On the relation of character codegrees and the minimal faithful quasi-permutation representation degree of p -groups

For a finite group G, we denote by c(G), the minimal degree of a faithful representation of G by quasi-permutation matrices over C. For an irreducible character χ of G, the codegree of χ is defined as cod(χ)=|G/ker(χ)|/χ(1). In this article, we establish equality between c(G) and a Q≥0-sum of codegrees of some irreducible characters of a non-abelian p-group G of odd order. We also study the relation between c(G) and irreducible character codegrees for various classes of non-abelian p-groups, such as, p-groups with cyclic center, maximal class p-groups, GVZ p-groups, and others.