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<a id="X8171B3798425E183" name="X8171B3798425E183"></a>
<div class="ChapSects"><a href="chap2.html#X8171B3798425E183">2 Tutorial for the AtlasRep Package</a>
<div class="ContSect"> <a href="chap2.html#X79CECBFE7A8EE2C2">2.1 Accessing a Specific Group in AtlasRep</a>

<div class="ContSSBlock">
 <a href="chap2.html#X87CD0FFB87D0BDD7">2.1-1 Accessing a Group in AtlasRep via its Name</a>

 <a href="chap2.html#X826C681B7EB3B67A">2.1-2 Accessing a Maximal Subgroup of a Group in AtlasRep</a>

</div></div>
<div class="ContSect"> <a href="chap2.html#X7F616D9685292471">2.2 Accessing Specific Generators in AtlasRep</a>

</div>
<div class="ContSect"> <a href="chap2.html#X7D3A29F879B140D3">2.3 Basic Concepts used in AtlasRep</a>

<div class="ContSSBlock">
 <a href="chap2.html#X7E2FB5E5852AD970">2.3-1 Groups, Generators, and Representations</a>

 <a href="chap2.html#X7DC99E4284093FBB">2.3-2 Straight Line Programs</a>

</div></div>
<div class="ContSect"> <a href="chap2.html#X87ACE06E82B68589">2.4 Examples of Using the AtlasRep Package</a>

<div class="ContSSBlock">
 <a href="chap2.html#X8563D96878AC685C">2.4-1 Example: Class Representatives</a>

 <a href="chap2.html#X81C9233778A3A817">2.4-2 Example: Permutation and Matrix Representations</a>

 <a href="chap2.html#X8284D7E87D38889C">2.4-3 Example: Outer Automorphisms</a>

 <a href="chap2.html#X794D669E7A507310">2.4-4 Example: Using Semi-presentations and Black Box Programs</a>

 <a href="chap2.html#X7CE7C2068017525C">2.4-5 Example: Using the GAP Library of Tables of Marks</a>

 <a href="chap2.html#X82550A9683E0DCA2">2.4-6 Example: Index 770 Subgroups in M_22</a>

 <a href="chap2.html#X84F9D163795B7DE1">2.4-7 Example: Index 462 Subgroups in M_22</a>

</div></div>
</div>

<h3>2 Tutorial for the AtlasRep Package</h3>

This chapter gives an overview of the basic functionality provided by the AtlasRep package. The main concepts and interface functions are presented in the first three sections, and Section <a href="chap2.html#X87ACE06E82B68589">2.4</a> shows a few small examples.

Let us first fix the setup for the examples shown in the package manual.

<ol>
<li>First of all, we load the AtlasRep package. Some of the examples require also the GAP packages CTblLib and TomLib, so we load also these packages.

<div class="example"><pre>
gap> LoadPackage( "AtlasRep", false );
true
gap> LoadPackage( "CTblLib", false );
true
gap> LoadPackage( "TomLib", false );
true
</pre></div>

</li>
<li>Depending on the terminal capabilities, the output of <code class="func">DisplayAtlasInfo</code> (<a href="chap3.html#X79DACFFA7E2D1A99">3.5-1</a>) may contain non-ASCII characters, which are not supported by the LaTeX and HTML versions of GAPDoc documents. The examples in this manual are used for tests of the package's functionality, thus we set the user preference DisplayFunction (see Section 4.2-11) to the value "Print" in order to produce output consisting only of ASCII characters, which is assumed to work in any terminal.

<div class="example"><pre>
gap> origpref:= UserPreference( "AtlasRep", "DisplayFunction" );;
gap> SetUserPreference( "AtlasRep", "DisplayFunction", "Print" );
</pre></div>

</li>
<li>The GAP output for the examples may look differently if data extensions have been loaded. In order to ignore these extensions in the examples, we unload them.

<div class="example"><pre>
gap> priv:= Difference(
> List( AtlasOfGroupRepresentationsInfo.notified, x -> x.ID ),
> [ "core", "internal" ] );;
gap> Perform( priv, AtlasOfGroupRepresentationsForgetData );
</pre></div>

</li>
<li>If the info level of <code class="func">InfoAtlasRep</code> (<a href="chap7.html#X8006BE167EB81E16">7.1-1</a>) is larger than zero then additional output appears on the screen. In order to avoid this output, we set the level to zero.

<div class="example"><pre>
gap> globallevel:= InfoLevel( InfoAtlasRep );;
gap> SetInfoLevel( InfoAtlasRep, 0 );
</pre></div>

</li>
</ol>
<a id="X79CECBFE7A8EE2C2" name="X79CECBFE7A8EE2C2"></a>

<h4>2.1 Accessing a Specific Group in AtlasRep</h4>

An important database to which the AtlasRep package gives access is the ATLAS of Group Representations <a href="chapBib.html#biBAGRv3">[WWT+]</a>. It contains generators and related data for several groups, mainly for extensions of simple groups (see Section <a href="chap2.html#X87CD0FFB87D0BDD7">2.1-1</a>) and for their maximal subgroups (see Section <a href="chap2.html#X826C681B7EB3B67A">2.1-2</a>).

In general, these data are not part of the package. They are downloaded as soon as they are needed for the first time, see Section <a href="chap4.html#X7C3293A98577EE68">4.2-1</a>.

<a id="X87CD0FFB87D0BDD7" name="X87CD0FFB87D0BDD7"></a>

<h5>2.1-1 Accessing a Group in AtlasRep via its Name</h5>

Each group that occurs in this database is specified by a name, which is a string similar to the name used in the ATLAS of Finite Groups <a href="chapBib.html#biBCCN85">[CCN+85]</a>. For those groups whose character tables are contained in the GAP Character Table Library <a href="chapBib.html#biBCTblLib">[Bre22]</a>, the names are equal to the <code class="func">Identifier</code> (<a href="../../../doc/ref/chap71.html#X79C40EE97890202F">Reference: Identifier for character tables</a>) values of these character tables. Examples of such names are <code class="code">"M24"</code> for the Mathieu group M_24, <code class="code">"2.A6"</code> for the double cover of the alternating group A_6, and <code class="code">"2.A6.2_1"</code> for the double cover of the symmetric group S_6. The names that actually occur are listed in the first column of the overview table that is printed by the function <code class="func">DisplayAtlasInfo</code> (<a href="chap3.html#X79DACFFA7E2D1A99">3.5-1</a>), called without arguments, see below. The other columns of the table describe the data that are available in the database.

For example, <code class="func">DisplayAtlasInfo</code> (<a href="chap3.html#X79DACFFA7E2D1A99">3.5-1</a>) may print the following lines. Omissions are indicated with <q><code class="code">...</code></q>.

<div class="example"><pre>
gap> DisplayAtlasInfo();
group | # | maxes | cl | cyc | out | fnd | chk | prs
-------------------------+----+-------+----+-----+-----+-----+-----+----
...
2.A5 | 26 | 3 | | | | | + | +
2.A5.2 | 11 | 4 | | | | | + | +
2.A6 | 18 | 5 | | | | | |
2.A6.2_1 | 3 | 6 | | | | | |
2.A7 | 24 | 2 | | | | | |
2.A7.2 | 7 | | | | | | |
...
M22 | 58 | 8 | + | + | | + | + | +
M22.2 | 46 | 7 | + | + | | + | + | +
M23 | 66 | 7 | + | + | | + | + | +
M24 | 62 | 9 | + | + | | + | + | +
McL | 46 | 12 | + | + | | + | + | +
McL.2 | 27 | 10 | | + | | + | + | +
O7(3) | 28 | | | | | | |
O7(3).2 | 3 | | | | | | |
...
Suz | 30 | 17 | | + | 2 | + | + |
...
</pre></div>

Called with a group name as the only argument, the function <code class="func">AtlasGroup</code> (<a href="chap3.html#X80AABEE783363B70">3.5-8</a>) returns a group isomorphic to the group with the given name, or <code class="keyw">fail</code>. If permutation generators are available in the database then a permutation group (of smallest available degree) is returned, otherwise a matrix group.

<div class="example"><pre>
gap> g:= AtlasGroup( "M24" );
Group([ (1,4)(2,7)(3,17)(5,13)(6,9)(8,15)(10,19)(11,18)(12,21)(14,16)
 (20,24)(22,23), (1,4,6)(2,21,14)(3,9,15)(5,18,10)(13,17,16)
 (19,24,23) ])
gap> IsPermGroup( g ); NrMovedPoints( g ); Size( g );
true
24
244823040
gap> AtlasGroup( "J5" );
fail
</pre></div>

<a id="X826C681B7EB3B67A" name="X826C681B7EB3B67A"></a>

<h5>2.1-2 Accessing a Maximal Subgroup of a Group in AtlasRep</h5>

Many maximal subgroups of extensions of simple groups can be constructed using the function <code class="func">AtlasSubgroup</code> (<a href="chap3.html#X7A3E460C82B3D9A3">3.5-9</a>). Given the name of the extension of the simple group and the number of the conjugacy class of maximal subgroups, this function returns a representative from this class.

<div class="example"><pre>
gap> g:= AtlasSubgroup( "M24", 1 );
Group([ (2,10)(3,12)(4,14)(6,9)(8,16)(15,18)(20,22)(21,24), (1,7,2,9)
 (3,22,10,23)(4,19,8,12)(5,14)(6,18)(13,16,17,24) ])
gap> IsPermGroup( g ); NrMovedPoints( g ); Size( g );
true
23
10200960
gap> AtlasSubgroup( "M24", 100 );
fail
</pre></div>

The classes of maximal subgroups are ordered w. r. t. decreasing subgroup order. So the first class contains maximal subgroups of smallest index.

Note that groups obtained by <code class="func">AtlasSubgroup</code> (<a href="chap3.html#X7A3E460C82B3D9A3">3.5-9</a>) may be not very suitable for computations in the sense that much nicer representations exist. For example, the sporadic simple O'Nan group O'N contains a maximal subgroup S isomorphic with the Janko group J_1; the smallest permutation representation of O'N has degree 122760, and restricting this representation to S yields a representation of J_1 of that degree. However, J_1 has a faithful permutation representation of degree 266, which admits much more efficient computations. If you are just interested in J_1 and not in its embedding into O'N then one possibility to get a <q>nicer</q> faithful representation is to call <code class="func">SmallerDegreePermutationRepresentation</code> (<a href="../../../doc/ref/chap43.html#X8086628878AFD3EA">Reference: SmallerDegreePermutationRepresentation</a>). In the abovementioned example, this works quite well; note that in general, we cannot expect that we get a representation of smallest degree in this way.

<div class="example"><pre>
gap> s:= AtlasSubgroup( "ON", 3 );
<permutation group of size 175560 with 2 generators>
gap> NrMovedPoints( s ); Size( s );
122760
175560
gap> hom:= SmallerDegreePermutationRepresentation( s );;
gap> NrMovedPoints( Image( hom ) ) < 2000;
true
</pre></div>

(Depending on random choices in the computations, one may or my not get the degree 266 representation.)

In this particular case, one could of course also ask directly for the group J_1.

<div class="example"><pre>
gap> j1:= AtlasGroup( "J1" );
<permutation group of size 175560 with 2 generators>
gap> NrMovedPoints( j1 );
266
</pre></div>

If you have a group G, say, and you are really interested in the embedding of a maximal subgroup of G into G then an easy way to get compatible generators is to create G with <code class="func">AtlasGroup</code> (<a href="chap3.html#X80AABEE783363B70">3.5-8</a>) and then to call <code class="func">AtlasSubgroup</code> (<a href="chap3.html#X7A3E460C82B3D9A3">3.5-9</a>) with first argument the group G.

<div class="example"><pre>
gap> g:= AtlasGroup( "ON" );
<permutation group of size 460815505920 with 2 generators>
gap> s:= AtlasSubgroup( g, 3 );
<permutation group of size 175560 with 2 generators>
gap> IsSubset( g, s );
true
gap> IsSubset( g, j1 );
false
</pre></div>

<a id="X7F616D9685292471" name="X7F616D9685292471"></a>

<h4>2.2 Accessing Specific Generators in AtlasRep</h4>

The function <code class="func">DisplayAtlasInfo</code> (<a href="chap3.html#X79DACFFA7E2D1A99">3.5-1</a>), called with an admissible name of a group as the only argument, lists the ATLAS data available for this group.

<div class="example"><pre>
gap> DisplayAtlasInfo( "A5" );
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.)
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
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
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

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)
</pre></div>

In order to fetch one of the listed permutation groups or matrix groups, you can call <code class="func">AtlasGroup</code> (<a href="chap3.html#X80AABEE783363B70">3.5-8</a>) with second argument the function <code class="func">Position</code> (<a href="../../../doc/ref/chap21.html#X79975EC6783B4293">Reference: Position</a>) and third argument the position in the list.

<div class="example"><pre>
gap> AtlasGroup( "A5", Position, 1 );
Group([ (1,2)(3,4), (1,3,5) ])
</pre></div>

Note that this approach may yield a different group after a data extension has been loaded.

Alternatively, you can describe the desired group by conditions, such as the degree in the case of a permutation group, and the dimension and the base ring in the case of a matrix group.

<div class="example"><pre>
gap> AtlasGroup( "A5", NrMovedPoints, 10 );
Group([ (2,4)(3,5)(6,8)(7,10), (1,2,3)(4,6,7)(5,8,9) ])
gap> AtlasGroup( "A5", Dimension, 4, Ring, GF(2) );
<matrix group of size 60 with 2 generators>
</pre></div>

The same holds for the restriction to maximal subgroups: Use <code class="func">AtlasSubgroup</code> (<a href="chap3.html#X7A3E460C82B3D9A3">3.5-9</a>) with the same arguments as <code class="func">AtlasGroup</code> (<a href="chap3.html#X80AABEE783363B70">3.5-8</a>), except that additionally the number of the class of maximal subgroups is entered as the last argument. Note that the conditions refer to the group, not to the subgroup; it may happen that the subgroup moves fewer points than the big group.

<div class="example"><pre>
gap> AtlasSubgroup( "A5", Dimension, 4, Ring, GF(2), 1 );
<matrix group of size 12 with 2 generators>
gap> g:= AtlasSubgroup( "A5", NrMovedPoints, 10, 3 );
Group([ (2,4)(3,5)(6,8)(7,10), (1,4)(3,8)(5,7)(6,10) ])
gap> Size( g ); NrMovedPoints( g );
6
9
</pre></div>

<a id="X7D3A29F879B140D3" name="X7D3A29F879B140D3"></a>

<h4>2.3 Basic Concepts used in AtlasRep</h4>

<a id="X7E2FB5E5852AD970" name="X7E2FB5E5852AD970"></a>

<h5>2.3-1 Groups, Generators, and Representations</h5>

Up to now, we have talked only about groups and subgroups. The AtlasRep package provides access to group generators, and in fact these generators have the property that mapping one set of generators to another set of generators for the same group defines an isomorphism. These generators are called standard generators, see Section <a href="chap3.html#X795DB7E486E0817D">3.3</a>.

So instead of thinking about several generating sets of a group G, say, we can think about one abstract group G, with one fixed set of generators, and mapping these generators to any set of generators provided by AtlasRep defines a representation of G. This viewpoint had motivated the name <q>ATLAS of Group Representations</q> for the core part of the database.

If you are interested in the generators provided by the database rather than in the groups they generate, you can use the function <code class="func">OneAtlasGeneratingSetInfo</code> (<a href="chap3.html#X841478AB7CD06D44">3.5-6</a>) instead of <code class="func">AtlasGroup</code> (<a href="chap3.html#X80AABEE783363B70">3.5-8</a>), with the same arguments. This will yield a record that describes the representation in question. Calling the function <code class="func">AtlasGenerators</code> (<a href="chap3.html#X7D1CCCF8852DFF39">3.5-3</a>) with this record will then yield a record with the additional component <code class="code">generators</code>, which holds the list of generators.

<div class="example"><pre>
gap> info:= OneAtlasGeneratingSetInfo( "A5", NrMovedPoints, 10 );
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" )
gap> info2:= AtlasGenerators( info );
rec( charactername := "1a+4a+5a", constituents := [ 1, 4, 5 ],
 contents := "core",
 generators := [ (2,4)(3,5)(6,8)(7,10), (1,2,3)(4,6,7)(5,8,9) ],
 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" )
gap> info2.generators;
[ (2,4)(3,5)(6,8)(7,10), (1,2,3)(4,6,7)(5,8,9) ]
</pre></div>

The record <code class="code">info</code> appears as the value of the attribute <code class="func">AtlasRepInfoRecord</code> (<a href="chap3.html#X87BC7D9C7BA2F27A">3.5-10</a>) in groups that are returned by <code class="func">AtlasGroup</code> (<a href="chap3.html#X80AABEE783363B70">3.5-8</a>).

<div class="example"><pre>
gap> g:= AtlasGroup( "A5", NrMovedPoints, 10 );;
gap> AtlasRepInfoRecord( g );
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" )
</pre></div>

<a id="X7DC99E4284093FBB" name="X7DC99E4284093FBB"></a>

<h5>2.3-2 Straight Line Programs</h5>

For computing certain group elements from standard generators, such as generators of a subgroup or class representatives, AtlasRep uses straight line programs, see <a href="../../../doc/ref/chap37.html#X7DC99E4284093FBB">Reference: Straight Line Programs</a>. Essentially this means to evaluate words in the generators, which is similar to <code class="func">MappedWord</code> (<a href="../../../doc/ref/chap36.html#X7EC17930781D104A">Reference: MappedWord</a>) but can be more efficient.

It can be useful to deal with these straight line programs, see <code class="func">AtlasProgram</code> (<a href="chap3.html#X801F2E657C8A79ED">3.5-4</a>). For example, an automorphism α, say, of the group G, if available in AtlasRep, is given by a straight line program that defines the images of standard generators of G. This way, one can for example compute the image of a subgroup U of G under α by first applying the straight line program for α to standard generators of G, and then applying the straight line program for the restriction from G to U.

<div class="example"><pre>
gap> prginfo:= AtlasProgramInfo( "A5", "maxes", 1 );
rec( groupname := "A5", identifier := [ "A5", "A5G1-max1W1", 1 ],
 size := 12, standardization := 1, subgroupname := "A4",
 version := "1" )
gap> prg:= AtlasProgram( prginfo.identifier );
rec( groupname := "A5", identifier := [ "A5", "A5G1-max1W1", 1 ],
 program := <straight line program>, size := 12,
 standardization := 1, subgroupname := "A4", version := "1" )
gap> Display( prg.program );
# input:
r:= [ g1, g2 ];
# program:
r[3]:= r[1]*r[2];
r[4]:= r[2]*r[1];
r[5]:= r[3]*r[3];
r[1]:= r[5]*r[4];
# return values:
[ r[1], r[2] ]
gap> ResultOfStraightLineProgram( prg.program, info2.generators );
[ (1,10)(2,3)(4,9)(7,8), (1,2,3)(4,6,7)(5,8,9) ]
</pre></div>

<a id="X87ACE06E82B68589" name="X87ACE06E82B68589"></a>

<h4>2.4 Examples of Using the AtlasRep Package</h4>

<a id="X8563D96878AC685C" name="X8563D96878AC685C"></a>

<h5>2.4-1 Example: Class Representatives</h5>

First we show the computation of class representatives of the Mathieu group M_11, in a 2-modular matrix representation. We start with the ordinary and Brauer character tables of this group.

<div class="example"><pre>
gap> tbl:= CharacterTable( "M11" );;
gap> modtbl:= tbl mod 2;;
gap> CharacterDegrees( modtbl );
[ [ 1, 1 ], [ 10, 1 ], [ 16, 2 ], [ 44, 1 ] ]
</pre></div>

The output of <code class="func">CharacterDegrees</code> (<a href="../../../doc/ref/chap71.html#X81FEFF768134481A">Reference: CharacterDegrees</a>) means that the 2-modular irreducibles of M_11 have degrees 1, 10, 16, 16, and 44.

Using <code class="func">DisplayAtlasInfo</code> (<a href="chap3.html#X79DACFFA7E2D1A99">3.5-1</a>), we find out that matrix generators for the irreducible 10-dimensional representation are available in the database.

<div class="example"><pre>
gap> DisplayAtlasInfo( "M11", Characteristic, 2 );
Representations for G = M11: (all refer to std. generators 1)
----------------------------
6: G <= GL(10,2) character 10a
7: G <= GL(32,2) character 16ab
8: G <= GL(44,2) character 44a
16: G <= GL(16a,4) character 16a
17: G <= GL(16b,4) character 16b
</pre></div>

So we decide to work with this representation. We fetch the generators and compute the list of class representatives of M_11 in the representation. The ordering of class representatives is the same as that in the character table of the ATLAS of Finite Groups (<a href="chapBib.html#biBCCN85">[CCN+85]</a>), which coincides with the ordering of columns in the GAP table we have fetched above.

<div class="example"><pre>
gap> info:= OneAtlasGeneratingSetInfo( "M11", Characteristic, 2,
> Dimension, 10 );;
gap> gens:= AtlasGenerators( info.identifier );;
gap> ccls:= AtlasProgram( "M11", gens.standardization, "classes" );
rec( groupname := "M11", identifier := [ "M11", "M11G1-cclsW1", 1 ],
 outputs := [ "1A", "2A", "3A", "4A", "5A", "6A", "8A", "8B", "11A",
 "11B" ], program := <straight line program>,
 standardization := 1, version := "1" )
gap> reps:= ResultOfStraightLineProgram( ccls.program, gens.generators );;
</pre></div>

If we would need only a few class representatives, we could use the GAP library function <code class="func">RestrictOutputsOfSLP</code> (<a href="../../../doc/ref/chap37.html#X7C9CABD17BE4850F">Reference: RestrictOutputsOfSLP</a>) to create a straight line program that computes only specified outputs. Here is an example where only the class representatives of order eight are computed.

<div class="example"><pre>
gap> ord8prg:= RestrictOutputsOfSLP( ccls.program,
> Filtered( [ 1 .. 10 ], i -> ccls.outputs[i][1] = '8' ) );
<straight line program>
gap> ord8reps:= ResultOfStraightLineProgram( ord8prg, gens.generators );;
gap> List( ord8reps, m -> Position( reps, m ) );
[ 7, 8 ]
</pre></div>

Let us check that the class representatives have the right orders.

<div class="example"><pre>
gap> List( reps, Order ) = OrdersClassRepresentatives( tbl );
true
</pre></div>

From the class representatives, we can compute the Brauer character we had started with. This Brauer character is defined on all classes of the 2-modular table. So we first pick only those representatives, using the GAP function <code class="func">GetFusionMap</code> (<a href="../../../doc/ref/chap73.html#X8464DD23879431D9">Reference: GetFusionMap</a>); in this situation, it returns the class fusion from the Brauer table into the ordinary table.

<div class="example"><pre>
gap> fus:= GetFusionMap( modtbl, tbl );
[ 1, 3, 5, 9, 10 ]
gap> modreps:= reps{ fus };;
</pre></div>

Then we call the GAP function <code class="func">BrauerCharacterValue</code> (<a href="../../../doc/ref/chap72.html#X8304B68E84511685">Reference: BrauerCharacterValue</a>), which computes the Brauer character value from the matrix given.

<div class="example"><pre>
gap> char:= List( modreps, BrauerCharacterValue );
[ 10, 1, 0, -1, -1 ]
gap> Position( Irr( modtbl ), char );
2
</pre></div>

<a id="X81C9233778A3A817" name="X81C9233778A3A817"></a>

<h5>2.4-2 Example: Permutation and Matrix Representations</h5>

The second example shows the computation of a permutation representation from a matrix representation. We work with the 10-dimensional representation used above, and consider the action on the 2^10 vectors of the underlying row space.

<div class="example"><pre>
gap> grp:= Group( gens.generators );;
gap> v:= GF(2)^10;;
gap> orbs:= Orbits( grp, AsList( v ) );;
gap> List( orbs, Length );
[ 1, 396, 55, 330, 66, 165, 11 ]
</pre></div>

We see that there are six nontrivial orbits, and we can compute the permutation actions on these orbits directly using <code class="func">Action</code> (<a href="../../../doc/ref/chap40.html#X86C2BE2481FDC8EE">Reference: Action homomorphisms</a>). However, for larger examples, one cannot write down all orbits on the row space, so one has to use another strategy if one is interested in a particular orbit.

Let us assume that we are interested in the orbit of length 11. The point stabilizer is the first maximal subgroup of M_11, thus the restriction of the representation to this subgroup has a nontrivial fixed point space. This restriction can be computed using the AtlasRep package.

<div class="example"><pre>
gap> gens:= AtlasGenerators( "M11", 6, 1 );;
</pre></div>

Now computing the fixed point space is standard linear algebra.

<div class="example"><pre>
gap> id:= IdentityMat( 10, GF(2) );;
gap> sub1:= Subspace( v, NullspaceMat( gens.generators[1] - id ) );;
gap> sub2:= Subspace( v, NullspaceMat( gens.generators[2] - id ) );;
gap> fix:= Intersection( sub1, sub2 );
<vector space of dimension 1 over GF(2)>
</pre></div>

The final step is of course the computation of the permutation action on the orbit.

<div class="example"><pre>
gap> orb:= Orbit( grp, Basis( fix )[1] );;
gap> act:= Action( grp, orb );; Print( act, "\n" );
Group( [ ( 1, 2)( 4, 6)( 5, 8)( 7,10), ( 1, 3, 5, 9)( 2, 4, 7,11) ] )
</pre></div>

Note that this group is not equal to the group obtained by fetching the permutation representation from the database. This is due to a different numbering of the points, thus the groups are permutation isomorphic, that is, they are conjugate in the symmetric group on eleven points.

<div class="example"><pre>
gap> permgrp:= Group( AtlasGenerators( "M11", 1 ).generators );;
gap> Print( permgrp, "\n" );
Group( [ ( 2,10)( 4,11)( 5, 7)( 8, 9), (1,4,3,8)(2,5,6,9) ] )
gap> permgrp = act;
false
gap> IsConjugate( SymmetricGroup(11), permgrp, act );
true
</pre></div>

<a id="X8284D7E87D38889C" name="X8284D7E87D38889C"></a>

<h5>2.4-3 Example: Outer Automorphisms</h5>

The straight line programs for applying outer automorphisms to standard generators can of course be used to define the automorphisms themselves as GAP mappings.

<div class="example"><pre>
gap> DisplayAtlasInfo( "G2(3)", IsStraightLineProgram );
Programs for G = G2(3): (all refer to std. generators 1)
-----------------------
- class repres.
- presentation
- repr. cyc. subg.
- std. gen. checker
- automorphisms:
 2
- maxes (all 10):
 1: U3(3).2
 2: U3(3).2
 3: (3^(1+2)+x3^2):2S4
 4: (3^(1+2)+x3^2):2S4
 5: L3(3).2
 6: L3(3).2
 7: L2(8).3
 8: 2^3.L3(2)
 9: L2(13)
 10: 2^(1+4)+:3^2.2
gap> prog:= AtlasProgram( "G2(3)", "automorphism", "2" ).program;;
gap> info:= OneAtlasGeneratingSetInfo( "G2(3)", Dimension, 7 );;
gap> gens:= AtlasGenerators( info ).generators;;
gap> imgs:= ResultOfStraightLineProgram( prog, gens );;
</pre></div>

If we are not suspicious whether the script really describes an automorphism then we should tell this to GAP, in order to avoid the expensive checks of the properties of being a homomorphism and bijective (see Section <a href="../../../doc/ref/chap40.html#X81A7BB0F7D81B247">Reference: Creating Group Homomorphisms</a>). This looks as follows.

<div class="example"><pre>
gap> g:= Group( gens );;
gap> aut:= GroupHomomorphismByImagesNC( g, g, gens, imgs );;
gap> SetIsBijective( aut, true );
</pre></div>

If we are suspicious whether the script describes an automorphism then we might have the idea to check it with GAP, as follows.

<div class="example"><pre>
gap> aut:= GroupHomomorphismByImages( g, g, gens, imgs );;
gap> IsBijective( aut );
true
</pre></div>

(Note that even for a comparatively small group such as G_2(3), this was a difficult task for GAP before version 4.3.)

Often one can form images under an automorphism α, say, without creating the homomorphism object. This is obvious for the standard generators of the group G themselves, but also for generators of a maximal subgroup M computed from standard generators of G, provided that the straight line programs in question refer to the same standard generators. Note that the generators of M are given by evaluating words in terms of standard generators of G, and their images under α can be obtained by evaluating the same words at the images under α of the standard generators of G.

<div class="example"><pre>
gap> max1:= AtlasProgram( "G2(3)", 1 ).program;;
gap> mgens:= ResultOfStraightLineProgram( max1, gens );;
gap> comp:= CompositionOfStraightLinePrograms( max1, prog );;
gap> mimgs:= ResultOfStraightLineProgram( comp, gens );;
</pre></div>

The list <code class="code">mgens</code> is the list of generators of the first maximal subgroup of G_2(3), <code class="code">mimgs</code> is the list of images under the automorphism given by the straight line program <code class="code">prog</code>. Note that applying the program returned by <code class="func">CompositionOfStraightLinePrograms</code> (<a href="../../../doc/ref/chap37.html#X8274C7948248C053">Reference: CompositionOfStraightLinePrograms</a>) means to apply first <code class="code">prog</code> and then <code class="code">max1</code>. Since we have already constructed the GAP object representing the automorphism, we can check whether the results are equal.

<div class="example"><pre>
gap> mimgs = List( mgens, x -> x^aut );
true
</pre></div>

However, it should be emphasized that using <code class="code">aut</code> requires a huge machinery of computations behind the scenes, whereas applying the straight line programs <code class="code">prog</code> and <code class="code">max1</code> involves only elementary operations with the generators. The latter is feasible also for larger groups, for which constructing the GAP automorphism might be too hard.

<a id="X794D669E7A507310" name="X794D669E7A507310"></a>

<h5>2.4-4 Example: Using Semi-presentations and Black Box Programs</h5>

Let us suppose that we want to restrict a representation of the Mathieu group M_12 to a non-maximal subgroup of the type L_2(11). The idea is that this subgroup can be found as a maximal subgroup of a maximal subgroup of the type M_11, which is itself maximal in M_12. For that, we fetch a representation of M_12 and use a straight line program for restricting it to the first maximal subgroup, which has the type M_11.

<div class="example"><pre>
gap> info:= OneAtlasGeneratingSetInfo( "M12", NrMovedPoints, 12 );
rec( charactername := "1a+11a", constituents := [ 1, 2 ],
 contents := "core", groupname := "M12", id := "a",
 identifier := [ "M12", [ "M12G1-p12aB0.m1", "M12G1-p12aB0.m2" ], 1,
 12 ], isPrimitive := true, maxnr := 1, p := 12, rankAction := 2,
 repname := "M12G1-p12aB0", repnr := 1, size := 95040,
 stabilizer := "M11", standardization := 1, transitivity := 5,
 type := "perm" )
gap> gensM12:= AtlasGenerators( info.identifier );;
gap> restM11:= AtlasProgram( "M12", "maxes", 1 );;
gap> gensM11:= ResultOfStraightLineProgram( restM11.program,
> gensM12.generators );
[ (3,9)(4,12)(5,10)(6,8), (1,4,11,5)(2,10,8,3) ]
</pre></div>

Now we cannot simply apply a straight line program for a group to some generators, since they are not necessarily standard generators of the group. We check this property using a semi-presentation for M_11, see <a href="chap6.html#X7C94ECAC8583CEAE">6.1-7</a>.

<div class="example"><pre>
gap> checkM11:= AtlasProgram( "M11", "check" );
rec( groupname := "M11", identifier := [ "M11", "M11G1-check1", 1, 1 ]
 , program := <straight line decision>, standardization := 1,
 version := "1" )
gap> ResultOfStraightLineDecision( checkM11.program, gensM11 );
true
</pre></div>

So we are lucky that applying the appropriate program for M_11 will give us the required generators for L_2(11).

<div class="example"><pre>
gap> restL211:= AtlasProgram( "M11", "maxes", 2 );;
gap> gensL211:= ResultOfStraightLineProgram( restL211.program, gensM11 );
[ (3,9)(4,12)(5,10)(6,8), (1,11,9)(2,12,8)(3,6,10) ]
gap> G:= Group( gensL211 );; Size( G ); IsSimple( G );
660
true
</pre></div>

In this case, we could also use the information that is stored about M_11, as follows.

<div class="example"><pre>
gap> DisplayAtlasInfo( "M11", IsStraightLineProgram );
Programs for G = M11: (all refer to std. generators 1)
---------------------
- presentation
- repr. cyc. subg.
- std. gen. finder
- class repres.:
 (direct)
 (composed)
- maxes (all 5):
 1: A6.2_3
 1: A6.2_3 (std. 1)
 2: L2(11)
 2: L2(11) (std. 1)
 3: 3^2:Q8.2
 4: S5
 4: S5 (std. 1)
 5: 2.S4
- standardizations of maxes:
 from 1st max., version 1 to A6.2_3, std. 1
 from 2nd max., version 1 to L2(11), std. 1
 from 4th max., version 1 to A5.2, std. 1
- std. gen. checker:
 (check)
 (pres)
</pre></div>

The entry <q>std.1</q> in the line about the maximal subgroup of type L_2(11) means that a straight line program for computing standard generators (in standardization 1) of the subgroup. This program can be fetched as follows.

<div class="example"><pre>
gap> restL211std:= AtlasProgram( "M11", "maxes", 2, 1 );;
gap> ResultOfStraightLineProgram( restL211std.program, gensM11 );
[ (3,9)(4,12)(5,10)(6,8), (1,11,9)(2,12,8)(3,6,10) ]
</pre></div>

We see that we get the same generators for the subgroup as above. (In fact the second approach first applies the same program as is given by <code class="code">restL211.program</code>, and then applies a program to the results that does nothing.)

Usually representations are not given in terms of standard generators. For example, let us take the M_11 type group returned by the GAP function <code class="func">MathieuGroup</code> (<a href="../../../doc/ref/chap50.html#X788FA7DE84E0FE6A">Reference: MathieuGroup</a>).

<div class="example"><pre>
gap> G:= MathieuGroup( 11 );;
gap> gens:= GeneratorsOfGroup( G );
[ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) ]
gap> ResultOfStraightLineDecision( checkM11.program, gens );
false
</pre></div>

If we want to compute an L_2(11) type subgroup of this group, we can use a black box program for computing standard generators, and then apply the straight line program for computing the restriction.

<div class="example"><pre>
gap> find:= AtlasProgram( "M11", "find" );
rec( groupname := "M11", identifier := [ "M11", "M11G1-find1", 1, 1 ],
 program := <black box program>, standardization := 1,
 version := "1" )
gap> stdgens:= ResultOfBBoxProgram( find.program, Group( gens ) );;
gap> List( stdgens, Order );
[ 2, 4 ]
gap> ResultOfStraightLineDecision( checkM11.program, stdgens );
true
gap> gensL211:= ResultOfStraightLineProgram( restL211.program, stdgens );;
gap> List( gensL211, Order );
[ 2, 3 ]
gap> G:= Group( gensL211 );; Size( G ); IsSimple( G );
660
true
</pre></div>

Note that applying the black box program several times may yield different group elements, because computations of random elements are involved, see <code class="func">ResultOfBBoxProgram</code> (<a href="chap6.html#X869BACFB80A3CC87">6.2-4</a>). All what the black box program promises is to construct standard generators, and these are defined only up to conjugacy in the automorphism group of the group in question.

<a id="X7CE7C2068017525C" name="X7CE7C2068017525C"></a>

<h5>2.4-5 Example: Using the GAP Library of Tables of Marks</h5>

The GAP Library of Tables of Marks (the GAP package TomLib, <a href="chapBib.html#biBTomLib">[NMP18]</a>) provides, for many almost simple groups, information for constructing representatives of all conjugacy classes of subgroups. If this information is compatible with the standard generators of the ATLAS of Group Representations then we can use it to restrict any representation from the ATLAS to prescribed subgroups. This is useful in particular for those subgroups for which the ATLAS of Group Representations itself does not contain a straight line program.

<div class="example"><pre>
gap> tom:= TableOfMarks( "A5" );
TableOfMarks( "A5" )
gap> info:= StandardGeneratorsInfo( tom );
[ rec( ATLAS := true, description := "|a|=2, |b|=3, |ab|=5",
 generators := "a, b",
 script := [ [ 1, 2 ], [ 2, 3 ], [ 1, 1, 2, 1, 5 ] ],
 standardization := 1 ) ]
</pre></div>

The <code class="keyw">true</code> value of the component <code class="code">ATLAS</code> indicates that the information stored on <code class="code">tom</code> refers to the standard generators of type 1 in the ATLAS of Group Representations.

We want to restrict a 4-dimensional integral representation of A_5 to a Sylow 2 subgroup of A_5, and use <code class="func">RepresentativeTomByGeneratorsNC</code> (<a href="../../../doc/ref/chap70.html#X7F625AB880B73AC3">Reference: RepresentativeTomByGeneratorsNC</a>) for that.

<div class="example"><pre>
gap> info:= OneAtlasGeneratingSetInfo( "A5", Ring, Integers, Dimension, 4 );;
gap> stdgens:= 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> orders:= OrdersTom( tom );
[ 1, 2, 3, 4, 5, 6, 10, 12, 60 ]
gap> pos:= Position( orders, 4 );
4
gap> sub:= RepresentativeTomByGeneratorsNC( tom, pos, stdgens.generators );
<matrix group of size 4 with 2 generators>
gap> GeneratorsOfGroup( sub );
[ [ [ 1, 0, 0, 0 ], [ -1, -1, -1, -1 ], [ 0, 0, 0, 1 ],
 [ 0, 0, 1, 0 ] ],
 [ [ 1, 0, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 1, 0, 0 ],
 [ -1, -1, -1, -1 ] ] ]
</pre></div>

<a id="X82550A9683E0DCA2" name="X82550A9683E0DCA2"></a>

<h5>2.4-6 Example: Index 770 Subgroups in M_22</h5>

The sporadic simple Mathieu group M_22 contains a unique class of subgroups of index 770 (and order 576). This can be seen for example using GAP's Library of Tables of Marks.

<div class="example"><pre>
gap> tom:= TableOfMarks( "M22" );
TableOfMarks( "M22" )
gap> subord:= Size( UnderlyingGroup( tom ) ) / 770;
576
gap> ord:= OrdersTom( tom );;
gap> tomstabs:= Filtered( [ 1 .. Length( ord ) ], i -> ord[i] = subord );
[ 144 ]
</pre></div>

The permutation representation of M_22 on the right cosets of such a subgroup S is contained in the ATLAS of Group Representations.

<div class="example"><pre>
gap> DisplayAtlasInfo( "M22", NrMovedPoints, 770 );
Representations for G = M22: (all refer to std. generators 1)
----------------------------
12: G <= Sym(770) rank 9, on cosets of (A4xA4):4 < 2^4:A6
</pre></div>

Now we verify the information shown about the point stabilizer and about the maximal overgroups of S in M_22.

<div class="example"><pre>
gap> maxtom:= MaximalSubgroupsTom( tom );
[ [ 155, 154, 153, 152, 151, 150, 146, 145 ],
 [ 22, 77, 176, 176, 231, 330, 616, 672 ] ]
gap> List( tomstabs, i -> List( maxtom[1], j -> ContainedTom( tom, i, j ) ) );
[ [ 0, 10, 0, 0, 0, 0, 0, 0 ] ]
</pre></div>

We see that the only maximal subgroups of M_22 that contain S have index 77 in M_22. According to the ATLAS of Finite Groups, these maximal subgroups have the structure 2^4:A_6. From that and from the structure of A_6, we conclude that S has the structure 2^4:(3^2:4).

Alternatively, we look at the permutation representation of degree 770. We fetch it from the ATLAS of Group Representations. There is exactly one nontrivial block system for this representation, with 77 blocks of length 10.

<div class="example"><pre>
gap> g:= AtlasGroup( "M22", NrMovedPoints, 770 );
<permutation group of size 443520 with 2 generators>
gap> allbl:= AllBlocks( g );;
gap> List( allbl, Length );
[ 10 ]
</pre></div>

Furthermore, GAP computes that the point stabilizer S has the structure (A_4 × A_4):4.

<div class="example"><pre>
gap> stab:= Stabilizer( g, 1 );;
gap> StructureDescription( stab : nice );
"(A4 x A4) : C4"
gap> blocks:= Orbit( g, allbl[1], OnSets );;
gap> act:= Action( g, blocks, OnSets );;
gap> StructureDescription( Stabilizer( act, 1 ) );
"(C2 x C2 x C2 x C2) : A6"
</pre></div>

<a id="X84F9D163795B7DE1" name="X84F9D163795B7DE1"></a>

<h5>2.4-7 Example: Index 462 Subgroups in M_22</h5>

The ATLAS of Group Representations contains three degree 462 permutation representations of the group M_22.

<div class="example"><pre>
gap> DisplayAtlasInfo( "M22", NrMovedPoints, 462 );
Representations for G = M22: (all refer to std. generators 1)
----------------------------
7: G <= Sym(462a) rank 5, on cosets of 2^4:A5 < 2^4:A6
8: G <= Sym(462b) rank 8, on cosets of 2^4:A5 < L3(4), 2^4:S5
9: G <= Sym(462c) rank 8, on cosets of 2^4:A5 < L3(4), 2^4:A6
</pre></div>

The point stabilizers in these three representations have the structure 2^4:A_5. Using GAP's Library of Tables of Marks, we can show that these stabilizers are exactly the three classes of subgroups of order 960 in M_22. For that, we first verify that the group generators stored in GAP's table of marks coincide with the standard generators used by the ATLAS of Group Representations.

<div class="example"><pre>
gap> tom:= TableOfMarks( "M22" );
TableOfMarks( "M22" )
gap> genstom:= GeneratorsOfGroup( UnderlyingGroup( tom ) );;
gap> checkM22:= AtlasProgram( "M22", "check" );
rec( groupname := "M22", identifier := [ "M22", "M22G1-check1", 1, 1 ]
 , program := <straight line decision>, standardization := 1,
 version := "1" )
gap> ResultOfStraightLineDecision( checkM22.program, genstom );
true
</pre></div>

There are indeed three classes of subgroups of order 960 in M_22.

<div class="example"><pre>
gap> ord:= OrdersTom( tom );;
gap> tomstabs:= Filtered( [ 1 .. Length( ord ) ], i -> ord[i] = 960 );
[ 147, 148, 149 ]
</pre></div>

Now we compute representatives of these three classes in the three representations <code class="code">462a</code>, <code class="code">462b</code>, and <code class="code">462c</code>. We see that each of the three classes occurs as a point stabilizer in exactly one of the three representations.

<div class="example"><pre>
gap> atlasreps:= AllAtlasGeneratingSetInfos( "M22", NrMovedPoints, 462 );
[ rec( charactername := "1a+21a+55a+154a+231a",
 constituents := [ 1, 2, 5, 7, 9 ], contents := "core",
 groupname := "M22", id := "a",
 identifier :=
 [ "M22", [ "M22G1-p462aB0.m1", "M22G1-p462aB0.m2" ], 1, 462 ],
 isPrimitive := false, p := 462, rankAction := 5,
 repname := "M22G1-p462aB0", repnr := 7, size := 443520,
 stabilizer := "2^4:A5 < 2^4:A6", standardization := 1,
 transitivity := 1, type := "perm" ),
 rec( charactername := "1a+21a^2+55a+154a+210a",
 constituents := [ 1, [ 2, 2 ], 5, 7, 8 ], contents := "core",
 groupname := "M22", id := "b",
 identifier :=
 [ "M22", [ "M22G1-p462bB0.m1", "M22G1-p462bB0.m2" ], 1, 462 ],
 isPrimitive := false, p := 462, rankAction := 8,
 repname := "M22G1-p462bB0", repnr := 8, size := 443520,
 stabilizer := "2^4:A5 < L3(4), 2^4:S5", standardization := 1,
--> --------------------

--> maximum size reached

--> --------------------

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