By a quasi-permutation matrix we mean a square matrix over the complex field C
with non-negative integral trace. Thus, every permutation matrix over C is a quasipermutation matrix. For a given finite group G, let p(G) denote the minimal degree of a faithful permutation representation of G (or of a faithful representation of G by permutation matrices), let q(G) denote the minimal degree of a faithful representation of G by quasi-permutation matrices over the rational field Q, and let c(G) be the minimal degree of a faithful representation of G by complex quasi-permutation matrices. In this paper, we will calculate the irreducible modules and characters of metacyclic 2-groups and we also find c(G), q(G) and p(G) for these groups.