Faithful irreducible representations of metacyclic groups (Q1340982)

The author proves some results on the representations of metacyclic groups. One of these is the following: A metacyclic group has a faithful irreducible representation over a field \(\mathbb{F}\) is and only if the center of the group is cyclic and the characteristic of \(\mathbb{F}\) does not divide the order of the Fitting group. As an application of these results, it is shown that, for an odd prime \(p\), over the field of complex numbers, there are exactly \(p - 1\) linear isomorphic types of linear groups abstractly isomorphic to some metacyclic group of order \(p^ 6\). The author seems unaware of the fact that there are related results in the literature.