CharTableWreathSymmetric( tbl, n )
returns the character table of the wreath product of an arbitrary group G with the full symmetric group S_n, where tbl is the character table of G.
gap> c3:= CharTable("Cyclic", 3);;
gap> wr:= CharTableWreathSymmetric(c3, 2);;
gap> PrintCharTable( wr );
rec( size := 18, identifier := "C3wrS2", centralizers :=
[ 18, 9, 9, 18, 9, 18, 6, 6, 6 ], classes :=
[ 1, 2, 2, 1, 2, 1, 3, 3, 3 ], orders := [ 1, 3, 3, 3, 3, 3, 2, 6, 6
], irredinfo := [ rec(
charparam := [ [ 1, 1 ], [ ], [ ] ] ), rec(
charparam := [ [ 1 ], [ 1 ], [ ] ] ), rec(
charparam := [ [ 1 ], [ ], [ 1 ] ] ), rec(
charparam := [ [ ], [ 1, 1 ], [ ] ] ), rec(
charparam := [ [ ], [ 1 ], [ 1 ] ] ), rec(
charparam := [ [ ], [ ], [ 1, 1 ] ] ), rec(
charparam := [ [ 2 ], [ ], [ ] ] ), rec(
charparam := [ [ ], [ 2 ], [ ] ] ), rec(
charparam := [ [ ], [ ], [ 2 ] ] )
], name := "C3wrS2", order := 18, classparam :=
[ [ [ 1, 1 ], [ ], [ ] ], [ [ 1 ], [ 1 ], [ ] ],
[ [ 1 ], [ ], [ 1 ] ], [ [ ], [ 1, 1 ], [ ] ],
[ [ ], [ 1 ], [ 1 ] ], [ [ ], [ ], [ 1, 1 ] ],
[ [ 2 ], [ ], [ ] ], [ [ ], [ 2 ], [ ] ], [ [ ], [ ], [ 2 ] ]
], powermap := [ , [ 1, 3, 2, 6, 5, 4, 1, 4, 6 ],
[ 1, 1, 1, 1, 1, 1, 7, 7, 7 ] ], irreducibles :=
[ [ 1, 1, 1, 1, 1, 1, -1, -1, -1 ],
[ 2, -E(3)^2, -E(3), 2*E(3), -1, 2*E(3)^2, 0, 0, 0 ],
[ 2, -E(3), -E(3)^2, 2*E(3)^2, -1, 2*E(3), 0, 0, 0 ],
[ 1, E(3), E(3)^2, E(3)^2, 1, E(3), -1, -E(3), -E(3)^2 ],
[ 2, -1, -1, 2, -1, 2, 0, 0, 0 ],
[ 1, E(3)^2, E(3), E(3), 1, E(3)^2, -1, -E(3)^2, -E(3) ],
[ 1, 1, 1, 1, 1, 1, 1, 1, 1 ],
[ 1, E(3), E(3)^2, E(3)^2, 1, E(3), 1, E(3), E(3)^2 ],
[ 1, E(3)^2, E(3), E(3), 1, E(3)^2, 1, E(3)^2, E(3) ]
], operations := CharTableOps )
medskip
gap> DisplayCharTable( wr );
C3wrS2
2 1 . . 1 . 1 1 1 1
3 2 2 2 2 2 2 1 1 1
1a 3a 3b 3c 3d 3e 2a 6a 6b
2P 1a 3b 3a 3e 3d 3c 1a 3c 3e
3P 1a 1a 1a 1a 1a 1a 2a 2a 2a
X.1 1 1 1 1 1 1 -1 -1 -1
X.2 2 A /A B -1 /B . . .
X.3 2 /A A /B -1 B . . .
X.4 1 -/A -A -A 1 -/A -1 /A A
X.5 2 -1 -1 2 -1 2 . . .
X.6 1 -A -/A -/A 1 -A -1 A /A
X.7 1 1 1 1 1 1 1 1 1
X.8 1 -/A -A -A 1 -/A 1 -/A -A
X.9 1 -A -/A -/A 1 -A 1 -A -/A
A = -E(3)^2
= (1+ER(-3))/2 = 1+b3
B = 2*E(3)
= -1+ER(-3) = 2b3
medskip
The record component classparam contains the sequences of partitions
that parametrize the classes as well as the characters of the wreath
product. Note that this parametrization prevents the principal character
from being the first one in the list irreducibles.
medskip
GAP 3.4.4