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 quasi-permutation matrices over the complex field C. In this paper we will assume that, G is a p-group 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)):