Writing projective representations over subfields

Let G= X be an absolutely irreducible subgroup of GL(d,K), and let F be a proper subfield of the finite field K. We present a practical algorithm to decide constructively whether or not G is conjugate to a subgroup of GL(d,F).K×, where K× denotes the centre of GL(d,K). If the derived group of G also acts absolutely irreducibly, then the algorithm is Las Vegas and costs O(Xd3+d2logF) arithmetic operations in K. This work forms part of a recognition project based on Aschbacherʹs classification of maximal subgroups of GL(d,K).