Computing irreducible representations of supersolvable groups over small finite fields

We present an algorithm to compute a full set of irreducible representations of a supersolvable group G over a finite field K, charK |G|, which is not assumed to be a splitting field of G. The main subroutines of our algorithm are a modification of the algorithm of Baum and Clausen (Math. Comp. 63 (1994), 351–359) to obtain information on algebraically conjugate representations, and an effective version of Speiser’s generalization of Hilbert’s Theorem 90 stating that H1(Gal(L/K),GL(n,L)) vanishes for all n ≥ 1.