80.1 ComplexReflectionGroup

ComplexReflectionGroup( STnumber )

ComplexReflectionGroup( p, q, r )

The first form of ComplexReflectionGroup returns the complex reflection group which has Shephard-Todd number STnumber, see ST54. The second form returns the imprimitive complex reflection group G(p,q,r).

    gap> G := ComplexReflectionGroup( 4 );
    ComplexReflectionGroup(4)
    gap> ReflectionDegrees( G );
    [ 4, 6 ]
    gap> Size( G );
    24
    gap> q := X( Cyclotomics );; q.name := "q";;
    gap> FakeDegrees( G, q );
    [ q^0, q^8, q^4, q^7 + q^5, q^3 + q, q^5 + q^3, q^6 + q^4 + q^2 ] 

Complex reflection groups are represented as permutation group records

with the following additional fields:

roots:

a set of complex roots in V, given as a matrix, on which W has a faithful permutation representation. roots[1..semisimpleRank] should be linearly independent. Roots are not always of same length, and sometimes the number of roots may be greater than the order of W!

semisimpleRank:

the dimension of the subspace of V generated by the roots (for an irreducible group, equal to the dimension of V).

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