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) VL = 0 for any normal subgroup L of G with prime power index. (2) dim VP > 2 dim VH 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.
"The gap hypothesis for finite groups which have an abelian quotient group not of order a power of 2." J. Math. Soc. Japan 64 (1) 91 - 106, January, 2012. https://doi.org/10.2969/jmsj/06410091