Characters, conjugacy classes and centrally large subgroups of p-groups of small rank

In this paper, we investigate the irreducible characters and conjugacy classes of a metacyclic p-group where p is an odd prime number. We show that, if |G′|=pn, then G has precisely |G:G′|(p−1)/pk+1 irreducible characters of degree pk, for k=1,…,n. We also show that G contains precisely |Z(G)|(pk−pk−2) conjugacy classes of length pk, for k=1,…,n. Moreover, we have |G|=|Z(G)|⋅2|G′|. We also investigate the concept of a centrally large subgroup, as introduced by Glauberman. In addition, we show that these results are, in general, false for p=2. Furthermore, we consider p-groups of small rank, for p>2.