GAP (AtlasRep) - Chapter 3: The User Interface of the AtlasRep Package

Goto Chapter: Top 1 2 3 4 5 6 7 Bib Ind
[Top of Book] [Contents] [Previous Chapter] [Next Chapter]

[MathJax on]

3 The User Interface of the AtlasRep Package
3.1 Accessing vs. Constructing Representations
3.2 Group Names Used in the AtlasRep Package
3.3 Standard Generators Used in the AtlasRep Package
3.4 Class Names Used in the AtlasRep Package
3.4-1 Definition of ATLAS Class Names
3.4-2 AtlasClassNames
3.4-3 AtlasCharacterNames
3.5 Accessing Data via AtlasRep
3.5-1 DisplayAtlasInfo
3.5-2 Examples for DisplayAtlasInfo
3.5-3 AtlasGenerators
3.5-4 AtlasProgram
3.5-5 AtlasProgramInfo
3.5-6 OneAtlasGeneratingSetInfo
3.5-7 AllAtlasGeneratingSetInfos
3.5-8 AtlasGroup
3.5-9 AtlasSubgroup
3.5-10 AtlasRepInfoRecord
3.5-11 EvaluatePresentation
3.5-12 StandardGeneratorsData
3.6 Browse Applications Provided by AtlasRep
3.6-1 BrowseMinimalDegrees
3.6-2 BrowseBibliographySporadicSimple

### 3 The User Interface of the AtlasRep Package

The user interface is the part of the GAP interface that allows one to display information about the current contents of the database and to access individual data (perhaps by downloading them, see Section 4.2-1). The corresponding functions are described in this chapter. See Section 2.4 for some small examples how to use the functions of the interface.

Data extensions of the AtlasRep package are regarded as another part of the GAP interface, they are described in Chapter 5. Finally, the low level part of the interface is described in Chapter 7.

#### 3.1 Accessing vs. Constructing Representations

Note that accessing the data means in particular that it is not the aim of this package to construct representations from known ones. For example, if at least one permutation representation for a group G is stored but no matrix representation in a positive characteristic p, say, then `OneAtlasGeneratingSetInfo` (3.5-6) returns `fail` when it is asked for a description of an available set of matrix generators for G in characteristic p, although such a representation can be obtained by reduction modulo p of an integral matrix representation, which in turn can be constructed from any permutation representation.

#### 3.2 Group Names Used in the AtlasRep Package

When you access data via the AtlasRep package, you specify the group in question by an admissible name. Thus it is essential to know these names, which are called the GAP names of the group in the following.

For a group G, say, whose character table is available in GAP's Character Table Library (see [Bre22]), the admissible names of G are the admissible names of this character table. One such name is the `Identifier` (Reference: Identifier for character tables) value of the character table, see CTblLib: Admissible Names for Character Tables in CTblLib. This name is usually very similar to the name used in the ATLAS of Finite Groups [CCN+85]. For example, `"M22"` is a GAP name of the Mathieu group M_22, `"12_1.U4(3).2_1"` is a GAP name of 12_1.U_4(3).2_1, the two names `"S5"` and `"A5.2"` are GAP names of the symmetric group S_5, and the two names `"F3+"` and `"Fi24'"` are GAP names of the simple Fischer group Fi_24^'.

When a GAP name is required as an input of a package function, this input is case insensitive. For example, both `"A5"` and `"a5"` are valid arguments of `DisplayAtlasInfo` (3.5-1).

Internally, for example as part of filenames (see Section 7.6), the package uses names that may differ from the GAP names; these names are called ATLAS-file names. For example, `"A5"`, `"TE62"`, and `"F24"` are ATLAS-file names. Of these, only `"A5"` is also a GAP name, but the other two are not; corresponding GAP names are `"2E6(2)"` and `"Fi24'"`, respectively.

#### 3.3 Standard Generators Used in the AtlasRep Package

For the general definition of standard generators of a group, see [Wil96].

Several different standard generators may be defined for a group, the definitions for each group that occurs in the ATLAS of Group Representations can be found at

http://atlas.math.rwth-aachen.de/Atlas/v3.

When one specifies the standardization, the i-th set of standard generators is denoted by the number i. Note that when more than one set of standard generators is defined for a group, one must be careful to use compatible standardization. For example, the straight line programs, straight line decisions and black box programs in the database refer to a specific standardization of their inputs. That is, a straight line program for computing generators of a certain subgroup of a group G is defined only for a specific set of standard generators of G, and applying the program to matrix or permutation generators of G but w. r. t. a different standardization may yield unpredictable results. Therefore the results returned by the functions described in this chapter contain information about the standardizations they refer to.

#### 3.4 Class Names Used in the AtlasRep Package

For each straight line program (see `AtlasProgram` (3.5-4)) that is used to compute lists of class representatives, it is essential to describe the classes in which these elements lie. Therefore, in these cases the records returned by the function `AtlasProgram` (3.5-4) contain a component `outputs` with value a list of class names.

Currently we define these class names only for simple groups and certain extensions of simple groups, see Section 3.4-1. The function `AtlasClassNames` (3.4-2) can be used to compute the list of class names from the character table in the GAP Library.

##### 3.4-1 Definition of ATLAS Class Names

For the definition of class names of an almost simple group, we assume that the ordinary character tables of all nontrivial normal subgroups are shown in the ATLAS of Finite Groups [CCN+85].

Each class name is a string consisting of the element order of the class in question followed by a combination of capital letters, digits, and the characters `'` and `-` (starting with a capital letter). For example, `1A`, `12A1`, and `3B'` denote the class that contains the identity element, a class of element order 12, and a class of element order 3, respectively.

1. For the table of a simple group, the class names are the same as returned by the two argument version of the GAP function `ClassNames` (Reference: ClassNames), cf. [CCN+85, Chapter 7, Section 5]: The classes are arranged w. r. t. increasing element order and for each element order w. r. t. decreasing centralizer order, the conjugacy classes that contain elements of order n are named n`A`, n`B`, n`C`, ...; the alphabet used here is potentially infinite, and reads `A`, `B`, `C`, ..., `Z`, `A1`, `B1`, ..., `A2`, `B2`, ....

   For example, the classes of the alternating group A_5 have the names `1A`, `2A`, `3A`, `5A`, and `5B`.

2. Next we consider the case of an upward extension G.A of a simple group G by a cyclic group of order A. The ATLAS defines class names for each element g of G.A only w. r. t. the group G.a, say, that is generated by G and g; namely, there is a power of g (with the exponent coprime to the order of g) for which the class has a name of the same form as the class names for simple groups, and the name of the class of g w. r. t. G.a is then obtained from this name by appending a suitable number of dashes `'`. So dashed class names refer exactly to those classes that are not printed in the ATLAS.

   For example, those classes of the symmetric group S_5 that do not lie in A_5 have the names `2B`, `4A`, and `6A`. The outer classes of the group L_2(8).3 have the names `3B`, `6A`, `9D`, and `3B'`, `6A'`, `9D'`. The outer elements of order 5 in the group Sz(32).5 lie in the classes with names `5B`, `5B'`, `5B''`, and `5B'''`.

   In the group G.A, the class of g may fuse with other classes. The name of the class of g in G.A is obtained from the names of the involved classes of G.a by concatenating their names after removing the element order part from all of them except the first one.

   For example, the elements of order 9 in the group L_2(27).6 are contained in the subgroup L_2(27).3 but not in L_2(27). In L_2(27).3, they lie in the classes `9A`, `9A'`, `9B`, and `9B'`; in L_2(27).6, these classes fuse to `9AB` and `9A'B'`.

3. Now we define class names for general upward extensions G.A of a simple group G. Each element g of such a group lies in an upward extension G.a by a cyclic group, and the class names w. r. t. G.a are already defined. The name of the class of g in G.A is obtained by concatenating the names of the classes in the orbit of G.A on the classes of cyclic upward extensions of G, after ordering the names lexicographically and removing the element order part from all of them except the first one. An exception is the situation where dashed and non-dashed class names appear in an orbit; in this case, the dashed names are omitted.

   For example, the classes `21A` and `21B` of the group U_3(5).3 fuse in U_3(5).S_3 to the class `21AB`, and the class `2B` of U_3(5).2 fuses with the involution classes `2B'`, `2B''` in the groups U_3(5).2^' and U_3(5).2^{''} to the class `2B` of U_3(5).S_3.

   It may happen that some names in the `outputs` component of a record returned by `AtlasProgram` (3.5-4) do not uniquely determine the classes of the corresponding elements. For example, the (algebraically conjugate) classes `39A` and `39B` of the group Co_1 have not been distinguished yet. In such cases, the names used contain a minus sign `-`, and mean "one of the classes in the range described by the name before and the name after the minus sign"; the element order part of the name does not appear after the minus sign. So the name `39A-B` for the group Co_1 means `39A` or `39B`, and the name `20A-B'''` for the group Sz(32).5 means one of the classes of element order 20 in this group (these classes lie outside the simple group Sz).

4. For a downward extension m.G.A of an almost simple group G.A by a cyclic group of order m, let π denote the natural epimorphism from m.G.A onto G.A. Each class name of m.G.A has the form `nX_0`, `nX_1` etc., where `nX` is the class name of the image under π, and the indices `0`, `1` etc. are chosen according to the position of the class in the lifting order rows for G, see [CCN+85, Chapter 7, Section 7, and the example in Section 8]).

   For example, if m = 6 then `1A_1` and `1A_5` denote the classes containing the generators of the kernel of π, that is, central elements of order 6.

##### 3.4-2 AtlasClassNames

`‣ AtlasClassNames`( tbl ) | ( function )

Returns: a list of class names.

Let tbl be the ordinary or modular character table of a group G, say, that is almost simple or a downward extension of an almost simple group and such that tbl is an ATLAS table from the GAP Character Table Library, according to its `InfoText` (Reference: InfoText) value. Then `AtlasClassNames` returns the list of class names for G, as defined in Section 3.4-1. The ordering of class names is the same as the ordering of the columns of tbl.

(The function may work also for character tables that are not ATLAS tables, but then clearly the class names returned are somewhat arbitrary.)

    gap> AtlasClassNames( CharacterTable( "L3(4).3" ) );
    [ "1A", "2A", "3A", "4ABC", "5A", "5B", "7A", "7B", "3B", "3B'",
      "3C", "3C'", "6B", "6B'", "15A", "15A'", "15B", "15B'", "21A",
      "21A'", "21B", "21B'" ]
    gap> AtlasClassNames( CharacterTable( "U3(5).2" ) );
    [ "1A", "2A", "3A", "4A", "5A", "5B", "5CD", "6A", "7AB", "8AB",
      "10A", "2B", "4B", "6D", "8C", "10B", "12B", "20A", "20B" ]
    gap> AtlasClassNames( CharacterTable( "L2(27).6" ) );
    [ "1A", "2A", "3AB", "7ABC", "13ABC", "13DEF", "14ABC", "2B", "4A",
      "26ABC", "26DEF", "28ABC", "28DEF", "3C", "3C'", "6A", "6A'",
      "9AB", "9A'B'", "6B", "6B'", "12A", "12A'" ]
    gap> AtlasClassNames( CharacterTable( "L3(4).3.2_2" ) );
    [ "1A", "2A", "3A", "4ABC", "5AB", "7A", "7B", "3B", "3C", "6B",
      "15A", "15B", "21A", "21B", "2C", "4E", "6E", "8D", "14A", "14B" ]
    gap> AtlasClassNames( CharacterTable( "3.A6" ) );
    [ "1A_0", "1A_1", "1A_2", "2A_0", "2A_1", "2A_2", "3A_0", "3B_0",
      "4A_0", "4A_1", "4A_2", "5A_0", "5A_1", "5A_2", "5B_0", "5B_1",
      "5B_2" ]
    gap> AtlasClassNames( CharacterTable( "2.A5.2" ) );
    [ "1A_0", "1A_1", "2A_0", "3A_0", "3A_1", "5AB_0", "5AB_1", "2B_0",
      "4A_0", "4A_1", "6A_0", "6A_1" ]

##### 3.4-3 AtlasCharacterNames

`‣ AtlasCharacterNames`( tbl ) | ( function )

Returns: a list of character names.

Let tbl be the ordinary or modular character table of a simple group. `AtlasCharacterNames` returns a list of strings, the i-th entry being the name of the i-th irreducible character of tbl; this name consists of the degree of this character followed by distinguishing lowercase letters.

    gap> AtlasCharacterNames( CharacterTable( "A5" ) );
    [ "1a", "3a", "3b", "4a", "5a" ]

#### 3.5 Accessing Data via AtlasRep

The examples shown in this section refer to the situation that no extensions have been notified, and to a perhaps outdated table of contents. That is, the current version of the database may contain more information than is shown here.

##### 3.5-1 DisplayAtlasInfo

`‣ DisplayAtlasInfo`( [listofnames][,] [std][,] ["contents", sources][,] [...] ) | ( function )

`‣ DisplayAtlasInfo`( gapname[, std][, ...] ) | ( function )

This function lists the information available via the AtlasRep package, for the given input.

There are essentially three ways of calling this function.

- If there is no argument or if the first argument is a list listofnames of strings that are GAP names of groups, `DisplayAtlasInfo` shows an overview of the known information.

- If the first argument is a string gapname that is a GAP name of a group, `DisplayAtlasInfo` shows an overview of the information that is available for this group.

- If the string `"contents"` is the only argument then the function shows which parts of the database are available; these are at least the `"core"` part, which means the data from the ATLAS of Group Representations, and the `"internal"` part, which means the data that are distributed with the AtlasRep package. Other parts can become available by calls to `AtlasOfGroupRepresentationsNotifyData` (5.1-1). Note that the shown numbers of locally available files depend on what has already been downloaded.

In each case, the information will be printed to the screen or will be fed into a pager, see Section 4.2-11. An interactive alternative to `DisplayAtlasInfo` is the function `BrowseAtlasInfo` (Browse: BrowseAtlasInfo), see [BL18].

The following paragraphs describe the structure of the output in the two cases. Examples can be found in Section 3.5-2.

Called without arguments, `DisplayAtlasInfo` shows a general overview for all groups. If some information is available for the group G, say, then one line is shown for G, with the following columns.

`group`

  the GAP name of G (see Section 3.2),

`#`

  the number of faithful representations stored for G that satisfy the additional conditions given (see below),

`maxes`

  the number of available straight line programs for computing generators of maximal subgroups of G,

`cl`

  a `+` sign if at least one program for computing representatives of conjugacy classes of elements of G is stored,

`cyc`

  a `+` sign if at least one program for computing representatives of classes of maximally cyclic subgroups of G is stored,

`out`

  descriptions of outer automorphisms of G for which at least one program is stored,

`fnd`

  a `+` sign if at least one program is available for finding standard generators,

`chk`

  a `+` sign if at least one program is available for checking whether a set of generators is a set of standard generators, and

`prs`

  a `+` sign if at least one program is available that encodes a presentation.

Called with a list listofnames of strings that are GAP names of some groups, `DisplayAtlasInfo` prints the overview described above but restricted to the groups in this list.

In addition to or instead of listofnames, the string `"contents"` and a description sources of the data may be given about which the overview is formed. See below for admissible values of sources.

Called with a string gapname that is a GAP name of a group, `DisplayAtlasInfo` prints an overview of the information that is available for this group. One line is printed for each faithful representation, showing the number of this representation (which can be used in calls of `AtlasGenerators` (3.5-3)), and a string of one of the following forms; in both cases, id is a (possibly empty) string.

`G <= Sym(`nid`)`

  denotes a permutation representation of degree n, for example `G <= Sym(40a)` and `G <= Sym(40b)` denote two (nonequivalent) representations of degree 40.

`G <= GL(`nid,descr`)`

  denotes a matrix representation of dimension n over a coefficient ring described by descr, which can be a prime power, `ℤ` (denoting the ring of integers), a description of an algebraic extension field, `ℂ` (denoting an unspecified algebraic extension field), or `ℤ/`m`ℤ` for an integer m (denoting the ring of residues mod m); for example, `G <= GL(2a,4)` and `G <= GL(2b,4)` denote two (nonequivalent) representations of dimension 2 over the field with four elements.

After the representations, the programs available for gapname are listed. The following optional arguments can be used to restrict the overviews.

std

  must be a positive integer or a list of positive integers; if it is given then only those representations are considered that refer to the std-th set of standard generators or the i-th set of standard generators, for i in std (see Section 3.3),

`"contents"` and sources

  for a string or a list of strings sources, restrict the data about which the overview is formed; if sources is the string `"core"` then only data from the ATLAS of Group Representations are considered, if sources is a string that denotes a data extension in the sense of a `dirid` argument of `AtlasOfGroupRepresentationsNotifyData` (5.1-1) then only the data that belong to this data extension are considered; also a list of such strings may be given, then the union of these data is considered,

`Identifier` and id

  restrict to representations with `id` component in the list id (note that this component is itself a list, entering this list is not admissible), or satisfying the function id,

`IsPermGroup` and `true` (or `false`)

  restrict to permutation representations (or to representations that are not permutation representations),

`NrMovedPoints` and n

  for a positive integer, a list of positive integers, or a property n, restrict to permutation representations of degree equal to n, or in the list n, or satisfying the function n,

`NrMovedPoints` and the string `"minimal"`

  restrict to faithful permutation representations of minimal degree (if this information is available),

`IsTransitive` and a boolean value

  restrict to transitive or intransitive permutation representations where this information is available (if the value `true` or `false` is given), or to representations for which this information is not available (if the value `fail` is given),

`IsPrimitive` and a boolean value

  restrict to primitive or imprimitive permutation representations where this information is available (if the value `true` or `false` is given), or to representations for which this information is not available (if the value `fail` is given),

`Transitivity` and n

  for a nonnegative integer, a list of nonnegative integers, or a property n, restrict to permutation representations for which the information is available that the transitivity is equal to n, or is in the list n, or satisfies the function n; if n is `fail` then restrict to all permutation representations for which this information is not available,

`RankAction` and n

  for a nonnegative integer, a list of nonnegative integers, or a property n, restrict to permutation representations for which the information is available that the rank is equal to n, or is in the list n, or satisfies the function n; if n is `fail` then restrict to all permutation representations for which this information is not available,

`IsMatrixGroup` and `true` (or `false`)

  restrict to matrix representations (or to representations that are not matrix representations),

`Characteristic` and p

  for a prime integer, a list of prime integers, or a property p, restrict to matrix representations over fields of characteristic equal to p, or in the list p, or satisfying the function p (representations over residue class rings that are not fields can be addressed by entering `fail` as the value of p),

`Dimension` and n

  for a positive integer, a list of positive integers, or a property n, restrict to matrix representations of dimension equal to n, or in the list n, or satisfying the function n,

`Characteristic`, p, `Dimension`, and the string `"minimal"`

  for a prime integer p, restrict to faithful matrix representations over fields of characteristic p that have minimal dimension (if this information is available),

`Ring` and R

  for a ring or a property R, restrict to matrix representations for which the information is available that the ring spanned by the matrix entries is contained in this ring or satisfies this property (note that the representation might be defined over a proper subring); if R is `fail` then restrict to all matrix representations for which this information is not available,

`Ring`, R, `Dimension`, and the string `"minimal"`

  for a ring R, restrict to faithful matrix representations over this ring that have minimal dimension (if this information is available),

`Character` and chi

  for a class function or a list of class functions chi, restrict to representations with these characters (note that the underlying characteristic of the class function, see Section Reference: UnderlyingCharacteristic, determines the characteristic of the representation),

`Character` and name

  for a string name, restrict to representations for which the character is known to have this name, according to the information shown by `DisplayAtlasInfo`; if the characteristic is not specified then it defaults to zero,

`Character` and n

  for a positive integer n, restrict to representations for which the character is known to be the n-th irreducible character in GAP's library character table of the group in question; if the characteristic is not specified then it defaults to zero,

`IsStraightLineProgram` and `true`

  restrict to straight line programs, straight line decisions (see Section 6.1), and black box programs (see Section 6.2), and

`IsStraightLineProgram` and `false`

  restrict to representations.

Note that the above conditions refer only to the information that is available without accessing the representations. For example, if it is not stored in the table of contents whether a permutation representation is primitive then this representation does not match an `IsPrimitive` condition in `DisplayAtlasInfo`.

If "minimality" information is requested and no available representation matches this condition then either no minimal representation is available or the information about the minimality is missing. See `MinimalRepresentationInfo` (6.3-1) for checking whether the minimality information is available for the group in question. Note that in the cases where the string `"minimal"` occurs as an argument, `MinimalRepresentationInfo` (6.3-1) is called with third argument `"lookup"`; this is because the stored information was precomputed just for the groups in the ATLAS of Group Representations, so trying to compute non-stored minimality information (using other available databases) will hardly be successful.

The representations are ordered as follows. Permutation representations come first (ordered according to their degrees), followed by matrix representations over finite fields (ordered first according to the field size and second according to the dimension), matrix representations over the integers, and then matrix representations over algebraic extension fields (both kinds ordered according to the dimension), the last representations are matrix representations over residue class rings (ordered first according to the modulus and second according to the dimension).

The maximal subgroups are ordered according to decreasing group order. For an extension G.p of a simple group G by an outer automorphism of prime order p, this means that G is the first maximal subgroup and then come the extensions of the maximal subgroups of G and the novelties; so the n-th maximal subgroup of G and the n-th maximal subgroup of G.p are in general not related. (This coincides with the numbering used for the `Maxes` (CTblLib: Maxes) attribute for character tables.)

##### 3.5-2 Examples for DisplayAtlasInfo

Here are some examples how `DisplayAtlasInfo` (3.5-1) can be called, and how its output can be interpreted.

    gap> DisplayAtlasInfo( "contents" );
    - AtlasRepAccessRemoteFiles: false

    - AtlasRepDataDirectory: /home/you/gap/pkg/atlasrep/

    ID       | address, version, files
    ---------+------------------------------------------------
    core     | http://atlas.math.rwth-aachen.de/Atlas/,
             | version 2019-04-08,
             | 10586 files locally available.
    ---------+------------------------------------------------
    internal | atlasrep/datapkg,
             | version 2019-05-06,
             | 276 files locally available.
    ---------+------------------------------------------------
    mfer     | http://www.math.rwth-aachen.de/~mfer/datagens/,
             | version 2015-10-06,
             | 34 files locally available.
    ---------+------------------------------------------------
    ctblocks | ctblocks/atlas/,
             | version 2019-04-08,
             | 121 files locally available.

Note: The above output does not fit to the rest of the manual examples, since data extensions except `internal` have been removed at the beginning of Chapter 2.

The output tells us that two data extensions have been notified in addition to the core data from the ATLAS of Group Representations and the (local) internal data distributed with the AtlasRep package. The files of the extension `mfer` must be downloaded before they can be read (but note that the access to remote files is disabled), and the files of the extension `ctblocks` are locally available in the `ctblocks/atlas` subdirectory of the GAP package directory. This table (in particular the numbers of locally available files) depends on your installation of the package and how many files you have already downloaded.

    gap> DisplayAtlasInfo( [ "M11", "A5" ] );
    group |  # | maxes | cl | cyc | out | fnd | chk | prs
    ------+----+-------+----+-----+-----+-----+-----+----
    M11   | 42 |     5 |  + |  +  |     |  +  |  +  |  +
    A5*   | 18 |     3 |  + |     |     |     |  +  |  +

The above output means that the database provides 42 representations of the Mathieu group M_11, straight line programs for computing generators of representatives of all five classes of maximal subgroups, for computing representatives of the conjugacy classes of elements and of generators of maximally cyclic subgroups, contains no straight line program for applying outer automorphisms (well, in fact M_11 admits no nontrivial outer automorphism), and contains straight line decisions that check a set of generators or a set of group elements for being a set of standard generators. Analogously, 18 representations of the alternating group A_5 are available, straight line programs for computing generators of representatives of all three classes of maximal subgroups, and no straight line programs for computing representatives of the conjugacy classes of elements, of generators of maximally cyclic subgroups, and no for computing images under outer automorphisms; straight line decisions for checking the standardization of generators or group elements are available.

    gap> DisplayAtlasInfo( [ "M11", "A5" ], NrMovedPoints, 11 );
    group | # | maxes | cl | cyc | out | fnd | chk | prs
    ------+---+-------+----+-----+-----+-----+-----+----
    M11   | 1 |     5 |  + |  +  |     |  +  |  +  |  +

The given conditions restrict the overview to permutation representations on 11 points. The rows for all those groups are omitted for which no such representation is available, and the numbers of those representations are shown that satisfy the given conditions. In the above example, we see that no representation on 11 points is available for A_5, and exactly one such representation is available for M_11.

    gap> DisplayAtlasInfo( "A5", IsPermGroup, true );
    Representations for G = A5:    (all refer to std. generators 1)
    ---------------------------
    1: G <= Sym(5)  3-trans., on cosets of A4 (1st max.)
    2: G <= Sym(6)  2-trans., on cosets of D10 (2nd max.)
    3: G <= Sym(10) rank 3, on cosets of S3 (3rd max.)
    gap> DisplayAtlasInfo( "A5", NrMovedPoints, [ 4 .. 9 ] );
    Representations for G = A5:    (all refer to std. generators 1)
    ---------------------------
    1: G <= Sym(5) 3-trans., on cosets of A4 (1st max.)
    2: G <= Sym(6) 2-trans., on cosets of D10 (2nd max.)

The first three representations stored for A_5 are (in fact primitive) permutation representations.

    gap> DisplayAtlasInfo( "A5", Dimension, [ 1 .. 3 ] );
    Representations for G = A5:    (all refer to std. generators 1)
    ---------------------------
     8: G <= GL(2a,4)                character 2a
     9: G <= GL(2b,4)                character 2b
    10: G <= GL(3,5)                 character 3a
    12: G <= GL(3a,9)                character 3a
    13: G <= GL(3b,9)                character 3b
    17: G <= GL(3a,Field([Sqrt(5)])) character 3a
    18: G <= GL(3b,Field([Sqrt(5)])) character 3b
    gap> DisplayAtlasInfo( "A5", Characteristic, 0 );
    Representations for G = A5:    (all refer to std. generators 1)
    ---------------------------
    14: G <= GL(4,Z)                 character 4a
    15: G <= GL(5,Z)                 character 5a
    16: G <= GL(6,Z)                 character 3ab
    17: G <= GL(3a,Field([Sqrt(5)])) character 3a
    18: G <= GL(3b,Field([Sqrt(5)])) character 3b

The representations with number between 4 and 13 are (in fact irreducible) matrix representations over various finite fields, those with numbers 14 to 16 are integral matrix representations, and the last two are matrix representations over the field generated by sqrt{5} over the rational number field.

    gap> DisplayAtlasInfo( "A5", Identifier, "a" );
    Representations for G = A5:    (all refer to std. generators 1)
    ---------------------------
     4: G <= GL(4a,2)                character 4a
     8: G <= GL(2a,4)                character 2a
    12: G <= GL(3a,9)                character 3a
    17: G <= GL(3a,Field([Sqrt(5)])) character 3a

Each of the representations with the numbers 4, 8, 12, and 17 is labeled with the distinguishing letter `a`.

    gap> DisplayAtlasInfo( "A5", NrMovedPoints, IsPrimeInt );
    Representations for G = A5:    (all refer to std. generators 1)
    ---------------------------
    1: G <= Sym(5) 3-trans., on cosets of A4 (1st max.)
    gap> DisplayAtlasInfo( "A5", Characteristic, IsOddInt );
    Representations for G = A5:    (all refer to std. generators 1)
    ---------------------------
     6: G <= GL(4,3)  character 4a
     7: G <= GL(6,3)  character 3ab
    10: G <= GL(3,5)  character 3a
    11: G <= GL(5,5)  character 5a
    12: G <= GL(3a,9) character 3a
    13: G <= GL(3b,9) character 3b
    gap> DisplayAtlasInfo( "A5", Dimension, IsPrimeInt );
    Representations for G = A5:    (all refer to std. generators 1)
    ---------------------------
     8: G <= GL(2a,4)                character 2a
     9: G <= GL(2b,4)                character 2b
    10: G <= GL(3,5)                 character 3a
    11: G <= GL(5,5)                 character 5a
    12: G <= GL(3a,9)                character 3a
    13: G <= GL(3b,9)                character 3b
    15: G <= GL(5,Z)                 character 5a
    17: G <= GL(3a,Field([Sqrt(5)])) character 3a
    18: G <= GL(3b,Field([Sqrt(5)])) character 3b
    gap> DisplayAtlasInfo( "A5", Ring, IsFinite and IsPrimeField );
    Representations for G = A5:    (all refer to std. generators 1)
    ---------------------------
     4: G <= GL(4a,2) character 4a
     5: G <= GL(4b,2) character 2ab
     6: G <= GL(4,3)  character 4a
     7: G <= GL(6,3)  character 3ab
    10: G <= GL(3,5)  character 3a
    11: G <= GL(5,5)  character 5a

The above examples show how the output can be restricted using a property (a unary function that returns either `true` or `false`) that follows `NrMovedPoints` (Reference: NrMovedPoints for a permutation), `Characteristic` (Reference: Characteristic), `Dimension` (Reference: Dimension), or `Ring` (Reference: Ring) in the argument list of `DisplayAtlasInfo` (3.5-1).

    gap> DisplayAtlasInfo( "A5", IsStraightLineProgram, true );
    Programs for G = A5:    (all refer to std. generators 1)
    --------------------
    - class repres.*
    - presentation
    - maxes (all 3):
      1:  A4
      2:  D10
      3:  S3
    - std. gen. checker:
      (check)
      (pres)

Straight line programs are available for computing generators of representatives of the three classes of maximal subgroups of A_5, and a straight line decision for checking whether given generators are in fact standard generators is available as well as a presentation in terms of standard generators, see `AtlasProgram` (3.5-4).

##### 3.5-3 AtlasGenerators

`‣ AtlasGenerators`( gapname, repnr[, maxnr] ) | ( function )

`‣ AtlasGenerators`( identifier ) | ( function )

Returns: a record containing generators for a representation, or `fail`.

In the first form, gapname must be a string denoting a GAP name (see Section 3.2) of a group, and repnr a positive integer. If at least repnr representations for the group with GAP name gapname are available then `AtlasGenerators`, when called with gapname and repnr, returns an immutable record describing the repnr-th representation; otherwise `fail` is returned. If a third argument maxnr, a positive integer, is given then an immutable record describing the restriction of the repnr-th representation to the maxnr-th maximal subgroup is returned.

The result record has at least the following components.

`contents`

  the identifier of the part of the database to which the generators belong, for example `"core"` or `"internal"`,

`generators`

  a list of generators for the group,

`groupname`

  the GAP name of the group (see Section 3.2),

`identifier`

  a GAP object (a list of filenames plus additional information) that uniquely determines the representation, see Section 7.7; the value can be used as `identifier` argument of `AtlasGenerators`.

`repname`

  a string that is an initial part of the filenames of the generators.

`repnr`

  the number of the representation in the current session, equal to the argument repnr if this is given.

`standardization`

  the positive integer denoting the underlying standard generators,

`type`

  a string that describes the type of the representation (`"perm"` for a permutation representation, `"matff"` for a matrix representation over a finite field, `"matint"` for a matrix representation over the ring of integers, `"matalg"` for a matrix representation over an algebraic number field).

Additionally, the following describing components may be available if they are known, and depending on the data type of the representation.

`size`

  the group order,

`id`

  the distinguishing string as described for `DisplayAtlasInfo` (3.5-1),

`charactername`

  a string that describes the character of the representation,

`constituents`

  a list of positive integers denoting the positions of the irreducible constituents of the character of the representation,

`p` (for permutation representations)

  for the number of moved points,

`dim` (for matrix representations)

  the dimension of the matrices,

`ring` (for matrix representations)

  the ring generated by the matrix entries,

`transitivity` (for permutation representations)

  a nonnegative integer, see `Transitivity` (Reference: Transitivity),

`orbits` (for intransitive permutation representations)

  the sorted list of orbit lengths on the set of moved points,

`rankAction` (for transitive permutation representations)

  the number of orbits of the point stabilizer on the set of moved points, see `RankAction` (Reference: RankAction),

`stabilizer` (for transitive permutation representations)

  a string that describes the structure of the point stabilizers,

`isPrimitive` (for transitive permutation representations)

  `true` if the point stabilizers are maximal subgroups, and `false` otherwise,

`maxnr` (for primitive permutation representations)

  the number of the class of maximal subgroups that contains the point stabilizers, w. r. t. the `Maxes` (CTblLib: Maxes) list.

It should be noted that the number repnr refers to the number shown by `DisplayAtlasInfo` (3.5-1) in the current session; it may be that after the addition of new representations (for example after loading a package that provides some), repnr refers to another representation.

The alternative form of `AtlasGenerators`, with only argument identifier, can be used to fetch the result record with `identifier` value equal to identifier. The purpose of this variant is to access the same representation also in different GAP sessions.

    gap> gens1:= AtlasGenerators( "A5", 1 );
    rec( charactername := "1a+4a", constituents := [ 1, 4 ],
      contents := "core", generators := [ (1,2)(3,4), (1,3,5) ],
      groupname := "A5", id := "",
      identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ],
      isPrimitive := true, maxnr := 1, p := 5, rankAction := 2,
      repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4",
      standardization := 1, transitivity := 3, type := "perm" )
    gap> gens8:= AtlasGenerators( "A5", 8 );
    rec( charactername := "2a", constituents := [ 2 ], contents := "core",
      dim := 2,
      generators := [ [ [ Z(2)^0, 0*Z(2) ], [ Z(2^2), Z(2)^0 ] ],
          [ [ 0*Z(2), Z(2)^0 ], [ Z(2)^0, Z(2)^0 ] ] ], groupname := "A5",
      id := "a",
      identifier := [ "A5", [ "A5G1-f4r2aB0.m1", "A5G1-f4r2aB0.m2" ], 1,
          4 ], repname := "A5G1-f4r2aB0", repnr := 8, ring := GF(2^2),
      size := 60, standardization := 1, type := "matff" )
    gap> gens17:= AtlasGenerators( "A5", 17 );
    rec( charactername := "3a", constituents := [ 2 ], contents := "core",
      dim := 3,
      generators :=
        [ [ [ -1, 0, 0 ], [ 0, -1, 0 ], [ -E(5)-E(5)^4, -E(5)-E(5)^4, 1 ]
             ], [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] ],
      groupname := "A5", id := "a",
      identifier := [ "A5", "A5G1-Ar3aB0.g", 1, 3 ],
      polynomial := [ -1, 1, 1 ], repname := "A5G1-Ar3aB0", repnr := 17,
      ring := NF(5,[ 1, 4 ]), size := 60, standardization := 1,
      type := "matalg" )

Each of the above pairs of elements generates a group isomorphic to A_5.

    gap> gens1max2:= AtlasGenerators( "A5", 1, 2 );
    rec( charactername := "1a+4a", constituents := [ 1, 4 ],
      contents := "core", generators := [ (1,2)(3,4), (2,3)(4,5) ],
      groupname := "D10", id := "",
      identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5, 2 ],
      isPrimitive := true, maxnr := 1, p := 5, rankAction := 2,
      repname := "A5G1-p5B0", repnr := 1, size := 10, stabilizer := "A4",
      standardization := 1, transitivity := 3, type := "perm" )
    gap> id:= gens1max2.identifier;;
    gap> gens1max2 = AtlasGenerators( id );
    true
    gap> max2:= Group( gens1max2.generators );;
    gap> Size( max2 );
    10
    gap> IdGroup( max2 ) = IdGroup( DihedralGroup( 10 ) );
    true

The elements stored in `gens1max2.generators` describe the restriction of the first representation of A_5 to a group in the second class of maximal subgroups of A_5 according to the list in the ATLAS of Finite Groups [CCN+85]; this subgroup is isomorphic to the dihedral group D_10.

##### 3.5-4 AtlasProgram

`‣ AtlasProgram`( gapname[, std][, "contents", sources][, "version", vers], ... ) | ( function )

`‣ AtlasProgram`( identifier ) | ( function )

Returns: a record containing a program, or `fail`.

In the first form, gapname must be a string denoting a GAP name (see Section 3.2) of a group G, say. If the database contains a straight line program (see Section Reference: Straight Line Programs) or straight line decision (see Section 6.1) or black box program (see Section 6.2) as described by the arguments indicated by ... (see below) then `AtlasProgram` returns an immutable record containing this program. Otherwise `fail` is returned.

If the optional argument std is given, only those straight line programs/decisions are considered that take generators from the std-th set of standard generators of G as input, see Section 3.3.

If the optional arguments `"contents"` and sources are given then the latter must be either a string or a list of strings, with the same meaning as described for `DisplayAtlasInfo` (3.5-1).

If the optional arguments `"version"` and vers are given then the latter must be either a number or a list of numbers, and only those straight line programs/decisions are considered whose version number fits to vers.

The result record has at least the following components.

`groupname`

  the string gapname,

`identifier`

  a GAP object (a list of filenames plus additional information) that uniquely determines the program; the value can be used as identifier argument of `AtlasProgram` (see below),

`program`

  the required straight line program/decision, or black box program,

`standardization`

  the positive integer denoting the underlying standard generators of G,

`version`

  the substring of the filename of the program that denotes the version of the program.

If the program computes generators of the restriction to a maximal subgroup then also the following components are present.

`size`

  the order of the maximal subgroup,

`subgroupname`

  a string denoting a name of the maximal subgroup.

In the first form, the arguments indicated by ... must be as follows.

(the string `"maxes"` and) a positive integer maxnr

  the required program computes generators of the maxnr-th maximal subgroup of the group with GAP name gapname.

  In this case, the result record of `AtlasProgram` also may contain a component `size`, whose value is the order of the maximal subgroup in question.

the string `"maxes"` and two positive integers maxnr and std2

  the required program computes standard generators of the maxnr-th maximal subgroup of the group with GAP name gapname, w. r. t. the standardization std2.

  A prescribed `"version"` parameter refers to the straight line program for computing the restriction, not to the program for standardizing the result of the restriction.

  The meaning of the component `size` in the result, if present, is the same as in the previous case.

the string `"maxstd"` and three positive integers maxnr, vers, substd

  the required program computes standard generators of the maxnr-th maximal subgroup of the group with GAP name gapname w. r. t. standardization substd; in this case, the inputs of the program are not standard generators of the group with GAP name gapname but the outputs of the straight line program with version vers for computing generators of its maxnr-th maximal subgroup.

the string `"kernel"` and a string factname

  the required program computes generators of the kernel of an epimorphism from G to a group with GAP name factname.

one of the strings `"classes"` or `"cyclic"`

  the required program computes representatives of conjugacy classes of elements or representatives of generators of maximally cyclic subgroups of G, respectively.

  See [BSWW01] and [SWW00] for the background concerning these straight line programs. In these cases, the result record of `AtlasProgram` also contains a component `outputs`, whose value is a list of class names of the outputs, as described in Section 3.4.

the string `"cyc2ccl"` (and the string vers)

  the required program computes representatives of conjugacy classes of elements from representatives of generators of maximally cyclic subgroups of G. Thus the inputs are the outputs of the program of type `"cyclic"` whose version is vers.

the strings `"cyc2ccl"`, vers1, `"version"`, vers2

  the required program computes representatives of conjugacy classes of elements from representatives of generators of maximally cyclic subgroups of G, where the inputs are the outputs of the program of type `"cyclic"` whose version is vers1 and the required program itself has version vers2.

the strings `"automorphism"` and autname

  the required program computes images of standard generators under the outer automorphism of G that is given by this string.

  Note that a value `"2"` of autname means that the square of the automorphism is an inner automorphism of G (not necessarily the identity mapping) but the automorphism itself is not.

the string `"check"`

  the required result is a straight line decision that takes a list of generators for G and returns `true` if these generators are standard generators of G w. r. t. the standardization std, and `false` otherwise.

the string `"presentation"`

  the required result is a straight line decision that takes a list of group elements and returns `true` if these elements are standard generators of G w. r. t. the standardization std, and `false` otherwise.

  See `StraightLineProgramFromStraightLineDecision` (6.1-9) for an example how to derive defining relators for G in terms of the standard generators from such a straight line decision.

the string `"find"`

  the required result is a black box program that takes G and returns a list of standard generators of G, w. r. t. the standardization std.

the string `"restandardize"` and an integer std2

  the required result is a straight line program that computes standard generators of G w. r. t. the std2-th set of standard generators of G; in this case, the argument std must be given.

the strings `"other"` and descr

  the required program is described by descr.

The second form of `AtlasProgram`, with only argument the list identifier, can be used to fetch the result record with `identifier` value equal to identifier.

    gap> prog:= AtlasProgram( "A5", 2 );
    rec( groupname := "A5", identifier := [ "A5", "A5G1-max2W1", 1 ],
      program := <straight line program>, size := 10,
      standardization := 1, subgroupname := "D10", version := "1" )
    gap> StringOfResultOfStraightLineProgram( prog.program, [ "a", "b" ] );
    "[ a, bbab ]"
    gap> gens1:= AtlasGenerators( "A5", 1 );
    rec( charactername := "1a+4a", constituents := [ 1, 4 ],
      contents := "core", generators := [ (1,2)(3,4), (1,3,5) ],
      groupname := "A5", id := "",
      identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ],
      isPrimitive := true, maxnr := 1, p := 5, rankAction := 2,
      repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4",
      standardization := 1, transitivity := 3, type := "perm" )
    gap> maxgens:= ResultOfStraightLineProgram( prog.program,
    >                  gens1.generators );
    [ (1,2)(3,4), (2,3)(4,5) ]
    gap> maxgens = gens1max2.generators;
    true

The above example shows that for restricting representations given by standard generators to a maximal subgroup of A_5, we can also fetch and apply the appropriate straight line program. Such a program (see Reference: Straight Line Programs) takes standard generators of a group –in this example A_5– as its input, and returns a list of elements in this group –in this example generators of the D_10 subgroup we had met above– which are computed essentially by evaluating structured words in terms of the standard generators.

    gap> prog:= AtlasProgram( "J1", "cyclic" );
    rec( groupname := "J1", identifier := [ "J1", "J1G1-cycW1", 1 ],
      outputs := [ "6A", "7A", "10B", "11A", "15B", "19A" ],
      program := <straight line program>, standardization := 1,
      version := "1" )
    gap> gens:= GeneratorsOfGroup( FreeGroup( "x", "y" ) );;
    gap> ResultOfStraightLineProgram( prog.program, gens );
    [ (x*y)^2*((y*x)^2*y^2*x)^2*y^2, x*y, (x*(y*x*y)^2)^2*y,
      (x*y*x*(y*x*y)^3*x*y^2)^2*x*y*x*(y*x*y)^2*y, x*y*x*(y*x*y)^2*y,
      (x*y)^2*y ]

The above example shows how to fetch and use straight line programs for computing generators of representatives of maximally cyclic subgroups of a given group.

##### 3.5-5 AtlasProgramInfo

`‣ AtlasProgramInfo`( gapname[, std][, "contents", sources][, "version", vers], ... ) | ( function )

Returns: a record describing a program, or `fail`.

`AtlasProgramInfo` takes the same arguments as `AtlasProgram` (3.5-4), and returns a similar result. The only difference is that the records returned by `AtlasProgramInfo` have no components `program` and `outputs`. The idea is that one can use `AtlasProgramInfo` for testing whether the program in question is available at all, but without downloading files. The `identifier` component of the result of `AtlasProgramInfo` can then be used to fetch the program with `AtlasProgram` (3.5-4).

    gap> AtlasProgramInfo( "J1", "cyclic" );
    rec( groupname := "J1", identifier := [ "J1", "J1G1-cycW1", 1 ],
      standardization := 1, version := "1" )

##### 3.5-6 OneAtlasGeneratingSetInfo

`‣ OneAtlasGeneratingSetInfo`( [gapname][,] [std][,] [...] ) | ( function )

Returns: a record describing a representation that satisfies the conditions, or `fail`.

Let gapname be a string denoting a GAP name (see Section 3.2) of a group G, say. If the database contains at least one representation for G with the required properties then `OneAtlasGeneratingSetInfo` returns a record r whose components are the same as those of the records returned by `AtlasGenerators` (3.5-3), except that the component `generators` is not contained, and an additional component `givenRing` is present if `Ring` is one of the arguments in the function call.

The information in `givenRing` can be used later to construct the matrices over the prescribed ring. Note that this ring may be for example a domain constructed with `AlgebraicExtension` (Reference: AlgebraicExtension) instead of a field of cyclotomics or of a finite field constructed with `GF` (Reference: GF for field size).

The component `identifier` of r can be used as input for `AtlasGenerators` (3.5-3) in order to fetch the generators. If no representation satisfying the given conditions is available then `fail` is returned.

If the argument std is given then it must be a positive integer or a list of positive integers, denoting the sets of standard generators w. r. t. which the representation shall be given (see Section 3.3).

The argument gapname can be missing (then all available groups are considered), or a list of group names can be given instead.

Further restrictions can be entered as arguments, with the same meaning as described for `DisplayAtlasInfo` (3.5-1). The result of `OneAtlasGeneratingSetInfo` describes the first generating set for G that matches the restrictions, in the ordering shown by `DisplayAtlasInfo` (3.5-1).

Note that even in the case that the user preference `AtlasRepAccessRemoteFiles` has the value `true` (see Section 4.2-1), `OneAtlasGeneratingSetInfo` does not attempt to transfer remote data files, just the table of contents is evaluated. So this function (as well as `AllAtlasGeneratingSetInfos` (3.5-7)) can be used to check for the availability of certain representations, and afterwards one can call `AtlasGenerators` (3.5-3) for those representations one wants to work with.

In the following example, we try to access information about permutation representations for the alternating group A_5.

    gap> info:= OneAtlasGeneratingSetInfo( "A5" );
    rec( charactername := "1a+4a", constituents := [ 1, 4 ],
      contents := "core", groupname := "A5", id := "",
      identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ],
      isPrimitive := true, maxnr := 1, p := 5, rankAction := 2,
      repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4",
      standardization := 1, transitivity := 3, type := "perm" )
    gap> gens:= AtlasGenerators( info.identifier );
    rec( charactername := "1a+4a", constituents := [ 1, 4 ],
      contents := "core", generators := [ (1,2)(3,4), (1,3,5) ],
      groupname := "A5", id := "",
      identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ],
      isPrimitive := true, maxnr := 1, p := 5, rankAction := 2,
      repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4",
      standardization := 1, transitivity := 3, type := "perm" )
    gap> info = OneAtlasGeneratingSetInfo( "A5", IsPermGroup, true );
    true
    gap> info = OneAtlasGeneratingSetInfo( "A5", NrMovedPoints, "minimal" );
    true
    gap> info = OneAtlasGeneratingSetInfo( "A5", NrMovedPoints, [ 1 .. 10 ] );
    true
    gap> OneAtlasGeneratingSetInfo( "A5", NrMovedPoints, 20 );
    fail

Note that a permutation representation of degree 20 could be obtained by taking twice the primitive representation on 10 points; however, the database does not store this imprimitive representation (cf. Section 3.1).

We continue this example. Next we access matrix representations of A_5.

    gap> info:= OneAtlasGeneratingSetInfo( "A5", IsMatrixGroup, true );
    rec( charactername := "4a", constituents := [ 4 ], contents := "core",
      dim := 4, groupname := "A5", id := "a",
      identifier := [ "A5", [ "A5G1-f2r4aB0.m1", "A5G1-f2r4aB0.m2" ], 1,
          2 ], repname := "A5G1-f2r4aB0", repnr := 4, ring := GF(2),
      size := 60, standardization := 1, type := "matff" )
    gap> gens:= AtlasGenerators( info.identifier );
    rec( charactername := "4a", constituents := [ 4 ], contents := "core",
      dim := 4,
      generators := [ <an immutable 4x4 matrix over GF2>,
          <an immutable 4x4 matrix over GF2> ], groupname := "A5",
      id := "a",
      identifier := [ "A5", [ "A5G1-f2r4aB0.m1", "A5G1-f2r4aB0.m2" ], 1,
          2 ], repname := "A5G1-f2r4aB0", repnr := 4, ring := GF(2),
      size := 60, standardization := 1, type := "matff" )
    gap> info = OneAtlasGeneratingSetInfo( "A5", Dimension, 4 );
    true
    gap> info = OneAtlasGeneratingSetInfo( "A5", Characteristic, 2 );
    true
    gap> info2:= OneAtlasGeneratingSetInfo( "A5", Ring, GF(2) );;
    gap> info.identifier = info2.identifier;
    true
    gap> OneAtlasGeneratingSetInfo( "A5", Characteristic, [2,5], Dimension, 2 );
    rec( charactername := "2a", constituents := [ 2 ], contents := "core",
      dim := 2, groupname := "A5", id := "a",
      identifier := [ "A5", [ "A5G1-f4r2aB0.m1", "A5G1-f4r2aB0.m2" ], 1,
          4 ], repname := "A5G1-f4r2aB0", repnr := 8, ring := GF(2^2),
      size := 60, standardization := 1, type := "matff" )
    gap> OneAtlasGeneratingSetInfo( "A5", Characteristic, [2,5], Dimension, 1 );
    fail
    gap> info:= OneAtlasGeneratingSetInfo( "A5", Characteristic, 0,
    >                                            Dimension, 4 );
    rec( charactername := "4a", constituents := [ 4 ], contents := "core",
      dim := 4, groupname := "A5", id := "",
      identifier := [ "A5", "A5G1-Zr4B0.g", 1, 4 ],
      repname := "A5G1-Zr4B0", repnr := 14, ring := Integers, size := 60,
      standardization := 1, type := "matint" )
    gap> gens:= AtlasGenerators( info.identifier );
    rec( charactername := "4a", constituents := [ 4 ], contents := "core",
      dim := 4,
      generators :=
        [
          [ [ 1, 0, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 1, 0, 0 ],
              [ -1, -1, -1, -1 ] ],
          [ [ 0, 1, 0, 0 ], [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ],
              [ 1, 0, 0, 0 ] ] ], groupname := "A5", id := "",
      identifier := [ "A5", "A5G1-Zr4B0.g", 1, 4 ],
      repname := "A5G1-Zr4B0", repnr := 14, ring := Integers, size := 60,
      standardization := 1, type := "matint" )
    gap> info = OneAtlasGeneratingSetInfo( "A5", Ring, Integers );
    true
    gap> info2:= OneAtlasGeneratingSetInfo( "A5", Ring, CF(37) );;
    gap> info = info2;
    false
    gap> Difference( RecNames( info2 ), RecNames( info ) );
    [ "givenRing" ]
    gap> info2.givenRing;
    CF(37)
    gap> OneAtlasGeneratingSetInfo( "A5", Ring, Integers mod 77 );
    fail
    gap> info:= OneAtlasGeneratingSetInfo( "A5", Ring, CF(5), Dimension, 3 );
    rec( charactername := "3a", constituents := [ 2 ], contents := "core",
      dim := 3, givenRing := CF(5), groupname := "A5", id := "a",
      identifier := [ "A5", "A5G1-Ar3aB0.g", 1, 3 ],
      polynomial := [ -1, 1, 1 ], repname := "A5G1-Ar3aB0", repnr := 17,
      ring := NF(5,[ 1, 4 ]), size := 60, standardization := 1,
      type := "matalg" )
    gap> gens:= AtlasGenerators( info );
    rec( charactername := "3a", constituents := [ 2 ], contents := "core",
      dim := 3,
      generators :=
        [ [ [ -1, 0, 0 ], [ 0, -1, 0 ], [ -E(5)-E(5)^4, -E(5)-E(5)^4, 1 ]
             ], [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] ],
      givenRing := CF(5), groupname := "A5", id := "a",
      identifier := [ "A5", "A5G1-Ar3aB0.g", 1, 3 ],
      polynomial := [ -1, 1, 1 ], repname := "A5G1-Ar3aB0", repnr := 17,
      ring := NF(5,[ 1, 4 ]), size := 60, standardization := 1,
      type := "matalg" )
    gap> gens2:= AtlasGenerators( info.identifier );;
    gap> Difference( RecNames( gens ), RecNames( gens2 ) );
    [ "givenRing" ]
    gap> OneAtlasGeneratingSetInfo( "A5", Ring, GF(17) );
    fail

##### 3.5-7 AllAtlasGeneratingSetInfos

`‣ AllAtlasGeneratingSetInfos`( [gapname][,] [std][,] [...] ) | ( function )

Returns: the list of all records describing representations that satisfy the conditions.

`AllAtlasGeneratingSetInfos` is similar to `OneAtlasGeneratingSetInfo` (3.5-6). The difference is that the list of all records describing the available representations with the given properties is returned instead of just one such component. In particular an empty list is returned if no such representation is available.

    gap> AllAtlasGeneratingSetInfos( "A5", IsPermGroup, true );
    [ rec( charactername := "1a+4a", constituents := [ 1, 4 ],
          contents := "core", groupname := "A5", id := "",
          identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ]
            , isPrimitive := true, maxnr := 1, p := 5, rankAction := 2,
          repname := "A5G1-p5B0", repnr := 1, size := 60,
          stabilizer := "A4", standardization := 1, transitivity := 3,
          type := "perm" ),
      rec( charactername := "1a+5a", constituents := [ 1, 5 ],
          contents := "core", groupname := "A5", id := "",
          identifier := [ "A5", [ "A5G1-p6B0.m1", "A5G1-p6B0.m2" ], 1, 6 ]
            , isPrimitive := true, maxnr := 2, p := 6, rankAction := 2,
          repname := "A5G1-p6B0", repnr := 2, size := 60,
          stabilizer := "D10", standardization := 1, transitivity := 2,
          type := "perm" ),
      rec( charactername := "1a+4a+5a", constituents := [ 1, 4, 5 ],
          contents := "core", groupname := "A5", id := "",
          identifier := [ "A5", [ "A5G1-p10B0.m1", "A5G1-p10B0.m2" ], 1,
              10 ], isPrimitive := true, maxnr := 3, p := 10,
          rankAction := 3, repname := "A5G1-p10B0", repnr := 3,
          size := 60, stabilizer := "S3", standardization := 1,
          transitivity := 1, type := "perm" ) ]

Note that a matrix representation in any characteristic can be obtained by reducing a permutation representation or an integral matrix representation; however, the database does not store such a representation (cf. Section 3.1).

##### 3.5-8 AtlasGroup

`‣ AtlasGroup`( [gapname][,] [std][,] [...] ) | ( function )

`‣ AtlasGroup`( identifier ) | ( function )

Returns: a group that satisfies the conditions, or `fail`.

`AtlasGroup` takes the same arguments as `OneAtlasGeneratingSetInfo` (3.5-6), and returns the group generated by the `generators` component of the record that is returned by `OneAtlasGeneratingSetInfo` (3.5-6) with these arguments; if `OneAtlasGeneratingSetInfo` (3.5-6) returns `fail` then also `AtlasGroup` returns `fail`.

    gap> g:= AtlasGroup( "A5" );
    Group([ (1,2)(3,4), (1,3,5) ])

Alternatively, it is possible to enter exactly one argument, a record identifier as returned by `OneAtlasGeneratingSetInfo` (3.5-6) or `AllAtlasGeneratingSetInfos` (3.5-7), or the `identifier` component of such a record.

    gap> info:= OneAtlasGeneratingSetInfo( "A5" );
    rec( charactername := "1a+4a", constituents := [ 1, 4 ],
      contents := "core", groupname := "A5", id := "",
      identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ],
      isPrimitive := true, maxnr := 1, p := 5, rankAction := 2,
      repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4",
      standardization := 1, transitivity := 3, type := "perm" )
    gap> AtlasGroup( info );
    Group([ (1,2)(3,4), (1,3,5) ])
    gap> AtlasGroup( info.identifier );
    Group([ (1,2)(3,4), (1,3,5) ])

In the groups returned by `AtlasGroup`, the value of the attribute `AtlasRepInfoRecord` (3.5-10) is set. This information is used for example by `AtlasSubgroup` (3.5-9) when this function is called with second argument a group created by `AtlasGroup`.

##### 3.5-9 AtlasSubgroup

`‣ AtlasSubgroup`( gapname[, std][, ...], maxnr ) | ( function )

`‣ AtlasSubgroup`( identifier, maxnr ) | ( function )

`‣ AtlasSubgroup`( G, maxnr ) | ( function )

Returns: a group that satisfies the conditions, or `fail`.

The arguments of `AtlasSubgroup`, except the last argument maxnr, are the same as for `AtlasGroup` (3.5-8). If the database provides a straight line program for restricting representations of the group with name gapname (given w. r. t. the std-th standard generators) to the maxnr-th maximal subgroup and if a representation with the required properties is available, in the sense that calling `AtlasGroup` (3.5-8) with the same arguments except maxnr yields a group, then `AtlasSubgroup` returns the restriction of this representation to the maxnr-th maximal subgroup.

In all other cases, `fail` is returned.

Note that the conditions refer to the group and not to the subgroup. It may happen that in the restriction of a permutation representation to a subgroup, fewer points are moved, or that the restriction of a matrix representation turns out to be defined over a smaller ring. Here is an example.

    gap> g:= AtlasSubgroup( "A5", NrMovedPoints, 5, 1 );
    Group([ (1,5)(2,3), (1,3,5) ])
    gap> NrMovedPoints( g );
    4

Alternatively, it is possible to enter exactly two arguments, the first being a record identifier as returned by `OneAtlasGeneratingSetInfo` (3.5-6) or `AllAtlasGeneratingSetInfos` (3.5-7), or the `identifier` component of such a record, or a group G constructed with `AtlasGroup` (3.5-8).

    gap> info:= OneAtlasGeneratingSetInfo( "A5" );
    rec( charactername := "1a+4a", constituents := [ 1, 4 ],
      contents := "core", groupname := "A5", id := "",
      identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ],
      isPrimitive := true, maxnr := 1, p := 5, rankAction := 2,
      repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4",
      standardization := 1, transitivity := 3, type := "perm" )
    gap> AtlasSubgroup( info, 1 );
    Group([ (1,5)(2,3), (1,3,5) ])
    gap> AtlasSubgroup( info.identifier, 1 );
    Group([ (1,5)(2,3), (1,3,5) ])
    gap> AtlasSubgroup( AtlasGroup( "A5" ), 1 );
    Group([ (1,5)(2,3), (1,3,5) ])

##### 3.5-10 AtlasRepInfoRecord

`‣ AtlasRepInfoRecord`( G ) | ( attribute )

`‣ AtlasRepInfoRecord`( name ) | ( attribute )

Returns: the record stored in the group G when this was constructed with `AtlasGroup` (3.5-8), or a record with information about the group with name name.

For a group G that has been constructed with `AtlasGroup` (3.5-8), the value of this attribute is the info record that describes G, in the sense that this record was the first argument of the call to `AtlasGroup` (3.5-8), or it is the result of the call to `OneAtlasGeneratingSetInfo` (3.5-6) with the conditions that were listed in the call to `AtlasGroup` (3.5-8).

    gap> AtlasRepInfoRecord( AtlasGroup( "A5" ) );
    rec( charactername := "1a+4a", constituents := [ 1, 4 ],
      contents := "core", groupname := "A5", id := "",
      identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ],
      isPrimitive := true, maxnr := 1, p := 5, rankAction := 2,
      repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4",
      standardization := 1, transitivity := 3, type := "perm" )

For a string name that is a GAP name of a group G, say, `AtlasRepInfoRecord` returns a record that contains information about G which is used by `DisplayAtlasInfo` (3.5-1). The following components may be bound in the record.

`name`

  the string name,

`nrMaxes`

  the number of conjugacy classes of maximal subgroups of G,

`size`

  the order of G,

`sizesMaxes`

  a list which contains at position i, if bound, the order of a subgroup in the i-th class of maximal subgroups of G,

`slpMaxes`

  a list of length two; the first entry is a list of positions i such that a straight line program for computing the restriction of representations of G to a subgroup in the i-th class of maximal subgroups is available via AtlasRep; the second entry is the corresponding list of standardizations of the generators of G for which these straight line programs are available,

`structureMaxes`

  a list which contains at position i, if bound, a string that describes the structure of the subgroups in the i-th class of maximal subgroups of G.

    gap> AtlasRepInfoRecord( "A5" );
    rec( name := "A5", nrMaxes := 3, size := 60,
      sizesMaxes := [ 12, 10, 6 ],
      slpMaxes := [ [ 1 .. 3 ], [ [ 1 ], [ 1 ], [ 1 ] ] ],
      structureMaxes := [ "A4", "D10", "S3" ] )
    gap> AtlasRepInfoRecord( "J5" );
    rec(  )

##### 3.5-11 EvaluatePresentation

`‣ EvaluatePresentation`( G, gapname[, std] ) | ( operation )

`‣ EvaluatePresentation`( gens, gapname[, std] ) | ( operation )

Returns: a list of group elements or `fail`.

The first argument must be either a group G or a list gens of group generators, and gapname must be a string that is a GAP name (see Section 3.2) of a group H, say. The optional argument std, if given, must be a positive integer that denotes a standardization of generators of H, the default is 1.

`EvaluatePresentation` returns `fail` if no presentation for H w. r. t. the standardization std is stored in the database, and otherwise returns the list of results of evaluating the relators of a presentation for H at gens or the `GeneratorsOfGroup` (Reference: GeneratorsOfGroup) value of G, respectively. (An error is signalled if the number of generators is not equal to the number of inputs of the presentation.)

The result can be used as follows. Let N be the normal closure of the the result in G. The factor group G/N is an epimorphic image of H. In particular, if all entries of the result have order 1 then G itself is an epimorphic image of H. Moreover, an epimorphism is given by mapping the std-th standard generators of H to the N-cosets of the given generators of G.

    gap> g:= MathieuGroup( 12 );;
    gap> gens:= GeneratorsOfGroup( g );;  # switch to 2 generators
    gap> g:= Group( gens[1] * gens[3], gens[2] * gens[3] );;
    gap> EvaluatePresentation( g, "J0" );  # no pres. for group "J0"
    fail
    gap> relimgs:= EvaluatePresentation( g, "M11" );;
    gap> List( relimgs, Order );  # wrong group
    [ 3, 1, 5, 4, 10 ]
    gap> relimgs:= EvaluatePresentation( g, "M12" );;
    gap> List( relimgs, Order );  # generators are not standard
    [ 3, 4, 5, 4, 4 ]
    gap> g:= AtlasGroup( "M12" );;
    gap> relimgs:= EvaluatePresentation( g, "M12", 1 );;
    gap> List( relimgs, Order );  # right group, std. generators
    [ 1, 1, 1, 1, 1 ]
    gap> g:= AtlasGroup( "2.M12" );;
    gap> relimgs:= EvaluatePresentation( g, "M12", 1 );;
    gap> List( relimgs, Order );  # std. generators for extension
    [ 1, 2, 1, 1, 2 ]
    gap> Size( NormalClosure( g, SubgroupNC( g, relimgs ) ) );
    2

##### 3.5-12 StandardGeneratorsData

`‣ StandardGeneratorsData`( G, gapname[, std] ) | ( operation )

`‣ StandardGeneratorsData`( gens, gapname[, std] ) | ( operation )

Returns: a record that describes standard generators of the group in question, or `fail`, or the string `"timeout"`.

The first argument must be either a group G or a list gens of group generators, and gapname must be a string that is a GAP name (see Section 3.2) of a group H, say. The optional argument std, if given, must be a positive integer that denotes a standardization of generators of H, the default is 1.

If the global option `projective` is given then the group elements must be matrices over a finite field, and the group must be a central extension of the group H by a normal subgroup that consists of scalar matrices. In this case, all computations will be carried out modulo scalar matrices (in particular, element orders will be computed using `ProjectiveOrder` (Reference: ProjectiveOrder)), and the returned standard generators will belong to H.

`StandardGeneratorsData` returns

`fail`

  if no black box program for computing standard generators of H w. r. t. the standardization std is stored in the database, or if the black box program returns `fail` because a runtime error occurred or the program has proved that the given group or generators cannot generate a group isomorphic to H,

`"timeout"`

  if the black box program returns `"timeout"`, typically because some elements of a given order were not found among a reasonable number of random elements, or

a record containing standard generators

  otherwise.

When the result is not a record then either the group is not isomorphic to H (modulo scalars if applicable), or we were unlucky with choosing random elements.

When a record is returned and G or the group generated by gens, respectively, is isomorphic to H (or to a central extension of H by a group of scalar matrices if the global option `projective` is given) then the result describes the desired standard generators.

If G or the group generated by gens, respectively, is not isomorphic to H then it may still happen that `StandardGeneratorsData` returns a record. For a proof that the returned record describes the desired standard generators, one can use a presentation of H whose generators correspond to the std-th standard generators, see `EvaluatePresentation` (3.5-11).

A returned record has the following components.

`gapname`

  the string gapname,

`givengens`

  the list of group generators from which standard generators were computed, either gens or the `GeneratorsOfGroup` (Reference: GeneratorsOfGroup) value of G,

`stdgens`

  a list of standard generators of the group,

`givengenstostdgens`

  a straight line program that takes `givengens` as inputs, and returns `stdgens`,

`std`

  the underlying standardization std.

The first examples show three cases of failure, due to the unavailability of a suitable black box program or to a wrong choice of gapname. (In the search for standard generators of M_11 in the group M_12, one may or may not find an element whose order does not appear in M_11; in the first case, the result is `fail`, whereas a record is returned in the second case. Both cases occur.)

    gap> StandardGeneratorsData( MathieuGroup( 11 ), "J0" );
    fail
    gap> StandardGeneratorsData( MathieuGroup( 11 ), "M12" );
    "timeout"
    gap> repeat
    >      res:= StandardGeneratorsData( MathieuGroup( 12 ), "M11" );
    >    until res = fail;

The next example shows a computation of standard generators for the Mathieu group M_12. Using a presentation of M_12 w. r. t. these standard generators, we prove that the given group is isomorphic to M_12.

    gap> gens:= GeneratorsOfGroup( MathieuGroup( 12 ) );;
    gap> std:= 1;;
    gap> res:= StandardGeneratorsData( gens, "M12", std );;
    gap> Set( RecNames( res ) );
    [ "gapname", "givengens", "givengenstostdgens", "std", "stdgens" ]
    gap> gens = res.givengens;
    true
    gap> ResultOfStraightLineProgram( res.givengenstostdgens, gens )
    >    = res.stdgens;
    true
    gap> evl:= EvaluatePresentation( res.stdgens, "M12", std );;
    gap> ForAll( evl, IsOne );
    true

The next example shows the use of the global option `projective`. We take an irreducible matrix representation of the double cover of the Mathieu group M_12 (thus the center is represented by scalar matrices) and compute standard generators of the factor group M_12. Using a presentation of M_12 w. r. t. these standard generators, we prove that the given group is modulo scalars isomorphic to M_12, and we get generators for the kernel.

    gap> g:= AtlasGroup( "2.M12", IsMatrixGroup, Characteristic, IsPosInt );;
    gap> gens:= Permuted( GeneratorsOfGroup( g ), (1,2) );;
    gap> res:= StandardGeneratorsData( gens, "M12", std : projective );;
    gap> gens = res.givengens;
    true
    gap> ResultOfStraightLineProgram( res.givengenstostdgens, gens )
    >    = res.stdgens;
    true
    gap> evl:= EvaluatePresentation( res.stdgens, "M12", std );;
    gap> ForAll( evl, IsOne );
    false
    gap> ForAll( evl, x -> IsCentral( g, x ) );
    true

#### 3.6 Browse Applications Provided by AtlasRep

The functions `BrowseMinimalDegrees` (3.6-1), `BrowseBibliographySporadicSimple` (3.6-2), and `BrowseAtlasInfo` (Browse: BrowseAtlasInfo) (an alternative to `DisplayAtlasInfo` (3.5-1)) are available only if the GAP package Browse (see [BL18]) is loaded.

##### 3.6-1 BrowseMinimalDegrees

`‣ BrowseMinimalDegrees`( [gapnames] ) | ( function )

Returns: the list of info records for the clicked representations.

If the GAP package Browse (see [BL18]) is loaded then this function is available. It opens a browse table whose rows correspond to the groups for which AtlasRep knows some information about minimal degrees, whose columns correspond to the characteristics that occur, and whose entries are the known minimal degrees.

    gap> if IsBound( BrowseMinimalDegrees ) then
    >   down:= NCurses.keys.DOWN;;  DOWN:= NCurses.keys.NPAGE;;
    >   right:= NCurses.keys.RIGHT;;  END:= NCurses.keys.END;;
    >   enter:= NCurses.keys.ENTER;;  nop:= [ 14, 14, 14 ];;
    >   # just scroll in the table
    >   BrowseData.SetReplay( Concatenation( [ DOWN, DOWN, DOWN,
    >          right, right, right ], "sedddrrrddd", nop, nop, "Q" ) );
    >   BrowseMinimalDegrees();;
    >   # restrict the table to the groups with minimal ordinary degree 6
    >   BrowseData.SetReplay( Concatenation( "scf6",
    >        [ down, down, right, enter, enter ] , nop, nop, "Q" ) );
    >   BrowseMinimalDegrees();;
    >   BrowseData.SetReplay( false );
    > fi;

If an argument gapnames is given then it must be a list of GAP names of groups. The browse table is then restricted to the rows corresponding to these group names and to the columns that are relevant for these groups. A perhaps interesting example is the subtable with the data concerning sporadic simple groups and their covering groups, which has been published in [Jan05]. This table can be shown as follows.

    gap> if IsBound( BrowseMinimalDegrees ) then
    >   # just scroll in the table
    >   BrowseData.SetReplay( Concatenation( [ DOWN, DOWN, DOWN, END ],
    >          "rrrrrrrrrrrrrr", nop, nop, "Q" ) );
    >   BrowseMinimalDegrees( BibliographySporadicSimple.groupNamesJan05 );;
    > fi;

The browse table does not contain rows for the groups 6.M_22, 12.M_22, 6.Fi_22. Note that in spite of the title of [Jan05], the entries in Table 1 of this paper are in fact the minimal degrees of faithful irreducible representations, and in the above three cases, these degrees are larger than the minimal degrees of faithful representations. The underlying data of the browse table is about the minimal faithful (but not necessarily irreducible) degrees.

The return value of `BrowseMinimalDegrees` is the list of `OneAtlasGeneratingSetInfo` (3.5-6) values for those representations that have been "clicked" in visual mode.

The variant without arguments of this function is also available in the menu shown by `BrowseGapData` (Browse: BrowseGapData).

##### 3.6-2 BrowseBibliographySporadicSimple

`‣ BrowseBibliographySporadicSimple`( ) | ( function )

Returns: a record as returned by `ParseBibXMLExtString` (GAPDoc: ParseBibXMLextString).

If the GAP package Browse (see [BL18]) is loaded then this function is available. It opens a browse table whose rows correspond to the entries of the bibliographies in the ATLAS of Finite Groups [CCN+85] and in the ATLAS of Brauer Characters [JLPW95].

The function is based on `BrowseBibliography` (Browse: BrowseBibliography), see the documentation of this function for details, e.g., about the return value.

The returned record encodes the bibliography entries corresponding to those rows of the table that are "clicked" in visual mode, in the same format as the return value of `ParseBibXMLExtString` (GAPDoc: ParseBibXMLextString), see the manual of the GAP package GAPDoc [LN18] for details.

`BrowseBibliographySporadicSimple` can be called also via the menu shown by `BrowseGapData` (Browse: BrowseGapData).

    gap> if IsBound( BrowseBibliographySporadicSimple ) then
    >   enter:= NCurses.keys.ENTER;;  nop:= [ 14, 14, 14 ];;
    >   BrowseData.SetReplay( Concatenation(
    >     # choose the application
    >     "/Bibliography of Sporadic Simple Groups", [ enter, enter ],
    >     # search in the title column for the Atlas of Finite Groups
    >     "scr/Atlas of finite groups", [ enter,
    >     # and quit
    >     nop, nop, nop, nop ], "Q" ) );
    >   BrowseGapData();;
    >   BrowseData.SetReplay( false );
    > fi;

The bibliographies contained in the ATLAS of Finite Groups [CCN+85] and in the ATLAS of Brauer Characters [JLPW95] are available online in HTML format, see http://www.math.rwth-aachen.de/~Thomas.Breuer/atlasrep/bibl/index.html.

The source data in BibXMLext format, which are used by `BrowseBibliographySporadicSimple`, are distributed with the AtlasRep package, in four files with suffix `xml` in the package's `bibl` directory. Note that each of the two books contains two bibliographies.

Details about the BibXMLext format, including information how to transform the data into other formats such as BibTeX, can be found in the GAP package GAPDoc (see [LN18]).

[Top of Book] [Contents] [Previous Chapter] [Next Chapter]
Goto Chapter: Top 1 2 3 4 5 6 7 Bib Ind

generated by GAPDoc2HTML