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 from a remote server, see Section 4.3-1). The corresponding functions are described in this chapter. See Section 2.4 for some small examples how to use the functions of the interface.

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 are described in Chapter 7.

For some of the examples in this chapter, the **GAP** packages **CTblLib** and **TomLib** are needed, so we load them.

gap> LoadPackage( "ctbllib" ); true gap> LoadPackage( "tomlib" ); true

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`

(3.5-5) 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.

For a group G, say, whose character table is available in **GAP**'s Character Table Library, the admissible names of G are the admissible names of this character table. If G is almost simple, one such name is the `Identifier`

(Reference: Identifier (for character tables)) value of the character table, see `Accessing a Character Table from the Library`

(CTblLib: Accessing a Character Table from the Library). This name is usually very similar to the name used in the **ATLAS** of Finite Groups [CCNPW85]. 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 `"A5"`

is also a `"2E6(2)"`

and `"Fi24'"`

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

Several *different* standard generators may be defined for a group, the definitions can be found at

http://brauer.maths.qmul.ac.uk/Atlas

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.

For each straight line program (see `AtlasProgram`

(3.5-3)) 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`

(3.5-3) 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.

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 [CCNPW85].

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.

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. [CCNPW85, 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`

.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'`

.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`

(3.5-3) 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).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 [CCNPW85, 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.

`‣ 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" ]

`‣ 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" ]

Note that the output of the examples in this section refers to a perhaps outdated table of contents; the current version of the database may contain more information than is shown here.

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

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

This function lists the information available via the **AtlasRep** package, for the given input. Depending on whether remote access to data is enabled (see Section 4.3-1), all the data provided by the **ATLAS** of Group Representations or only those in the local installation are considered.

An interactive alternative to `DisplayAtlasInfo`

is the function `BrowseAtlasInfo`

(Browse: BrowseAtlasInfo), see [BL14].

Called without arguments, `DisplayAtlasInfo`

prints an overview what information the **ATLAS** of Group Representations provides. One line is printed for each group 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.

(The list can be printed to the screen or can be fed into a pager, see Section 4.3-5.)

Called with a list `listofnames` of strings that are **GAP** names for a group from the **ATLAS** of Group Representations, `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 for a group from the **ATLAS** of Group Representations, `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`

(3.5-2)), and a string of one of the following forms; in both cases, `id` is a (possibly empty) string.

`G <= Sym(`

`n``id`)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(`

`n``id`,`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/`

for an integer`m`ℤ`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`"public"`

then only non-private data (see Chapter 5) are considered, if`sources`is a string that denotes a private extension in the sense of a`dirid`argument of`AtlasOfGroupRepresentationsNotifyPrivateDirectory`

(5.1-1) then only the data that belong to this private 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

`identifier`

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`

restrict to 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`true`

or`false`

restrict to transitive or intransitive permutation representations (if this information is available),

`IsPrimitive`

and`true`

or`false`

restrict to primitive or imprimitive permutation representations (if this information is available),

`Transitivity`

and`n`for a nonnegative integer, a list of nonnegative integers, or a property

`n`, restrict to permutation representations of transitivity equal to`n`, or in the list`n`, or satisfying the function`n`(if this information is available),`RankAction`

and`n`for a nonnegative integer, a list of nonnegative integers, or a property

`n`, restrict to permutation representations of rank equal to`n`, or in the list`n`, or satisfying the function`n`(if this information is available),`IsMatrixGroup`

and`true`

restrict to 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 over this ring or satisfying this function (note that the representation might be defined over a proper subring of`R`),`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 matrix representations with these characters (note that the underlying characteristic of the class function, see Section Reference: UnderlyingCharacteristic, determines the characteristic of the matrices), and`IsStraightLineProgram`

and`true`

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

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.)

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

The above output means that the **ATLAS** of Group Representations contains 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( "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) 9: G <= GL(2b,4) 10: G <= GL(3,5) 12: G <= GL(3a,9) 13: G <= GL(3b,9) 17: G <= GL(3a,Field([Sqrt(5)])) 18: G <= GL(3b,Field([Sqrt(5)])) gap> DisplayAtlasInfo( "A5", Characteristic, 0 ); Representations for G = A5: (all refer to std. generators 1) --------------------------- 14: G <= GL(4,Z) 15: G <= GL(5,Z) 16: G <= GL(6,Z) 17: G <= GL(3a,Field([Sqrt(5)])) 18: G <= GL(3b,Field([Sqrt(5)]))

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) 8: G <= GL(2a,4) 12: G <= GL(3a,9) 17: G <= GL(3a,Field([Sqrt(5)]))

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

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`

.

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

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`

(3.5-3).

`‣ 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 the **ATLAS** of Group Representations contains at least `repnr` representations for the group with **GAP** name `gapname` 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.

`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; the value can be used as`identifier`argument of`AtlasGenerators`

.`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,

Additionally, the group order may be stored in the component `size`

, and describing components may be available that depend on the data type of the representation: For permutation representations, these are `p`

for the number of moved points, `id`

for the distinguishing string as described for `DisplayAtlasInfo`

(3.5-1), and information about primitivity, point stabilizers etc. if available; for matrix representations, these are `dim`

for the dimension of the matrices, `ring`

(if known) for the ring generated by the matrix entries, `id`

for the distinguishing string, and information about the character if available.

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, `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( 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( 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( 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 ], 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( generators := [ (1,2)(3,4), (2,3)(4,5) ], groupname := "D10", identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5, 2 ], repnr := 1, size := 10, standardization := 1 ) 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 [CCNPW85]; this subgroup is isomorphic to the dihedral group D_10.

`‣ AtlasProgram` ( gapname[, std], ... ) | ( 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 **ATLAS** of Group Representations 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 remaining arguments (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.

The result record has the following components.

`program`

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

`standardization`

the positive integer denoting the underlying standard generators of G,

`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).

In the first form, the last arguments 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.**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 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" ) gap> StringOfResultOfStraightLineProgram( prog.program, [ "a", "b" ] ); "[ a, bbab ]" gap> gens1:= AtlasGenerators( "A5", 1 ); rec( 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 ) 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.

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

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

.

`AtlasProgramInfo`

takes the same arguments as `AtlasProgram`

(3.5-3), 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 transferring it from a remote server. The `identifier`

component of the result of `AtlasProgramInfo`

can then be used to fetch the program with `AtlasProgram`

(3.5-3).

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

`‣ 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 **ATLAS** of Group Representations 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`

(3.5-2), except that the component `generators`

is not contained; the component `identifier`

of `r` can be used as input for `AtlasGenerators`

(3.5-2) 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 parameter "remote" has the value `true`

(see Section 4.3-1), `OneAtlasGeneratingSetInfo`

does *not* attempt to *transfer* remote data files, just the table of contents is evaluated. So this function (as well as `AllAtlasGeneratingSetInfos`

(3.5-6)) can be used to check for the availability of certain representations, and afterwards one can call `AtlasGenerators`

(3.5-2) 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( 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( 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 **ATLAS** of Group Representations does not store this imprimitive representation (cf. Section 3.1).

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

gap> info:= OneAtlasGeneratingSetInfo( "A5", IsMatrixGroup, true ); rec( 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( 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> info = OneAtlasGeneratingSetInfo( "A5", Ring, GF(2) ); true gap> OneAtlasGeneratingSetInfo( "A5", Characteristic, [2,5], Dimension, 2 ); rec( 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( 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( 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> info = OneAtlasGeneratingSetInfo( "A5", Ring, CF(37) ); true gap> OneAtlasGeneratingSetInfo( "A5", Ring, Integers mod 77 ); fail gap> info:= OneAtlasGeneratingSetInfo( "A5", Ring, CF(5), Dimension, 3 ); rec( dim := 3, groupname := "A5", id := "a", identifier := [ "A5", "A5G1-Ar3aB0.g", 1, 3 ], repname := "A5G1-Ar3aB0", repnr := 17, ring := NF(5,[ 1, 4 ]), size := 60, standardization := 1, type := "matalg" ) gap> gens:= AtlasGenerators( info.identifier ); rec( 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 ], repname := "A5G1-Ar3aB0", repnr := 17, ring := NF(5,[ 1, 4 ]), size := 60, standardization := 1, type := "matalg" ) gap> OneAtlasGeneratingSetInfo( "A5", Ring, GF(17) ); fail

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

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

`AllAtlasGeneratingSetInfos`

is similar to `OneAtlasGeneratingSetInfo`

(3.5-5). 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( 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( 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( 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 **ATLAS** of Group Representations does not *store* such a representation (cf. Section 3.1).

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

`‣ AtlasGroup` ( identifier ) | ( function ) |

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

.

`AtlasGroup`

takes the same arguments as `OneAtlasGeneratingSetInfo`

(3.5-5), and returns the group generated by the `generators`

component of the record that is returned by `OneAtlasGeneratingSetInfo`

(3.5-5) with these arguments; if `OneAtlasGeneratingSetInfo`

(3.5-5) 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`

(3.5-5) or `AllAtlasGeneratingSetInfos`

(3.5-6), or the `identifier`

component of such a record.

gap> info:= OneAtlasGeneratingSetInfo( "A5" ); rec( 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`

(3.5-9) is set. This information is used for example by `AtlasSubgroup`

(3.5-8) when this function is called with second argument a group created by `AtlasGroup`

.

`‣ 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 `maxn`, are the same as for `AtlasGroup`

(3.5-7). If the **ATLAS** of Group Representations 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`

(3.5-7) 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`

(3.5-5) or `AllAtlasGeneratingSetInfos`

(3.5-6), or the `identifier`

component of such a record, or a group `G` constructed with `AtlasGroup`

(3.5-7).

gap> info:= OneAtlasGeneratingSetInfo( "A5" ); rec( 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) ])

`‣ AtlasRepInfoRecord` ( G ) | ( attribute ) |

Returns: the record stored in the group `G` when this was constructed with `AtlasGroup`

(3.5-7).

For a group `G` that has been constructed with `AtlasGroup`

(3.5-7), 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`

(3.5-7), or it is the result of the call to `OneAtlasGeneratingSetInfo`

(3.5-5) with the conditions that were listed in the call to `AtlasGroup`

(3.5-7).

gap> AtlasRepInfoRecord( AtlasGroup( "A5" ) ); rec( 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" )

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 [BL14]) is loaded.

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

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

If the **GAP** package **Browse** (see [BL14]) is loaded then this function is available. It opens a browse table whose rows correspond to the groups for which the **ATLAS** of Group Representations contains 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 `groupnames` is given then it must be a list of group names of the **ATLAS** of Group Representations; 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`

(3.5-5) 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).

`‣ BrowseBibliographySporadicSimple` ( ) | ( function ) |

Returns: a record as returned by `ParseBibXMLExtString`

(GAPDoc: ParseBibXMLextString).

If the **GAP** package **Browse** (see [BL14]) 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 [CCNPW85] 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** [LN12] 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 [CCNPW85] 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`

, is part of 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 [LN12]).

generated by GAPDoc2HTML