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 (Formula presented.), the minimal degree of a faithful representation of G by quasi-permutation matrices over (Formula presented.). For an irreducible character (Formula presented.) of G, the codegree of (Formula presented.) is defined as (Formula presented.). In this article, we establish equality between (Formula presented.) and a (Formula presented.) -sum of codegrees of some irreducible characters of a non-abelian p-group G of odd order. We also study the relation between (Formula presented.) 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.