerivedSubgroupsTomPossible` list.

The `DerivedSubgroupsTomUnique` value is automatically set when the last missing unique value is entered in the `DerivedSubgroupsTomPossible` list by `DerivedSubgroupTom` (70.9-2).

##### 70.9-4 NormalizerTom

`‣ NormalizerTom`( tom, sub ) | ( operation )

`‣ NormalizersTom`( tom ) | ( attribute )

Let tom be the table of marks of a group \(G\), say. `NormalizerTom` tries to find the conjugacy class of the normalizer \(N\) in \(G\) of a subgroup \(U\) in the sub-th class of tom. 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 tom knows the subgroup lattice of \(G\) (see `IsTableOfMarksWithGens` (70.10-3)) then all normalizers are uniquely determined. `NormalizerTom` should never return an empty list.

`NormalizersTom` returns the list of positions of the classes of normalizers of subgroups in tom. In addition to the criteria for a single class of subgroup used by `NormalizerTom`, the approximations of normalizers for several classes are used and thus `NormalizersTom` may return better approximations than `NormalizerTom`.

    gap> NormalizerTom( a5, 4 );
    8
    gap> NormalizersTom( a5 );
    [ 9, 4, 6, 8, 7, 6, 7, 8, 9 ]

The example shows that a subgroup with class number 4 in \(A_5\) (which is a Kleinian four group) is normalized by a subgroup in class 8. This class contains the subgroups of \(A_5\) which are isomorphic to \(A_4\).

##### 70.9-5 ContainedTom

`‣ ContainedTom`( tom, sub1, sub2 ) | ( operation )

`ContainedTom` returns the number of subgroups in class sub1 of the table of marks tom that are contained in one fixed member of the class sub2.

    gap> ContainedTom( a5, 3, 5 );  ContainedTom( a5, 3, 8 );
    0
    4

##### 70.9-6 ContainingTom

`‣ ContainingTom`( tom, sub1, sub2 ) | ( operation )

`ContainingTom` returns the number of subgroups in class sub2 of the table of marks tom that contain one fixed member of the class sub1.

    gap> ContainingTom( a5, 3, 5 );  ContainingTom( a5, 3, 8 );
    0
    2

##### 70.9-7 CyclicExtensionsTom

`‣ CyclicExtensionsTom`( tom, p ) | ( operation )

`‣ CyclicExtensionsTom`( tom[, list] ) | ( attribute )

According to A. Dress [Dre69], two columns of the table of marks tom are equal modulo the prime p if and only if the corresponding subgroups are connected by a chain of normal extensions of order p.

Called with tom and p, `CyclicExtensionsTom` returns the classes of this equivalence relation.

In the second form, list 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 list. The default value for list is the set of prime divisors of the order of the group of tom.

(This information is not used by `NormalizerTom` (70.9-4) although it might give additional restrictions in the search of normalizers.)

    gap> CyclicExtensionsTom( a5, 2 );
    [ [ 1, 2, 4 ], [ 3, 6 ], [ 5, 7 ], [ 8 ], [ 9 ] ]

##### 70.9-8 DecomposedFixedPointVector

`‣ DecomposedFixedPointVector`( tom, fix ) | ( operation )

Let tom be the table of marks of the group \(G\), say, and let fix be a vector of fixed point numbers w.r.t. an action of \(G\), i.e., a vector which contains for each class of subgroups the number of fixed points under the given action. `DecomposedFixedPointVector` returns the decomposition of fix into rows of the table of marks. This decomposition corresponds to a decomposition of the action into transitive constituents. Trailing zeros in fix may be omitted.

    gap> DecomposedFixedPointVector( a5, [ 16, 4, 1, 0, 1, 1, 1 ] );
    [ 0, 0, 0, 0, 0, 1, 1 ]

The vector fix may be any vector of integers. The resulting decomposition, however, will not be integral, in general.

    gap> DecomposedFixedPointVector( a5, [ 0, 0, 0, 0, 1, 1 ] );
    [ 2/5, -1, -1/2, 0, 1/2, 1 ]

##### 70.9-9 EulerianFunctionByTom

`‣ EulerianFunctionByTom`( tom, n[, sub] ) | ( operation )

Called with two arguments, `EulerianFunctionByTom` computes the Eulerian function (see `EulerianFunction` (39.16-3)) of the underlying group \(G\) of the table of marks tom, that is, the number of n-tuples of elements in \(G\) that generate \(G\). If the optional argument sub is given then `EulerianFunctionByTom` computes the Eulerian function of each subgroup in the sub-th class of subgroups of tom.

For a group \(G\) whose table of marks is known, `EulerianFunctionByTom` is installed as a method for `EulerianFunction` (39.16-3).

    gap> EulerianFunctionByTom( a5, 2 );
    2280
    gap> EulerianFunctionByTom( a5, 3 );
    200160
    gap> EulerianFunctionByTom( a5, 2, 3 );
    8

##### 70.9-10 IntersectionsTom

`‣ IntersectionsTom`( tom, sub1, sub2 ) | ( operation )

The intersections of the groups in the sub1-th conjugacy class of subgroups of the table of marks tom with the groups in the sub2-th conjugacy classes of subgroups of tom are determined up to conjugacy by the decomposition of the tensor product of their rows of marks. `IntersectionsTom` returns a list \(l\) that describes this decomposition. The \(i\)-th entry in \(l\) is the multiplicity of groups in the \(i\)-th conjugacy class as an intersection.

    gap> IntersectionsTom( a5, 8, 8 );
    [ 0, 0, 1, 0, 0, 0, 0, 1 ]

Any two subgroups of class number 8 (\(A_4\)) of \(A_5\) are either equal and their intersection has again class number 8, or their intersection has class number \(3\), and is a cyclic subgroup of order 3.

##### 70.9-11 FactorGroupTom

`‣ FactorGroupTom`( tom, n ) | ( operation )

For a table of marks tom of the group \(G\), say, and the normal subgroup \(N\) of \(G\) corresponding to the n-th class of subgroups of tom, `FactorGroupTom` returns the table of marks of the factor group \(G / N\).

    gap> s4:= TableOfMarks( SymmetricGroup( 4 ) );
    TableOfMarks( Sym( [ 1 .. 4 ] ) )
    gap> LengthsTom( s4 );
    [ 1, 3, 6, 4, 1, 3, 3, 4, 3, 1, 1 ]
    gap> OrdersTom( s4 );
    [ 1, 2, 2, 3, 4, 4, 4, 6, 8, 12, 24 ]
    gap> s3:= FactorGroupTom( s4, 5 );
    TableOfMarks( Group([ f1, f2 ]) )
    gap> Display( s3 );
    1:  6
    2:  3 1
    3:  2 . 2
    4:  1 1 1 1

##### 70.9-12 MaximalSubgroupsTom

`‣ MaximalSubgroupsTom`( tom[, sub] ) | ( attribute )

Called with a table of marks tom, `MaximalSubgroupsTom` 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 tom, the second entry being the list of class lengths of these groups.

Called with a table of marks tom and a position sub, the same information for the sub-th class of subgroups is returned.

    gap> MaximalSubgroupsTom( s4 );
    [ [ 10, 9, 8 ], [ 1, 3, 4 ] ]
    gap> MaximalSubgroupsTom( s4, 10 );
    [ [ 5, 4 ], [ 1, 4 ] ]

##### 70.9-13 MinimalSupergroupsTom

`‣ MinimalSupergroupsTom`( tom, sub ) | ( operation )

For a table of marks tom, `MinimalSupergroupsTom` 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 sub-th class of subgroups of tom, the second entry being the list of class lengths of these groups.

    gap> MinimalSupergroupsTom( s4, 5 );
    [ [ 9, 10 ], [ 3, 1 ] ]

#### 70.10 Accessing Subgroups via Tables of Marks

Let tom be the table of marks of the group \(G\), and assume that tom has access to \(G\) via the `UnderlyingGroup` (70.7-7) value. Then it makes sense to use tom and its ordering of conjugacy classes of subgroups of \(G\) for storing information for constructing representatives of these classes. The group \(G\) is in general not sufficient for this, tom needs more information; this is available if and only if the `IsTableOfMarksWithGens` (70.10-3) value of tom is `true`. In this case, `RepresentativeTom` (70.10-4) can be used to get a subgroup of the \(i\)-th class, for all \(i\).

GAP provides two different possibilities to store generators of the representatives of classes of subgroups. The first is implemented by the attribute `GeneratorsSubgroupsTom` (70.10-1), which uses explicit generators of the subgroups. The second, more general, possibility is implemented by the attribute `StraightLineProgramsTom` (70.10-2), which encodes the generators as straight line programs (see 37.8) that evaluate to the generators in question when applied to standard generators of \(G\). This means that on the one hand, standard generators of \(G\) must be known in order to use `StraightLineProgramsTom` (70.10-2). On the other hand, the straight line programs allow one to compute easily generators not only of a subgroup \(U\) of \(G\) but also generators of the image of \(U\) in any representation of \(G\), provided that one knows standard generators of the image of \(G\) under this representation. See the manual of the package TomLib for details and an example.

##### 70.10-1 GeneratorsSubgroupsTom

`‣ GeneratorsSubgroupsTom`( tom ) | ( attribute )

Let tom be a table of marks with `IsTableOfMarksWithGens` (70.10-3) value `true`. Then `GeneratorsSubgroupsTom` returns a list of length two, the first entry being a list \(l\) of elements of the group stored as `UnderlyingGroup` (70.7-7) value of tom, the second entry being a list that contains at position \(i\) a list of positions in \(l\) of generators of a representative of a subgroup in class \(i\).

The `GeneratorsSubgroupsTom` value is known for all tables of marks that have been computed with `TableOfMarks` (70.3-1) from a group, and there is a method to compute the value for a table of marks that admits `RepresentativeTom` (70.10-4).

##### 70.10-2 StraightLineProgramsTom

`‣ StraightLineProgramsTom`( tom ) | ( attribute )

For a table of marks tom with `IsTableOfMarksWithGens` (70.10-3) value `true`, `StraightLineProgramsTom` returns a list that contains at position \(i\) either a list of straight line programs or a straight line program (see 37.8), encoding the generators of a representative of the \(i\)-th conjugacy class of subgroups of `UnderlyingGroup( tom )`; in the former case, each straight line program returns a generator, in the latter case, the program returns the list of generators.

There is no default method to compute the `StraightLineProgramsTom` 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 GAP library of tables of marks (see 70.13).

##### 70.10-3 IsTableOfMarksWithGens

`‣ IsTableOfMarksWithGens`( tom ) | ( filter )

This filter shall express the union of the filters `IsTableOfMarks and HasStraightLineProgramsTom` and `IsTableOfMarks and HasGeneratorsSubgroupsTom`. If a table of marks tom has this filter set then tom 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 `DerivedSubgroupTom` (70.9-2), `NormalizerTom` (70.9-4)).

    gap> a5:= TableOfMarks( "A5" );;  IsTableOfMarksWithGens( a5 );
    true
    gap> HasGeneratorsSubgroupsTom( a5 );  HasStraightLineProgramsTom( a5 );
    false
    true
    gap> alt5:= TableOfMarks( AlternatingGroup( 5 ) );;
    gap> IsTableOfMarksWithGens( alt5 );
    true
    gap> HasGeneratorsSubgroupsTom(alt5); HasStraightLineProgramsTom(alt5);
    true
    false
    gap> progs:= StraightLineProgramsTom( a5 );;
    gap> OrdersTom( a5 );
    [ 1, 2, 3, 4, 5, 6, 10, 12, 60 ]
    gap> IsCyclicTom( a5, 4 );
    false
    gap> Length( progs[4] );
    2
    gap> progs[4][1];
    <straight line program>
    gap> # first generator of an el. ab group of order 4:
    gap> Display( progs[4][1] );
    # input:
    r:= [ g1, g2 ];
    # program:
    r[3]:= r[2]*r[1];
    r[4]:= r[3]*r[2]^-1*r[1]*r[3]*r[2]^-1*r[1]*r[2];
    # return value:
    r[4]
    gap> x:= [ [ Z(2)^0, 0*Z(2) ], [ Z(2^2), Z(2)^0 ] ];;
    gap> y:= [ [ Z(2^2), Z(2)^0 ], [ 0*Z(2), Z(2^2)^2 ] ];;
    gap> res1:= ResultOfStraightLineProgram( progs[4][1], [ x, y ] );
    [ [ Z(2)^0, 0*Z(2) ], [ Z(2^2)^2, Z(2)^0 ] ]
    gap> res2:= ResultOfStraightLineProgram( progs[4][2], [ x, y ] );
    [ [ Z(2)^0, 0*Z(2) ], [ Z(2^2), Z(2)^0 ] ]
    gap> w:= y*x;;
    gap> res1 = w*y^-1*x*w*y^-1*x*y;
    true
    gap> subgrp:= Group( res1, res2 );;  Size( subgrp );  IsCyclic( subgrp );
    4
    false

##### 70.10-4 RepresentativeTom

`‣ RepresentativeTom`( tom, sub ) | ( operation )

`‣ RepresentativeTomByGenerators`( tom, sub, gens ) | ( operation )

`‣ RepresentativeTomByGeneratorsNC`( tom, sub, gens ) | ( operation )

Let tom be a table of marks with `IsTableOfMarksWithGens` (70.10-3) value `true`, and sub a positive integer. `RepresentativeTom` returns a representative of the sub-th conjugacy class of subgroups of tom.

If the attribute `StraightLineProgramsTom` (70.10-2) is set in tom then methods for the operations `RepresentativeTomByGenerators` and `RepresentativeTomByGeneratorsNC` are available, which return a representative of the sub-th conjugacy class of subgroups of tom, as a subgroup of the group generated by gens. This means that the standard generators of tom are replaced by gens.

`RepresentativeTomByGenerators` checks whether mapping the standard generators of tom to gens extends to a group isomorphism, and returns `fail` if not. `RepresentativeTomByGeneratorsNC` omits all checks. So `RepresentativeTomByGenerators` 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 `RepresentativeTom` under this isomorphism. The idea behind `RepresentativeTomByGeneratorsNC`, however, is to avoid the overhead of using isomorphisms when gens are known to be standard generators. In order to proceed like this, the attribute `StraightLineProgramsTom` (70.10-2) is needed.

    gap> RepresentativeTom( a5, 4 );
    Group([ (2,3)(4,5), (2,4)(3,5) ])

#### 70.11 The Interface between Tables of Marks and Character Tables

The following examples require the GAP Character Table Library to be available. If it is not yet loaded then we load it now.

    gap> LoadPackage( "ctbllib" );
    true

##### 70.11-1 FusionCharTableTom

`‣ FusionCharTableTom`( tbl, tom ) | ( operation )

`‣ PossibleFusionsCharTableTom`( tbl, tom[, options] ) | ( operation )

Let tbl be the ordinary character table of the group \(G\), say, and tom the table of marks of \(G\). `FusionCharTableTom` determines the fusion of the classes of elements from tbl to the classes of cyclic subgroups on tom, that is, a list that contains at position \(i\) the position of the class of cyclic subgroups in tom that are generated by elements in the \(i\)-th conjugacy class of elements in tbl.

Three cases are handled differently.

1. The fusion is explicitly stored on tbl. Then nothing has to be done. This happens only if both tbl and tom are tables from the GAP library (see 70.13 and the manual of the GAP Character Table Library).

2. The `UnderlyingGroup` (70.7-7) values of tbl and tom are known and equal. Then the group is used to compute the fusion.

3. There 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 `fail` is returned by `FusionCharTableTom`.

`PossibleFusionsCharTableTom` computes the list of possible fusions from tbl to tom, according to the criteria that have been checked. So if `FusionCharTableTom` returns a unique fusion then the list returned by `PossibleFusionsCharTableTom` for the same arguments contains exactly this fusion, and if `FusionCharTableTom` returns `fail` then the length of this list is different from \(1\).

The optional argument options must be a record that may have the following components.

`fusionmap`

  a parametrized map which is an approximation of the desired map,

`quick`

  a Boolean; if `true` then as soon as only one possibility remains this possibility is returned immediately; the default value is `false`.

    gap> a5c:= CharacterTable( "A5" );;
    gap> fus:= FusionCharTableTom( a5c, a5 );
    [ 1, 2, 3, 5, 5 ]

##### 70.11-2 PermCharsTom

`‣ PermCharsTom`( fus, tom ) | ( operation )

`‣ PermCharsTom`( tbl, tom ) | ( operation )

`PermCharsTom` returns the list of transitive permutation characters from the table of marks tom. In the first form, fus must be the fusion map from the ordinary character table of the group of tom to tom (see `FusionCharTableTom` (70.11-1)). In the second form, tbl must be the character table of the group of which tom is the table of marks. If the fusion map is not uniquely determined (see `FusionCharTableTom` (70.11-1)) then `fail` is returned.

If the fusion map fus is given as first argument then each transitive permutation character is represented by its values list. If the character table tbl is given then the permutation characters are class function objects (see Chapter 72).

    gap> PermCharsTom( a5c, a5 );
    [ Character( CharacterTable( "A5" ), [ 60, 0, 0, 0, 0 ] ),
      Character( CharacterTable( "A5" ), [ 30, 2, 0, 0, 0 ] ),
      Character( CharacterTable( "A5" ), [ 20, 0, 2, 0, 0 ] ),
      Character( CharacterTable( "A5" ), [ 15, 3, 0, 0, 0 ] ),
      Character( CharacterTable( "A5" ), [ 12, 0, 0, 2, 2 ] ),
      Character( CharacterTable( "A5" ), [ 10, 2, 1, 0, 0 ] ),
      Character( CharacterTable( "A5" ), [ 6, 2, 0, 1, 1 ] ),
      Character( CharacterTable( "A5" ), [ 5, 1, 2, 0, 0 ] ),
      Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ) ]
    gap> PermCharsTom( fus, a5 )[1];
    [ 60, 0, 0, 0, 0 ]

#### 70.12 Generic Construction of Tables of Marks

The following three operations construct a table of marks only from the data given, i.e., without underlying group.

##### 70.12-1 TableOfMarksCyclic

`‣ TableOfMarksCyclic`( n ) | ( operation )

`TableOfMarksCyclic` returns the table of marks of the cyclic group of order n.

A cyclic group of order n has as its subgroups for each divisor \(d\) of n a cyclic subgroup of order \(d\).

    gap> Display( TableOfMarksCyclic( 6 ) );
    1:  6
    2:  3 3
    3:  2 . 2
    4:  1 1 1 1

##### 70.12-2 TableOfMarksDihedral

`‣ TableOfMarksDihedral`( n ) | ( operation )

`TableOfMarksDihedral` returns the table of marks of the dihedral group of order m.

For each divisor \(d\) of m, a dihedral group of order \(m = 2n\) contains subgroups of order \(d\) according to the following rule. If \(d\) is odd and divides \(n\) then there is only one cyclic subgroup of order \(d\). If \(d\) is even and divides \(n\) then there are a cyclic subgroup of order \(d\) and two classes of dihedral subgroups of order \(d\) (which are cyclic, too, in the case \(d = 2\), see the example below). Otherwise (i.e., if \(d\) does not divide \(n\)) there is just one class of dihedral subgroups of order \(d\).

    gap> Display( TableOfMarksDihedral( 12 ) );
     1:  12
     2:   6 6
     3:   6 . 2
     4:   6 . . 2
     5:   4 . . . 4
     6:   3 3 1 1 . 1
     7:   2 2 . . 2 . 2
     8:   2 . 2 . 2 . . 2
     9:   2 . . 2 2 . . . 2
    10:   1 1 1 1 1 1 1 1 1 1

##### 70.12-3 TableOfMarksFrobenius

`‣ TableOfMarksFrobenius`( p, q ) | ( operation )

`TableOfMarksFrobenius` computes the table of marks of a Frobenius group of order \(p q\), where \(p\) is a prime and \(q\) divides \(p-1\).

    gap> Display( TableOfMarksFrobenius( 5, 4 ) );
    1:  20
    2:  10 2
    3:   5 1 1
    4:   4 . . 4
    5:   2 2 . 2 2
    6:   1 1 1 1 1 1

#### 70.13 The Library of Tables of Marks

The GAP package TomLib 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 `TableOfMarks` (70.3-1) with argument a string. If also the GAP 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 `TableOfMarks` (70.3-1) with argument the character table.

A list of all names of tables of marks that are provided by the TomLib package can be obtained via `AllLibTomNames` (TomLib: AllLibTomNames).

    gap> names:= AllLibTomNames();;
    gap> "A5" in names;
    true

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