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#@local g,t,lin
gap> START_TEST("ctbl.tst");
# `ClassPositionsOf...' for the trivial group (which usually causes trouble)
gap> g:= TrivialGroup( IsPermGroup );;
gap> t:= CharacterTable( g );;
gap> ClassPositionsOfAgemo( t, 2 );
[ 1 ]
gap> ClassPositionsOfCentre( t );
[ 1 ]
gap> ClassPositionsOfDerivedSubgroup( t );
[ 1 ]
gap> ClassPositionsOfDirectProductDecompositions( t );
[ ]
gap> ClassPositionsOfElementaryAbelianSeries( t );
[ [ 1 ] ]
gap> ClassPositionsOfFittingSubgroup( t );
[ 1 ]
gap> ClassPositionsOfLowerCentralSeries( t );
[ [ 1 ] ]
gap> ClassPositionsOfMaximalNormalSubgroups( t );
[ ]
gap> ClassPositionsOfNormalClosure( t, [ 1 ] );
[ 1 ]
gap> ClassPositionsOfNormalSubgroups( t );
[ [ 1 ] ]
gap> ClassPositionsOfUpperCentralSeries( t );
[ [ 1 ] ]
gap> ClassPositionsOfSolvableResiduum( t );
[ 1 ]
gap> ClassPositionsOfSupersolvableResiduum( t );
[ 1 ]
gap> ClassPositionsOfCentre( TrivialCharacter( t ) );
[ 1 ]
gap> ClassPositionsOfKernel( TrivialCharacter( t ) );
[ 1 ]
# Display for the table of the trivial group
gap> Display( CharacterTable( CyclicGroup( 1 ) ) );
CT1
1a
X.1 1
# Display with unusual parameters
gap> t:= CharacterTable( SymmetricGroup( 3 ) );; Irr( t );;
gap> Display( t, rec( centralizers:= false ) );
CT2
1a 2a 3a
2P 1a 1a 3a
3P 1a 2a 1a
X.1 1 -1 1
X.2 2 . -1
X.3 1 1 1
gap> Display( t, rec( centralizers:= "ATLAS" ) );
CT2
6 2 3
1a 2a 3a
2P 1a 1a 3a
3P 1a 2a 1a
X.1 1 -1 1
X.2 2 . -1
X.3 1 1 1
gap> Display( t, rec( chars:= 1 ) );
CT2
2 1 1 .
3 1 . 1
1a 2a 3a
2P 1a 1a 3a
3P 1a 2a 1a
X.1 1 -1 1
gap> Display( t, rec( chars:= [ 2, 3 ] ) );
CT2
2 1 1 .
3 1 . 1
1a 2a 3a
2P 1a 1a 3a
3P 1a 2a 1a
X.2 2 . -1
X.3 1 1 1
gap> Display( t, rec( chars:= PermChars( t ) ) );
CT2
2 1 1 .
3 1 . 1
1a 2a 3a
2P 1a 1a 3a
3P 1a 2a 1a
Y.1 1 1 1
Y.2 2 . 2
Y.3 3 1 .
Y.4 6 . .
gap> Display( t, rec( chars:= PermChars( t ), letter:= "P" ) );
CT2
2 1 1 .
3 1 . 1
1a 2a 3a
2P 1a 1a 3a
3P 1a 2a 1a
P.1 1 1 1
P.2 2 . 2
P.3 3 1 .
P.4 6 . .
gap> Display( t, rec( classes:= 1 ) );
CT2
2 1
3 1
1a
2P 1a
3P 1a
X.1 1
X.2 2
X.3 1
gap> Display( t, rec( classes:= [ 2, 3 ] ) );
CT2
2 1 .
3 . 1
2a 3a
2P 1a 3a
3P 2a 1a
X.1 -1 1
X.2 . -1
X.3 1 1
gap> Display( t, rec( indicator:= true ) );
CT2
2 1 1 .
3 1 . 1
1a 2a 3a
2P 1a 1a 3a
3P 1a 2a 1a
2
X.1 + 1 -1 1
X.2 + 2 . -1
X.3 + 1 1 1
gap> Display( t, rec( indicator:= [ 2, 3 ] ) );
CT2
2 1 1 .
3 1 . 1
1a 2a 3a
2P 1a 1a 3a
3P 1a 2a 1a
2 3
X.1 + 0 1 -1 1
X.2 + 1 2 . -1
X.3 + 1 1 1 1
gap> Display( t, rec( indicator:= [ 1, 2 ] ) );
CT2
2 1 1 .
3 1 . 1
1a 2a 3a
2P 1a 1a 3a
3P 1a 2a 1a
1 2
X.1 0 + 1 -1 1
X.2 0 + 2 . -1
X.3 1 + 1 1 1
gap> Display( t, rec( powermap:= false ) );
CT2
2 1 1 .
3 1 . 1
1a 2a 3a
X.1 1 -1 1
X.2 2 . -1
X.3 1 1 1
gap> Display( t, rec( powermap:= 2 ) );
CT2
2 1 1 .
3 1 . 1
1a 2a 3a
2P 1a 1a 3a
X.1 1 -1 1
X.2 2 . -1
X.3 1 1 1
gap> Display( t, rec( powermap:= [ 2, 3 ] ) );
CT2
2 1 1 .
3 1 . 1
1a 2a 3a
2P 1a 1a 3a
3P 1a 2a 1a
X.1 1 -1 1
X.2 2 . -1
X.3 1 1 1
gap> # Note that the 'ATLAS' option for power maps has the desired effect
gap> # only if the function 'CambridgeMaps' is bound during the tests,
gap> # which depends on the loaded packages; we omit this test.
gap> # Display( t, rec( powermap:= "ATLAS" ) );
gap> Display( t,
> rec( charnames:= List( CharacterParameters( t ), String ) ) );
CT2
2 1 1 .
3 1 . 1
1a 2a 3a
2P 1a 1a 3a
3P 1a 2a 1a
[ 1, [ 1, 1, 1 ] ] 1 -1 1
[ 1, [ 2, 1 ] ] 2 . -1
[ 1, [ 3 ] ] 1 1 1
gap> Display( t,
> rec( classnames:= List( ClassParameters( t ), String ) ) );
CT2
2 1 1 .
3 1 . 1
[ 1, [ 1, 1, 1 ] ] [ 1, [ 2, 1 ] ] [ 1, [ 3 ] ]
2P [ 1, [ 1, 1, 1 ] ] [ 1, [ 1, 1, 1 ] ] [ 1, [ 3 ] ]
3P [ 1, [ 1, 1, 1 ] ] [ 1, [ 2, 1 ] ] [ 1, [ 1, 1, 1 ] ]
X.1 1 -1 1
X.2 2 . -1
X.3 1 1 1
gap> Display( t,
> rec( characterField:= true ) );
CT2
2 1 1 .
3 1 . 1
1a 2a 3a
2P 1a 1a 3a
3P 1a 2a 1a
d
X.1 1 1 -1 1
X.2 1 2 . -1
X.3 1 1 1 1
# viewing and printing of character tables with stored groups
gap> t:= CharacterTable( DihedralGroup( 8 ) );;
gap> View( t ); Print( "\n" );
CharacterTable( <pc group of size 8 with 3 generators> )
gap> Print( t, "\n" );
CharacterTable( Group( [ f1, f2, f3 ] ) )
gap> ViewString( t );
"CharacterTable( <group of size 8 with \>3\< generators> )"
gap> PrintString( t );
"CharacterTable( \"Group( \>[ f1, f2, f3 ]\<\> )\< )"
gap> t:= CharacterTable( SymmetricGroup( 5 ) );;
gap> View( t ); Print( "\n" );
CharacterTable( Sym( [ 1 .. 5 ] ) )
gap> Print( t, "\n" );
CharacterTable( SymmetricGroup( [ 1 .. 5 ] ) )
gap> ViewString( t );
"CharacterTable( Sym( [ 1 .. 5 ] ) )"
gap> PrintString( t );
"CharacterTable( \"Group( \>[ (1,2,3,4,5), (1,2) ]\<\> )\< )"
# entries of mutable attributes are immutable
gap> t:= CharacterTable( SymmetricGroup( 5 ) );
CharacterTable( Sym( [ 1 .. 5 ] ) )
gap> PowerMap( t, 2 );; PowerMap( t, 3 );;
gap> Length( ComputedPowerMaps( t ) );
3
gap> IsMutable( ComputedPowerMaps( t ) );
true
gap> ForAny( ComputedPowerMaps( t ), IsMutable );
false
gap> Indicator( t, 2 );;
gap> Length( ComputedIndicators( t ) );
2
gap> IsMutable( ComputedIndicators( t ) );
true
gap> ForAny( ComputedIndicators( t ), IsMutable );
false
gap> t:= CharacterTable( CyclicGroup( 2 ) );;
gap> SetIdentifier( t, "C2" );
gap> ComputedIndicators( t )[2]:= [ Unknown(), Unknown() ];;
gap> Display( t, rec( indicator:= true ) );
C2
2 1 1
1a 2a
2P 1a 1a
2
X.1 ? 1 1
X.2 ? 1 -1
gap> PrimeBlocks( t, 2 );;
gap> Length( ComputedPrimeBlockss( t ) );
2
gap> IsMutable( ComputedPrimeBlockss( t ) );
true
gap> ForAny( ComputedPrimeBlockss( t ), IsMutable );
false
# create certain Brauer tables ...
# ... of p-solvable groups
gap> t:= CharacterTable( SymmetricGroup( 4 ) );;
gap> IsCharacterTable( t mod 2 );
true
gap> IsCharacterTable( t mod 3 );
true
# ... where all Brauer characters lift to characteristic zero
gap> g:= PSL(2,5);;
gap> t:= CharacterTable( g );;
gap> IsCharacterTable( t mod 3 );
true
gap> IsCharacterTable( t mod 5 );
true
# ... where all Brauer characters lift to characteristic zero,
# and the p-core is nontrivial
gap> g:= DirectProduct( SymmetricGroup(5), SymmetricGroup( 3 ) );;
gap> t:= CharacterTable( g );;
gap> IsCharacterTable( t mod 3 );
true
# ... where the Brauer tables of the factors of a product can be computed
gap> g:= AlternatingGroup( 5 );;
gap> t:= CharacterTable( g );;
gap> t:= CharacterTableDirectProduct( t, t );;
gap> IsCharacterTable( t mod 5 );
true
# test a bugfix
gap> g:= SmallGroup( 96, 3 );;
gap> t:= CharacterTable( g );;
gap> ClassPositionsOfLowerCentralSeries( t );
[ [ 1 .. 12 ], [ 1, 3, 4, 5, 6, 9, 10, 11 ] ]
gap> g:= SmallGroup( 3^5, 22 );;
gap> t:= CharacterTable( g );;
gap> ClassPositionsOfLowerCentralSeries( t );
[ [ 1 .. 35 ], [ 1, 4, 6, 12, 15 ], [ 1, 6, 15 ], [ 1 ] ]
gap> g:= SmallGroup( 96, 66 );;
gap> t:= CharacterTable( g );;
gap> ClassPositionsOfSupersolvableResiduum( t );
[ 1, 5, 6 ]
# test another bugfix ('IsSimple' does not imply 'IsPerfect')
gap> t:= CharacterTable( CyclicGroup( 2 ) );;
gap> IsSimpleCharacterTable( t );
true
gap> IsPerfectCharacterTable( t );
false
# compute indicators
gap> t:= CharacterTable( SymmetricGroup( 4 ) );;
gap> Indicator( t, 2 );
[ 1, 1, 1, 1, 1 ]
gap> Indicator( t mod 3, 2 );
[ 1, 1, 1, 1 ]
gap> Indicator( t mod 2, 2 );
[ 1, 1 ]
# linear characters
gap> lin:= LinearCharacters( SmallGroup( 24, 12 ) );;
gap> Length( lin );
2
gap> ForAll( lin, HasIsIrreducibleCharacter );
true
gap> lin:= LinearCharacters( SymmetricGroup( 4 ) );;
gap> Length( lin );
2
gap> ForAll( lin, HasIsIrreducibleCharacter );
true
gap> lin:= LinearCharacters( SymmetricGroup( 4 ), 2 );;
gap> Length( lin );
1
gap> ForAll( lin, HasIsIrreducibleCharacter );
true
gap> lin:= LinearCharacters( CharacterTable( SymmetricGroup( 4 ) ) );;
gap> Length( lin );
2
gap> ForAll( lin, HasIsIrreducibleCharacter );
true
gap> lin:= LinearCharacters( CharacterTable( SymmetricGroup( 4 ), 2 ) );;
gap> Length( lin );
1
gap> ForAll( lin, HasIsIrreducibleCharacter );
true
# irreducibility flag
gap> ForAll( Irr( SymmetricGroup( 4 ) ), HasIsIrreducibleCharacter );
true
gap> ForAll( Irr( SymmetricGroup( 4 ), 2 ), HasIsIrreducibleCharacter );
true
gap> ForAll( Irr( CharacterTable( SymmetricGroup( 4 ) ) ),
> HasIsIrreducibleCharacter );
true
gap> ForAll( Irr( CharacterTable( SymmetricGroup( 4 ), 2 ) ),
> HasIsIrreducibleCharacter );
true
gap> HasIsIrreducibleCharacter( TrivialCharacter( SymmetricGroup( 4 ) ) );
true
##
gap> STOP_TEST( "ctbl.tst" );
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