CharTableDirectProduct( tbl1, tbl2 )
returns the character table of the direct product of the groups given by the character tables tbl1 and tbl2.
The matrix of irreducibles is the Kronecker product (see
KroneckerProduct) of tbl1.irreducibles with tbl2.irreducibles.
gap> c2:= CharTable( "Cyclic", 2 );; s2:= CharTable( "Symmetric", 2 );;
gap> SortCharactersCharTable( s2 );;
gap> v4:= CharTableDirectProduct( c2, s2 );;
gap> PrintCharTable( v4 );
rec( size := 4, identifier := "C2xS2", centralizers :=
[ 4, 4, 4, 4 ], order := 4, name := "C2xS2", classparam :=
[ [ [ 1, 0 ], [ 1, [ 1, 1 ] ] ], [ [ 1, 0 ], [ 1, [ 2 ] ] ],
[ [ 1, 1 ], [ 1, [ 1, 1 ] ] ], [ [ 1, 1 ], [ 1, [ 2 ] ] ]
], orders := [ 1, 2, 2, 2 ], powermap := [ , [ 1, 1, 1, 1 ]
], irreducibles := [ [ 1, 1, 1, 1 ], [ 1, -1, 1, -1 ],
[ 1, 1, -1, -1 ], [ 1, -1, -1, 1 ] ], irredinfo := [ rec(
charparam := [ [ 1, 0 ], [ 1, [ 2 ] ] ] ), rec(
charparam := [ [ 1, 0 ], [ 1, [ 1, 1 ] ] ] ), rec(
charparam := [ [ 1, 1 ], [ 1, [ 2 ] ] ] ), rec(
charparam := [ [ 1, 1 ], [ 1, [ 1, 1 ] ] ] ) ], charparam :=
[ ], fusionsource := [ [ 'C', '2' ], "S2" ], fusions := [ rec(
name := [ 'C', '2' ],
map := [ 1, 1, 2, 2 ],
type := "factor" ), rec(
name := "S2",
map := [ 1, 2, 1, 2 ],
type := "factor" ) ], classes :=
[ 1, 1, 1, 1 ], operations := CharTableOps )
gap> c2.fusions;
[ rec(
map := [ 1, 3 ],
type := "normal",
name := "C2xS2" ) ]
Note: The result will contain those p-th powermaps for primes p where both tbl1 and tbl2 contain the p-th powermap. Additionally, if one of the tables contains it, and p does not divide the order of the other table, and the p-th powermap is uniquely determined (see Powermap), it will be computed; then the table of the direct product will contain the p-th powermap, too.
GAP 3.4.4