GAP Manual: 48.18. CharTableWreathSymmetric

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

medskipThe record component classtext 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

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