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<a id="X8354C98179CDB193" name="X8354C98179CDB193"></a>
<div class="ChapSects"><a href="chap1.html#X8354C98179CDB193">1 Maintenance Issues for the GAP Character Table Library</a>
<div class="ContSect"> <a href="chap1.html#X7ECA800587320C2C">1.1 Disproving Possible Character Tables (November 2006)</a>

<div class="ContSSBlock">
 <a href="chap1.html#X795DCCEA7F4D187A">1.1-1 A Perfect Pseudo Character Table (November 2006)</a>

 <a href="chap1.html#X80F0B4E07B0B2277">1.1-2 An Error in the Character Table of E_6(2) (March 2016)</a>

 <a href="chap1.html#X7D7982CD87413F76">1.1-3 An Error in a Power Map of the Character Table of 2.F_4(2).2 (November 2015)</a>

 <a href="chap1.html#X836E4B6184F32EF5">1.1-4 A Character Table with a Wrong Name (May 2017)</a>

</div></div>
<div class="ContSect"> <a href="chap1.html#X8159D79C7F071B33">1.2 Some finite factor groups of perfect space groups (February 2014)</a>

<div class="ContSSBlock">
 <a href="chap1.html#X8710D4947AEB366F">1.2-1 Constructing the space groups in question</a>

 <a href="chap1.html#X84E7FE70843422B0">1.2-2 Constructing the factor groups in question</a>

 <a href="chap1.html#X79109A20873E76DA">1.2-3 Examples with point group A_5</a>

 <a href="chap1.html#X83523D1E792F9E01">1.2-4 Examples with point group L_3(2)</a>

 <a href="chap1.html#X7A01A9BC846BE39A">1.2-5 Example with point group SL_2(7)</a>

 <a href="chap1.html#X7D3100B58093F37D">1.2-6 Example with point group 2^3.L_3(2)</a>

 <a href="chap1.html#X80800F3B7D6EF06C">1.2-7 Examples with point group A_6</a>

 <a href="chap1.html#X7D43452C79B0EAE1">1.2-8 Examples with point group L_2(8)</a>

 <a href="chap1.html#X8575CE147A9819BF">1.2-9 Example with point group M_11</a>

 <a href="chap1.html#X7C0201B77DA1682A">1.2-10 Example with point group U_3(3)</a>

 <a href="chap1.html#X85D9C329792E58F3">1.2-11 Examples with point group U_4(2)</a>

 <a href="chap1.html#X8635EE0B78A66120">1.2-12 A remark on one of the example groups</a>

</div></div>
<div class="ContSect"> <a href="chap1.html#X8448022280E82C52">1.3 Generality problems (December 2004/October 2015)</a>

<div class="ContSSBlock">
 <a href="chap1.html#X7D1A66C3844D09B1">1.3-1 Listing possible generality problems</a>

 <a href="chap1.html#X80EB5D827A78975A">1.3-2 A generality problem concerning the group J_3 (April 2015)</a>

 <a href="chap1.html#X82C37532783168AA">1.3-3 A generality problem concerning the group HN (August 2022)</a>

</div></div>
<div class="ContSect"> <a href="chap1.html#X7D8C6D1883C9CECA">1.4 Brauer Tables that can be derived from Known Tables</a>

<div class="ContSSBlock">
 <a href="chap1.html#X7DF018B77E722CA7">1.4-1 Brauer Tables via Construction Information</a>

 <a href="chap1.html#X795419A287BD228E">1.4-2 Liftable Brauer Characters (May 2017)</a>

</div></div>
<div class="ContSect"> <a href="chap1.html#X864EFF897A854F89">1.5 Information about certain subgroups of the Monster group</a>

<div class="ContSSBlock">
 <a href="chap1.html#X82C7A03684DD7C6E">1.5-1 The Monster group does not contain subgroups of the type 2.U_4(2) (August 2023)</a>

 <a href="chap1.html#X87EC0C48866D1BDE">1.5-2 Perfect central extensions of L_3(4) (August 2023)</a>

 <a href="chap1.html#X7F605CA28441687F">1.5-3 The character table of (2 × O_8^+(3)).S_4 ≤ 2.B (October 2023)</a>

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

<h3>1 Maintenance Issues for the GAP Character Table Library</h3>

This chapter collects examples of computations that arose in the context of maintaining the GAP Character Table Library. The sections have been added when the issues in question arose; the dates of the additions are shown in the section titles.

<a id="X7ECA800587320C2C" name="X7ECA800587320C2C"></a>

<h4>1.1 Disproving Possible Character Tables (November 2006)</h4>

I do not know a necessary and sufficient criterion for checking whether a given matrix together with a list of power maps describes the character table of a finite group. Examples of pseudo character tables (tables which satisfy certain necessary conditions but for which actually no group exists) have been given in <a href="chapBib.html#biBGag86">[Gag86]</a>. Another such example is described in Section <a href="chap2.html#X7E0C603880157C4E">2.4-17</a>. The tables in the GAP Character Table Library satisfy the usual tests. However, there are table candidates for which these tests are not good enough. Another question would be whether a given character table belongs to the group for which it is claimed to belong, see Section <a href="chap1.html#X836E4B6184F32EF5">1.1-4</a> for an example.

<a id="X795DCCEA7F4D187A" name="X795DCCEA7F4D187A"></a>

<h5>1.1-1 A Perfect Pseudo Character Table (November 2006)</h5>

(This example arose from a discussion with Jack Schmidt.)

Up to version 1.1.3 of the GAP Character Table Library, the table with identifier <code class="code">"P41/G1/L1/V4/ext2"</code> was not correct. The problem occurs already in the microfiches that are attached to <a href="chapBib.html#biBHP89">[HP89]</a>.

In the following, we show that this table is not the character table of a finite group, using the GAP library of perfect groups. Currently we do not know how to prove this inconsistency alone from the table.

We start with the construction of the inconsistent table; apart from a little editing, the following input equals the data formerly stored in the file <code class="file">data/ctoholpl.tbl</code> of the GAP Character Table Library.

<div class="example"><pre>
gap> tbl:= rec(
> Identifier:= "P41/G1/L1/V4/ext2",
> InfoText:= Concatenation( [
> "origin: Hanrath library,\n",
> "structure is 2^7.L2(8),\n",
> "characters sorted with permutation (12,14,15,13)(19,20)" ] ),
> UnderlyingCharacteristic:= 0,
> SizesCentralizers:= [64512,1024,1024,64512,64,64,64,64,128,128,64,
> 64,128,128,18,18,14,14,14,14,14,14,18,18,18,18,18,18],
> ComputedPowerMaps:= [,[1,1,1,1,2,3,3,2,3,2,2,1,3,2,16,16,20,20,22,
> 22,18,18,26,26,27,27,23,23],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,4,
> 1,21,22,17,18,19,20,16,15,15,16,16,15],,,,[1,2,3,4,5,6,7,8,9,10,
> 11,12,13,14,15,16,4,1,4,1,4,1,26,25,28,27,23,24]],
> Irr:= 0,
> AutomorphismsOfTable:= Group( [(23,26,27)(24,25,28),(9,13)(10,14),
> (17,19,21)(18,20,22)] ),
> ConstructionInfoCharacterTable:= ["ConstructClifford",[[[1,2,3,4,
> 5,6,7,8,9],[1,7,8,3,9,2],[1,4,5,6,2],[1,2,2,2,2,2,2,2]],
> [["L2(8)"],["Dihedral",18],["Dihedral",14],["2^3"]],[[[1,2,3,4],
> [1,1,1,1],["elab",4,25]],[[1,2,3,4,4,4,4,4,4,4],[2,6,5,2,3,4,5,
> 6,7,8],["elab",10,17]],[[1,2],[3,4],[[1,1],[-1,1]]],[[1,3],[4,
> 2],[[1,1],[-1,1]]],[[1,3],[5,3],[[1,1],[-1,1]]],[[1,3],[6,4],
> [[1,1],[-1,1]]],[[1,2],[7,2],[[1,1],[1,-1]]],[[1,2],[8,3],[[1,
> 1],[-1,1]]],[[1,2],[9,5],[[1,1],[1,-1]]]]]],
> );;
gap> ConstructClifford( tbl, tbl.ConstructionInfoCharacterTable[2] );
gap> ConvertToLibraryCharacterTableNC( tbl );;
</pre></div>

Suppose that there is a group G, say, with this table. Then G is perfect since the table has only one linear character.

<div class="example"><pre>
gap> Length( LinearCharacters( tbl ) );
1
gap> IsPerfectCharacterTable( tbl );
true
</pre></div>

The table satisfies the orthogonality relations, the structure constants are nonnegative integers, and symmetrizations of the irreducibles decompose into the irreducibles, with nonnegative integral coefficients.

<div class="example"><pre>
gap> IsInternallyConsistent( tbl );
true
gap> irr:= Irr( tbl );;
gap> test:= Concatenation( List( [ 2 .. 7 ],
> n -> Symmetrizations( tbl, irr, n ) ) );;
gap> Append( test, Set( Tensored( irr, irr ) ) );
gap> fail in Decomposition( irr, test, "nonnegative" );
false
gap> if ForAny( Tuples( [ 1 .. NrConjugacyClasses( tbl ) ], 3 ),
> t -> not ClassMultiplicationCoefficient( tbl, t[1], t[2], t[3] )
> in NonnegativeIntegers ) then
> Error( "contradiction" );
> fi;
</pre></div>

The GAP Library of Perfect Groups contains representatives of the four isomorphism types of perfect groups of order |G| = 64512.

<div class="example"><pre>
gap> n:= Size( tbl );
64512
gap> NumberPerfectGroups( n );
4
gap> grps:= List( [ 1 .. 4 ], i -> PerfectGroup( IsPermGroup, n, i ) );
[ L2(8) 2^6 E 2^1, L2(8) N 2^6 E 2^1 I, L2(8) N 2^6 E 2^1 II,
 L2(8) N 2^6 E 2^1 III ]
</pre></div>

If we believe that the classification of perfect groups of order |G| is correct then all we have to do is to show that none of the character tables of these four groups is equivalent to the given table.

<div class="example"><pre>
gap> tbls:= List( grps, CharacterTable );;
gap> List( tbls,
> x -> TransformingPermutationsCharacterTables( x, tbl ) );
[ fail, fail, fail, fail ]
</pre></div>

In fact, already the matrices of irreducible characters of the four groups do not fit to the given table.

<div class="example"><pre>
gap> List( tbls,
> t -> TransformingPermutations( Irr( t ), Irr( tbl ) ) );
[ fail, fail, fail, fail ]
</pre></div>

Let us look closer at the tables in question. Each character table of a perfect group of order 64512 has exactly one irreducible character of degree 63 that takes exactly the values -1, 0, 7, and 63; moreover, the value 7 occurs in exactly two classes.

<div class="example"><pre>
gap> testchars:= List( tbls,
> t -> Filtered( Irr( t ),
> x -> x[1] = 63 and Set( x ) = [ -1, 0, 7, 63 ] ) );;
gap> List( testchars, Length );
[ 1, 1, 1, 1 ]
gap> List( testchars, l -> Number( l[1], x -> x = 7 ) );
[ 2, 2, 2, 2 ]
</pre></div>

(Another way to state this is that in each of the four tables t in question, there are ten preimage classes of the involution class in the simple factor group L_2(8), there are eight preimage classes of this class in the factor group 2^6.L_2(8), and that the unique class in which an irreducible degree 63 character of this factor group takes the value 7 splits in t.)

In the erroneous table, however, there is only one class with the value 7 in this character.

<div class="example"><pre>
gap> testchars:= List( [ tbl ],
> t -> Filtered( Irr( t ),
> x -> x[1] = 63 and Set( x ) = [ -1, 0, 7, 63 ] ) );;
gap> List( testchars, Length );
[ 1 ]
gap> List( testchars, l -> Number( l[1], x -> x = 7 ) );
[ 1 ]
</pre></div>

This property can be checked easily for the displayed table stored in fiche 2, row 4, column 7 of <a href="chapBib.html#biBHP89">[HP89]</a>, with the name <code class="code">6L1<>Z^7<>L2(8); V4; MOD 2</code>, and it turns out that this table is not correct.

Note that these microfiches contain two tables of order 64512, and there were three tables of groups of that order in the GAP Character Table Library that contain <code class="code">origin: Hanrath library</code> in their <code class="func">InfoText</code> (<a href="../../../doc/ref/chap12.html#X871562FD7F982C12">Reference: InfoText</a>) value. Besides the incorrect table, these library tables are the character tables of the groups <code class="code">PerfectGroup( 64512, 1 )</code> and <code class="code">PerfectGroup( 64512, 3 )</code>, respectively. (The matrices of irreducible characters of these tables are equivalent.)

<div class="example"><pre>
gap> Filtered( [ 1 .. 4 ], i ->
> TransformingPermutationsCharacterTables( tbls[i],
> CharacterTable( "P41/G1/L1/V1/ext2" ) ) <> fail );
[ 1 ]
gap> Filtered( [ 1 .. 4 ], i ->
> TransformingPermutationsCharacterTables( tbls[i],
> CharacterTable( "P41/G1/L1/V2/ext2" ) ) <> fail );
[ 3 ]
gap> TransformingPermutations( Irr( tbls[1] ), Irr( tbls[3] ) ) <> fail;
true
</pre></div>

Since version 1.2 of the GAP Character Table Library, the character table with the <code class="func">Identifier</code> (<a href="../../../doc/ref/chap70.html#X810E53597B5BB4F8">Reference: Identifier for tables of marks</a>) value <code class="code">"P41/G1/L1/V4/ext2"</code> corresponds to the group <code class="code">PerfectGroup( 64512, 4 )</code>. The choice of this group was somewhat arbitrary since the vector system <code class="code">V4</code> seems to be not defined in <a href="chapBib.html#biBHP89">[HP89]</a>; anyhow, this group and the remaining perfect group, <code class="code">PerfectGroup( 64512, 2 )</code>, have equivalent matrices of irreducibles.

<div class="example"><pre>
gap> Filtered( [ 1 .. 4 ], i ->
> TransformingPermutationsCharacterTables( tbls[i],
> CharacterTable( "P41/G1/L1/V4/ext2" ) ) <> fail );
[ 4 ]
gap> TransformingPermutations( Irr( tbls[2] ), Irr( tbls[4] ) ) <> fail;
true
</pre></div>

<a id="X80F0B4E07B0B2277" name="X80F0B4E07B0B2277"></a>

<h5>1.1-2 An Error in the Character Table of E_6(2) (March 2016)</h5>

In March 2016, Bill Unger computed the character table of the simple group E_6(2) with Magma (see <a href="chapBib.html#biBCP96">[CP96]</a>) and compared it with the table that was contained in the GAP Character Table Library since 2000. It turned out that the two tables did not coincide.

The differences concern irrational character values on classes of element order 91 and power map values on these classes. (The character values and power maps fit to each other in both tables; thus it may be that the assumption of a wrong power has implied the wrong character values, or vice versa.) Specifically, the 11th power map in the GAP table fixed all elements of order 91. Using the smallest matrix representation of E_6(2) over the field with two elements, one can easily find an element g of order 91, and show that the characteristic polynomials of g and g^11 differ. Hence these two elements cannot be conjugate in E_6(2). In other words, the GAP table was wrong.

<div class="example"><pre>
gap> g:= AtlasGroup( "E6(2)" );;
gap> repeat x:= PseudoRandom( g ); until Order( x ) = 91;
gap> CharacteristicPolynomial( x ) = CharacteristicPolynomial( x^11 );
false
</pre></div>

The wrong GAP table has been corrected in version 1.3.0 of the GAP Character Table Library.

<div class="example"><pre>
gap> t:= CharacterTable( "E6(2)" );;
gap> ord91:= Positions( OrdersClassRepresentatives( t ), 91 );
[ 163, 164, 165, 166, 167, 168 ]
gap> PowerMap( t, 11 ){ ord91 };
[ 167, 168, 163, 164, 165, 166 ]
</pre></div>

<a id="X7D7982CD87413F76" name="X7D7982CD87413F76"></a>

<h5>1.1-3 An Error in a Power Map of the Character Table of 2.F_4(2).2 (November 2015)</h5>

As a part of the computations for <a href="chapBib.html#biBBMO17">[BMO17]</a>, the character table of the group 2.F_4(2).2 was computed automatically from a representation of the group, using Magma (see <a href="chapBib.html#biBCP96">[CP96]</a>). It turned out that the 2-nd power map that had been stored on the library character table of 2.F_4(2).2 had been wrong.

In fact, this was the one and only case of a power map for an Atlas group which was not determined by the character table, and the <code class="func">InfoText</code> (<a href="../../../doc/ref/chap12.html#X871562FD7F982C12">Reference: InfoText</a>) value of the character table had mentioned the two alternatives.

Note that the ambiguity is not present in the table of the factor group F_4(2).2, and only four faithful irreducible characters of 2.F_4(2).2 distinguish the four relevant conjugacy classes.

<div class="example"><pre>
gap> t:= CharacterTable( "2.F4(2).2" );;
gap> f:= CharacterTable( "F4(2).2" );;
gap> map:= PowerMap( t, 2 );
[ 1, 1, 1, 1, 1, 1, 1, 1, 9, 9, 11, 11, 3, 3, 3, 5, 5, 5, 3, 6, 6, 5,
 5, 7, 7, 5, 8, 7, 29, 29, 9, 9, 9, 9, 11, 11, 9, 9, 9, 9, 11, 11,
 43, 43, 20, 20, 20, 14, 14, 13, 13, 20, 21, 24, 28, 28, 57, 57, 29,
 29, 29, 29, 33, 33, 35, 37, 37, 37, 37, 33, 33, 37, 37, 35, 41, 41,
 42, 42, 79, 79, 43, 43, 83, 83, 45, 45, 47, 47, 53, 53, 91, 91, 57,
 57, 61, 61, 61, 98, 98, 70, 70, 63, 63, 81, 81, 83, 83, 1, 6, 7,
 11, 16, 17, 24, 24, 21, 27, 27, 25, 26, 29, 41, 53, 53, 53, 46, 56,
 56, 56, 56, 62, 75, 75, 78, 78, 77, 77, 79, 79, 86, 86, 85, 85, 88,
 88, 88, 88, 95, 95, 96, 96 ]
gap> PositionSublist( map, [ 86, 86, 85, 85 ] );
140
gap> OrdersClassRepresentatives( t ){ [ 140 .. 143 ] };
[ 32, 32, 32, 32 ]
gap> SizesCentralizers( t ){ [ 140 .. 143 ] };
[ 64, 64, 64, 64 ]
gap> GetFusionMap( t, f ){ [ 140 ..143 ] };
[ 86, 86, 87, 87 ]
gap> PowerMap( f, 2 ){ [ 86, 87 ] };
[ 50, 50 ]
gap> pos:= PositionsProperty( Irr( t ),
> x -> x[1] <> x[2] and Length( Set( x{ [ 140 .. 143 ] } ) ) > 1 );
[ 144, 145, 146, 147 ]
gap> List( pos, i -> Irr(t)[i]{ [ 140 .. 143 ] } );
[ [ 2*E(16)-2*E(16)^7, -2*E(16)+2*E(16)^7, 2*E(16)^3-2*E(16)^5,
 -2*E(16)^3+2*E(16)^5 ],
 [ -2*E(16)+2*E(16)^7, 2*E(16)-2*E(16)^7, -2*E(16)^3+2*E(16)^5,
 2*E(16)^3-2*E(16)^5 ],
 [ -2*E(16)^3+2*E(16)^5, 2*E(16)^3-2*E(16)^5, 2*E(16)-2*E(16)^7,
 -2*E(16)+2*E(16)^7 ],
 [ 2*E(16)^3-2*E(16)^5, -2*E(16)^3+2*E(16)^5, -2*E(16)+2*E(16)^7,
 2*E(16)-2*E(16)^7 ] ]
</pre></div>

I had not found a suitable subgroup of 2.F_4(2).2 whose character table could be used to decide the question which of the two alternatives is the correct one.

<a id="X836E4B6184F32EF5" name="X836E4B6184F32EF5"></a>

<h5>1.1-4 A Character Table with a Wrong Name (May 2017)</h5>

(This example is much older.)

The character table that is shown in <a href="chapBib.html#biBOst86">[Ost86, p. 126 f.]</a> is claimed to be the table of a Sylow 2 subgroup P of the sporadic simple Lyons group Ly. This table had been contained in the character table library of the CAS system (see <a href="chapBib.html#biBNPP84">[NPP84]</a>), which was one of the predecessors of GAP.

It is easy to see that no subgroup of Ly can have this character table. Namely, the group of that table contains elements of order eight with centralizer order 2^6, and this does not occur in Ly.

<div class="example"><pre>
gap> tbl:= CharacterTable( "Ly" );;
gap> orders:= OrdersClassRepresentatives( tbl );;
gap> order8:= Filtered( [ 1 .. Length( orders ) ], x -> orders[x] = 8 );
[ 12, 13 ]
gap> SizesCentralizers( tbl ){ order8 } / 2^6;
[ 15/2, 3/2 ]
</pre></div>

The table of P has been computed in <a href="chapBib.html#biBBre91">[Bre91]</a> with character theoretic methods. Nowadays it would be no problem to take a permutation representation of Ly, to compute its Sylow 2 subgroup, and use this group to compute its character table. However, the task is even easier if we assume that Ly has a subgroup of the structure 3.McL.2. This subgroup is of odd index, hence it contains a conjugate of P. Clearly the Sylow 2 subgroups in the factor group McL.2 are isomorphic with P. Thus we can start with a rather small permutation representation.

<div class="example"><pre>
gap> g:= AtlasGroup( "McL.2" );;
gap> NrMovedPoints( g );
275
gap> syl:= SylowSubgroup( g, 2 );;
gap> pc:= Image( IsomorphismPcGroup( syl ) );;
gap> t:= CharacterTable( pc );;
</pre></div>

The character table coincides with the one which is stored in the Character Table Library.

<div class="example"><pre>
gap> IsRecord( TransformingPermutationsCharacterTables( t,
> CharacterTable( "LyN2" ) ) );
true
</pre></div>

<a id="X8159D79C7F071B33" name="X8159D79C7F071B33"></a>

<h4>1.2 Some finite factor groups of perfect space groups (February 2014)</h4>

If one wants to find a group to which a given character table from the GAP Character Table Library belongs, one can try the function <code class="func">GroupInfoForCharacterTable</code> (<a href="../doc/chap3.html#X78DCD38B7D96D8A4">CTblLib: GroupInfoForCharacterTable</a>). For a long time, this was not successful in the case of 16 character tables that had been computed by W. Hanrath (see Section <q>Ordinary and Brauer Tables in the GAP Character Table Library</q> in the CTblLib manual).

Using the information from <a href="chapBib.html#biBHP89">[HP89]</a>, it is straightforward to construct such groups as factor groups of infinite groups. Since version 1.3.0 of the CTblLib package, calling <code class="func">GroupInfoForCharacterTable</code> (<a href="../doc/chap3.html#X78DCD38B7D96D8A4">CTblLib: GroupInfoForCharacterTable</a>) for the 16 library tables in question yields nonempty lists and thus allows one to access the results of these constructions, via the function <code class="code">CTblLib.FactorGroupOfPerfectSpaceGroup</code>. This is an undocumented auxiliary function that becomes available automatically when <code class="func">GroupInfoForCharacterTable</code> (<a href="../doc/chap3.html#X78DCD38B7D96D8A4">CTblLib: GroupInfoForCharacterTable</a>) has been called for the first time.

<div class="example"><pre>
gap> GroupInfoForCharacterTable( "A5" );;
gap> IsBound( CTblLib.FactorGroupOfPerfectSpaceGroup );
true
</pre></div>

Below we list the 16 group constructions. In each case, an epimorphism from the space group in question is defined by mapping the generators returned by by the function <code class="code">generatorsOfPerfectSpaceGroup</code> defined below to the generators stored in the attribute <code class="func">GeneratorsOfGroup</code> (<a href="../../../doc/ref/chap39.html#X79C44528864044C5">Reference: GeneratorsOfGroup</a>) of the group returned by <code class="code">CTblLib.FactorGroupOfPerfectSpaceGroup</code>.

<a id="X8710D4947AEB366F" name="X8710D4947AEB366F"></a>

<h5>1.2-1 Constructing the space groups in question</h5>

In <a href="chapBib.html#biBHP89">[HP89]</a>, a space group S is described as a subgroup { M(g, t); g ∈ P, t ∈ T } of GL(d+1, ℤ), where

<div class="pcenter"><table> <tr> <td class="tdright">M(g, t)</td> <td class="tdcenter"> = </td> <td class="tdleft"> <table class="GAPDocTable"> <tr> <td class="tdright">g</td> <td class="tdright">0</td> </tr> <tr> <td class="tdright">V(g)+t</td> <td class="tdright">1</td> </tr> </table> </td> </tr> </table> </div>

the point group P of S is a finite subgroup of GL(d, ℤ), the translation lattice T of S is a sublattice of ℤ^d, and the vector system V of S is a map from P to ℤ^d. Note that V maps the identity matrix I ∈ GL(d, ℤ) to the zero vector, and M(T):= { M(I, t); t ∈ T } is a normal subgroup of S that is isomorphic with T. More generally, M(n T) is a normal subgroup of S, for any positive integer n.

Specifically, P is given by generators g_1, g_2, ..., g_k, T is given by a ℤ-basis B = { b_1, b_2, ..., b_d } of T, and V is given by the vectors V(g_1), V(g_2), ..., V(g_k).

In the examples below, the matrix representation of P is irreducible, so we need just the following k+1 elements to generate S:

<div class="pcenter"> <table> <tr> <td class="tdleft"> <table class="GAPDocTable"> <tr> <td class="tdright">g_1</td> <td class="tdright">0</td> </tr> <tr> <td class="tdright">V(g_1)</td> <td class="tdright">1</td> </tr> </table> <td class="tdleft"> , </td> </td> <td class="tdleft"> <table class="GAPDocTable"> <tr> <td class="tdright">g_2</td> <td class="tdright">0</td> </tr> <tr> <td class="tdright">V(g_2)</td> <td class="tdright">1</td> </tr> </table> </td> <td class="tdleft"> , ..., </td> <td class="tdleft"> <table class="GAPDocTable"> <tr> <td class="tdright">g_k</td> <td class="tdright">0</td> </tr> <tr> <td class="tdright">V(g_k)</td> <td class="tdright">1</td> </tr> </table> </td> <td class="tdleft"> , </td> <td class="tdleft"> <table class="GAPDocTable"> <tr> <td class="tdright">I</td> <td class="tdright">0</td> </tr> <tr> <td class="tdright">b_1</td> <td class="tdright">1</td> </tr> </table> </td> <td class="tdleft"> . </td> </tr> </table> </div>

These generators are returned by the function <code class="code">generatorsOfPerfectSpaceGroup</code>, when the inputs are [ g_1, g_2, ..., g_k ], [ V(g_1), V(g_2), ..., V(g_k) ], and b_1.

<div class="example"><pre>
gap> generatorsOfPerfectSpaceGroup:= function( Pgens, V, t )
> local d, result, i, m;
> d:= Length( Pgens[1] );
> result:= [];
> for i in [ 1 .. Length( Pgens ) ] do
> m:= IdentityMat( d+1 );
> m{ [ 1 .. d ] }{ [ 1 .. d ] }:= Pgens[i];
> m[ d+1 ]{ [ 1 .. d ] }:= V[i];
> result[i]:= m;
> od;
> m:= IdentityMat( d+1 );
> m[ d+1 ]{ [ 1 .. d ] }:= t;
> Add( result, m );
> return result;
> end;;
</pre></div>

<a id="X84E7FE70843422B0" name="X84E7FE70843422B0"></a>

<h5>1.2-2 Constructing the factor groups in question</h5>

The space group S acts on ℤ^d, via v ⋅ M(g, t) = v g + V(g) + t. A (not necessarily faithful) representation of S/M(n T) can be obtained from the corresponding action of S on ℤ^d/(n ℤ^d), that is, by reducing the vectors modulo n. For the GAP computations, we work instead with vectors of length d+1, extending each vector in ℤ^d by 1 in the last position, and acting on these vectors by right multiplicaton with elements of S. Multiplication followed by reduction modulo n is implemented by the action function returned by <code class="code">multiplicationModulo</code> when this is called with argument n.

<div class="example"><pre>
gap> multiplicationModulo:= n -> function( v, g )
> return List( v * g, x -> x mod n ); end;;
</pre></div>

In some of the examples, the representation of P given in <a href="chapBib.html#biBHP89">[HP89]</a> is the action on the factor of a permutation module modulo its trivial submodule. For that, we provide the function <code class="code">deletedPermutationMat</code>, cf. <a href="chapBib.html#biBHP89">[HP89, p. 269]</a>.

<div class="example"><pre>
gap> deletedPermutationMat:= function( pi, n )
> local mat, j, i;
> mat:= PermutationMat( pi, n );
> mat:= mat{ [ 1 .. n-1 ] }{ [ 1 .. n-1 ] };
> j:= n ^ pi;
> if j <> n then
> for i in [ 1 .. n-1 ] do
> mat[i][j]:= -1;
> od;
> fi;
> return mat;
> end;;
</pre></div>

After constructing permutation generators for the example groups, we verify that the groups fit to the character tables from the GAP Character Table Library and to the permutation generators stored for the construction of the group via <code class="code">CTblLib.FactorGroupOfPerfectSpaceGroup</code>.

<div class="example"><pre>
gap> verifyFactorGroup:= function( gens, id )
> local sm, act, stored, hom;
> sm:= SmallerDegreePermutationRepresentation( Group( gens ) );
> gens:= List( gens, x -> x^sm );
> act:= Images( sm );
> if not IsRecord( TransformingPermutationsCharacterTables(
> CharacterTable( act ),
> CharacterTable( id ) ) ) then
> return "wrong character table";
> fi;
> GroupInfoForCharacterTable( id );
> stored:= CTblLib.FactorGroupOfPerfectSpaceGroup( id );
> hom:= GroupHomomorphismByImages( stored, act,
> GeneratorsOfGroup( stored ), gens );
> if hom = fail or not IsBijective( hom ) then
> return "wrong group";
> fi;
> return true;
> end;;
</pre></div>

<a id="X79109A20873E76DA" name="X79109A20873E76DA"></a>

<h5>1.2-3 Examples with point group A_5</h5>

There are two examples with d = 5. The generators of the point group are as follows (see <a href="chapBib.html#biBHP89">[HP89, p. 272]</a>).

<div class="example"><pre>
gap> a:= deletedPermutationMat( (1,3)(2,4), 6 );;
gap> b:= deletedPermutationMat( (1,2,3)(4,5,6), 6 );;
</pre></div>

In both cases, the vector system is V_2.

<div class="example"><pre>
gap> v:= [ [ 2, 2, 0, 0, 1 ], 0 * b[1] ];;
</pre></div>

In the first example, the translation lattice is the sublattice L = 2 L_1 of the full lattice L_1 = ℤ^d.

<div class="example"><pre>
gap> t:= [ 2, 0, 0, 0, 0 ];;
</pre></div>

The library character table with identifier <code class="code">"P1/G2/L1/V2/ext4"</code> belongs to the factor group of S modulo the normal subgroup M(4 L), so we compute the action on an orbit modulo 8.

<div class="example"><pre>
gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;
gap> g:= Group( sgens );;
gap> fun:= multiplicationModulo( 8 );;
gap> orb:= Orbit( g, [ 1, 0, 0, 0, 0, 1 ], fun );;
gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
gap> verifyFactorGroup( permgens, "P1/G2/L1/V2/ext4" );
true
</pre></div>

In the second example, the translation lattice is the sublattice 2 L_2 of ℤ^d where L_2 has the following basis.

<div class="example"><pre>
gap> bas:= [ [-1,-1, 1, 1, 1 ],
> [-1, 1,-1, 1, 1 ],
> [ 1, 1, 1,-1,-1 ],
> [ 1, 1,-1,-1, 1 ],
> [-1, 1, 1,-1, 1 ] ];;
</pre></div>

For the sake of simplicity, we rewrite the action of the point group to one on L_2, and we adjust also the vector system.

<div class="example"><pre>
gap> B:= Basis( Rationals^Length( bas ), bas );;
gap> abas:= List( bas, x -> Coefficients( B, x * a ) );;
gap> bbas:= List( bas, x -> Coefficients( B, x * b ) );;
gap> vbas:= List( v, x -> Coefficients( B, x ) );
[ [ 3/2, 1, 2, 3/2, -1 ], [ 0, 0, 0, 0, 0 ] ]
</pre></div>

In order to work with integral matrices (which is necessary because <code class="code">multiplicationModulo</code> uses GAP's mod operator), we double both the vector system and the translation lattice.

<div class="example"><pre>
gap> vbas:= vbas * 2;
[ [ 3, 2, 4, 3, -2 ], [ 0, 0, 0, 0, 0 ] ]
gap> t:= 2 * t;
[ 4, 0, 0, 0, 0 ]
</pre></div>

The library character table with identifier <code class="code">"P1/G2/L2/V2/ext4"</code> belongs to the factor group of S modulo the normal subgroup M(8 L_2); since we have doubled the lattice, we compute the action on an orbit modulo 16.

<div class="example"><pre>
gap> sgens:= generatorsOfPerfectSpaceGroup( [ abas, bbas ], vbas, t );;
gap> g:= Group( sgens );;
gap> fun:= multiplicationModulo( 16 );;
gap> orb:= Orbit( g, [ 0, 0, 0, 0, 0, 1 ], fun );;
gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
gap> verifyFactorGroup( permgens, "P1/G2/L2/V2/ext4" );
true
</pre></div>

<a id="X83523D1E792F9E01" name="X83523D1E792F9E01"></a>

<h5>1.2-4 Examples with point group L_3(2)</h5>

There are three examples with d = 6 and one example with d = 8. The generators of the point group for the first three examples are as follows (see <a href="chapBib.html#biBHP89">[HP89, p. 290]</a>).

<div class="example"><pre>
gap> a:= [ [ 0, 1, 0, 1, 0, 0 ],
> [ 1, 0, 1, 1, 1, 1 ],
> [-1,-1,-1,-1, 0, 0 ],
> [ 0, 0,-1,-1,-1,-1 ],
> [ 1, 1, 1, 1, 0, 1 ],
> [ 0, 0, 1, 0, 1, 0 ] ];;
gap> b:= [ [-1, 0, 0, 0, 0,-1 ],
> [ 0, 0,-1, 0,-1, 0 ],
> [ 1, 1, 1, 1, 1, 1 ],
> [ 0, 0, 1, 0, 0, 0 ],
> [-1,-1,-1, 0, 0, 0 ],
> [ 1, 0, 0, 0, 0, 0 ] ];;
</pre></div>

The first vector system is the trivial vector system V_1 (that is, the space group S is a split extension of the point group and the translation lattice), and the translation lattice is the full lattice L_1 = ℤ^d.

The library character table with identifier <code class="code">"P11/G1/L1/V1/ext4"</code> belongs to the factor group of S modulo the normal subgroup M(4 L_1), so we compute the action on an orbit modulo 4.

<div class="example"><pre>
gap> v:= List( [ 1, 2 ], i -> 0 * a[1] );;
gap> t:= [ 1, 0, 0, 0, 0, 0 ];;
gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;
gap> g:= Group( sgens );;
gap> fun:= multiplicationModulo( 4 );;
gap> seed:= [ 1, 0, 0, 0, 0, 0, 1 ];;
gap> orb:= Orbit( g, seed, fun );;
gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
gap> verifyFactorGroup( permgens, "P11/G1/L1/V1/ext4" );
true
</pre></div>

The second vector system is V_2, and the translation lattice is 2 L_1.

The library character table with identifier <code class="code">"P11/G1/L1/V2/ext4"</code> belongs to the factor group of S modulo the normal subgroup M(8 L_1), so we compute the action on an orbit modulo 8.

<div class="example"><pre>
gap> v:= [ [ 1, 0, 1, 0, 0, 0 ], 0 * a[1] ];;
gap> t:= [ 2, 0, 0, 0, 0, 0 ];;
gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;
gap> g:= Group( sgens );;
gap> fun:= multiplicationModulo( 8 );;
gap> orb:= Orbit( g, [ 1, 0, 0, 0, 0, 0, 1 ], fun );;
gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
gap> verifyFactorGroup( permgens, "P11/G1/L1/V2/ext4" );
true
</pre></div>

The third vector system is V_3, and the translation lattice is 2 L_1.

The library character table with identifier <code class="code">"P11/G1/L1/V3/ext4"</code> belongs to the factor group of S modulo the normal subgroup M(8 L_1), so we compute the action on an orbit modulo 8.

<div class="example"><pre>
gap> v:= [ [ 0, 1, 0, 0, 1, 0 ], 0 * a[1] ];;
gap> t:= [ 2, 0, 0, 0, 0, 0 ];;
gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;
gap> g:= Group( sgens );;
gap> fun:= multiplicationModulo( 8 );;
gap> orb:= Orbit( g, [ 1, 0, 0, 0, 0, 0, 1 ], fun );;
gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
gap> verifyFactorGroup( permgens, "P11/G1/L1/V3/ext4" );
true
</pre></div>

The generators of the point group for the fourth example are as follows (see <a href="chapBib.html#biBHP89">[HP89, p. 293]</a>).

<div class="example"><pre>
gap> a:= [ [ 1, 0, 0, 1, 0,-1, 0, 1 ],
> [ 0,-1, 1, 0,-1, 0, 0, 0 ],
> [ 1, 0, 0, 1, 0,-1, 0, 0 ],
> [ 0,-1, 0,-1, 0, 1, 1,-1 ],
> [ 1, 0,-1, 1, 1,-1, 0, 0 ],
> [ 1,-1,-1, 0, 0, 0, 1, 0 ],
> [ 0,-1, 1, 0,-1, 1, 0,-1 ],
> [ 1, 0,-1, 0, 0, 0, 0, 0 ] ];;
gap> b:= [ [ 1, 0,-2, 0, 1,-1, 1, 0 ],
> [ 0,-1, 0, 0, 0, 0, 1,-1 ],
> [ 1, 0,-1, 0, 1,-1, 0, 0 ],
> [-1,-1, 1,-1,-1, 2, 0,-1 ],
> [ 0, 0, 0,-1, 0, 0, 0, 0 ],
> [ 0,-1, 0,-1,-1, 1, 1,-1 ],
> [ 1,-1, 0, 0, 0, 0, 0, 0 ],
> [ 1, 0, 0, 0, 0, 0, 0, 0 ] ];;
</pre></div>

The vector system is the trivial vector system V_1, and the translation lattice is the full lattice L_1 = ℤ^d.

The library character table with identifier <code class="code">"P11/G4/L1/V1/ext3"</code> belongs to the factor group of S modulo the normal subgroup M(3 L_1), so we compute the action on an orbit modulo 3.

<div class="example"><pre>
gap> v:= List( [ 1, 2 ], i -> 0 * a[1] );;
gap> t:= [ 1, 0, 0, 0, 0, 0, 0, 0 ];;
gap> sgens:= generatorsOfPerfectSpaceGroup( [ a, b ], v, t );;
gap> g:= Group( sgens );;
gap> fun:= multiplicationModulo( 3 );;
gap> seed:= [ 1, 0, 0, 0, 0, 0, 0, 0, 1 ];;
gap> orb:= Orbit( g, seed, fun );;
gap> permgens:= List( sgens, x -> Permutation( x, orb, fun ) );;
gap> verifyFactorGroup( permgens, "P11/G4/L1/V1/ext3" );
true
</pre></div>

<a id="X7A01A9BC846BE39A" name="X7A01A9BC846BE39A"></a>

<h5>1.2-5 Example with point group SL_2(7)</h5>

There is one example with d = 8. The generators of the point group are as follows (see <a href="chapBib.html#biBHP89">[HP89, p. 295]</a>).

<div class="example"><pre>
gap> a:= KroneckerProduct( IdentityMat( 4 ), [ [ 0, 1 ], [ -1, 0 ] ] );;
gap> b:= [ [ 0,-1, 0, 0, 0, 0, 0, 0 ],
> [ 0, 0, 1, 0, 0, 0, 0, 0 ],
> [-1, 0, 0, 0, 0, 0, 0, 0 ],
> [ 0, 0, 0, 0, 0, 0,-1, 0 ],
> [ 0, 0, 0,-1, 0, 0, 0, 0 ],
> [ 0, 0, 0, 0, 0, 1, 0, 0 ],
> [ 0, 0, 0, 0, 1, 0, 0, 0 ],
> [ 0, 0, 0, 0, 0, 0, 0, 1 ] ];;
</pre></div>

The vector system is the trivial vector system V_1, and the translation lattice is the sublattice L_2 of ℤ^d that has the following basis, which is called B(2,8) in <a href="chapBib.html#biBHP89">[HP89, p. 269]</a>.

<div class="example"><pre>
gap> bas:= [ [ 1, 1, 0, 0, 0, 0, 0, 0 ],
> [ 0, 1, 1, 0, 0, 0, 0, 0 ],
> [ 0, 0, 1, 1, 0, 0, 0, 0 ],
> [ 0, 0, 0, 1, 1, 0, 0, 0 ],
> [ 0, 0, 0, 0, 1, 1, 0, 0 ],
> [ 0, 0, 0, 0, 0, 1, 1, 0 ],
> [ 0, 0, 0, 0, 0, 0, 1, 1 ],
> [ 0, 0, 0, 0, 0, 0,-1, 1 ] ];;
</pre></div>

For the sake of simplicity, we rewrite the action to one on L_2.

<div class="example"><pre>
gap> B:= Basis( Rationals^Length( bas ), bas );;
gap> abas:= List( bas, x -> Coefficients( B, x * a ) );;
gap> bbas:= List( bas, x -> Coefficients( B, x * b ) );;
</pre></div>

The library character table with identifier <code class="code">"P12/G1/L2/V1/ext2"</code> belongs to the factor group of S modulo the normal subgroup M(2 L_2). The action on an orbit modulo 2 is not faithful, its kernel contains the centre of SL(2,7). We can compute a faithful representation by acting on pairs: One entry is the usual vector and the other entry carries the action of the point group.

<div class="example"><pre>
gap> v:= List( [ 1, 2 ], i -> 0 * a[1] );;
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