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 $K$, ${\text {\rm char}} K\nmid |G|$, 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 $H^{1}({\text {\rm Gal}}(L/K), {\text {\rm GL}}(n,L))$ vanishes for all $n\ge 1$.