Quasi-permutation representations of 2-groups satisfying the Hasse principle

Abstract

In Behravesh (J Lond Math Soc 55(2):251–260, 1997), c(G), q(G) and p(G) are defined for a finite group G. In this paper, we will calculate c(G), q(G) and p(G) for some 2-groups G satisfying the Hasse principle in Fuma and Ninomiya (Math J Okayama Univ 46:31–38, 2004). We will consider

$$G=\langle x, y, z: x^{2^{m-2}}=y^{2}=z^2=1, [x, y]=[y, z]=1, x^{z}=xy \rangle$$

where m ≥ 4. By comparing the character tables and Galois conjugacy classes of Irr(G) and Irr(Z(G)), we will show that

$$c(G)=q(G)=p(G)= 2c(Z(G))=2^{m-2}+4.$$