MatRepresentationsPGroup( G )
MatRepresentationsPGroup( G [, int ] )
MatRepresentationsPGroup( G ) returns a list of homomorphisms from
the finite polycyclic group G to irreducible complex matrix groups.
These matrix groups form a system of representatives of the complex
irreducible representations of G.
MatRepresentationsPGroup( G, int ) returns only the int-th
Let G be a finite polycyclic group with an abelian normal subgroup N
such that the factorgroup <G> / <N> is supersolvable.
MatRepresentationsPGroup uses the algorithm described in Bau91.
Note that for such groups all such representations are equivalent to
monomial ones, and in fact
MatRepresentationsPGroup only returns
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.
the irreducible representations of G with kernel containing M then.
gap> g:= SolvableGroup( 6, 2 ); S3 gap> MatRepresentationsPGroup( g ); [ GroupHomomorphismByImages( S3, Group( [ [ 1 ] ] ), [ a, b ], [ [ [ 1 ] ], [ [ 1 ] ] ] ), GroupHomomorphismByImages( S3, Group( [ [ -1 ] ] ), [ a, b ], [ [ [ -1 ] ], [ [ 1 ] ] ] ), GroupHomomorphismByImages( S3, Group( [ [ 0, 1 ], [ 1, 0 ] ], [ [ E(3), 0 ], [ 0, E(3)^2 ] ] ), [ a, b ], [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ E(3), 0 ], [ 0, E(3)^2 ] ] ] ) ]
CharTablePGroup can be used to compute the character table of a group
with the above properties (see CharTablePGroup).
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