Quasi-permutation representations of metacyclic \(2\)-groups with cyclic center (Q1305312)

A square matrix over \(\mathbb{C}\) is said to be a quasi-permutation matrix, if its trace is a nonnegative rational integer. For a finite group \(G\), let \(p(G)\), \(q(G)\), \(c(G)\) be the minimal degree of a faithful permutation, quasi-permutation over \(\mathbb{Q}\), quasi-permutation complex representation of \(G\), respectively. In this paper these invariants are calculated for metacyclic 2-groups \(G\) with cyclic center. These groups are classified by \textit{Y. Iida} and \textit{T. Yamada} [SUT J. Math. 28, No. 1, 23-46 (1992; Zbl 0805.20006)]. If \(G\) is not generalized quaternion, all these invariants are equal. If \(G\) is generalized quaternion of order \(2^{n+1}\), then \(c(G)=2^n\) and \(p(G)=q(G)=2^{n+1}\).