Groups with Two Extreme Character Degrees and their Minimal Faithful Representations

for a finite group G, we denote by p(G) the minimal degree of faithful permutation representations of G, and denote by c(G), the minimal degree of faithful representation of G by quasipermutation matrices over the complex field C. In this paper we will assume that, G is a pgroup of exponent p and class 2, where p is prime and cd(G) = {1, |G : Z(G)|^1/2}. Then we will show that c(G)≤ |G : Z(G)|^{1/2} c(Z(G)) , p(G) ≤ |G : Z(G)|^{1/2}p(Z(G)):