The gap hypothesis for finite groups which have an abelian quotient group not of order a power of 2

For a finite group G, an L(G)-free gap G-module V is a finite dimensional real G-representation space satisfying the two conditions: (1) V ^L = 0 for any normal subgroup L of G with prime power index. (2) dim V ^P > 2 dim V ^H for any P < H ≤ G such that P is of prime power order. A finite group G not of prime power order is called a gap group if there is an L (G)-free gap G-module. We give a necessary and sufficient condition for that G is a gap group for a finite group G satisfying that G/[G, G] is not a 2-group, where [G, G] is the commutator subgroup of G.