Gap modules for semidirect product groups

Let G be a finite group not of prime power order. A gap G-module V is a finite-dimensional real G-representation space satisfying the following two conditions. The first is the condition dim V^p >2dim V^H for all P<H ≤ G such that P is of prime power order and the other is the condition that V has only one H-fixed point 0 for all large subgroups H: precisely to say, H ? L(G). If there exists a gap G-module, then G is called a gap group. We study G-modules induced from C-modules for subgroups C of G and obtain a sufficient condition for G to become a gap group. Consequently, we show that non-solvable general linear groups and the automorphism groups of sporadic groups are all gap groups.