GAP Manual: 48.26. CharTablePGroup

CharTablePGroup( G )

CharTablePGroup returns the character table of the finite polycyclic group G, and stores it in G.charTable. Do not change the order of G.conjugacyClasses after having called CharTablePGroup.

Let G be a finite polycyclic group with an abelian normal subgroup N such that the factorgroup G / N is supersolvable. CharTablePGroup uses the algorithm described in Bau91.

If G has not the property stated above, a system of representatives of irreducible representations and characters only for the factor group G / M can be computed using this algorithm, where M is the derived subgroup of the supersolvable residuum of G. In this case first a warning is printed. CharTablePGroup returns an incomplete table containing exactly those irreducibles with kernel containing M.

    gap> t:= CharTablePGroup( SolvableGroup( 8, 4 ) );;
    gap> PrintCharTable( t );
    rec( size := 8, centralizers := [ 8, 8, 4, 4, 4 ], classes :=
    [ 1, 1, 2, 2, 2 ], orders := [ 1, 2, 2, 2, 4 ], irreducibles :=
    [ [ 1, 1, 1, 1, 1 ], [ 1, 1, -1, 1, -1 ], [ 1, 1, 1, -1, -1 ],
      [ 1, 1, -1, -1, 1 ], [ 2, -2, 0, 0, 0 ]
     ], operations := CharTableOps, order := 8, powermap :=
    [ , [ 1, 1, 1, 1, 2 ]
     ], identifier := "D8", name := "D8", group := D8 )

MatRepresentationsPGroup can be used to compute representatives of the complex irreducible representations (see MatRepresentationsPGroup).