Computing the exterior center of metacyclic p-groups of nilpotency class at least three

Abstract

A group G is metacyclic if it contains a cyclic normal subgroup K such that GK is also cyclic. Metacyclic p-groups classified by dfferent authors. A group is called capable if it is a central factor group. The purpose of this study is to compute the exterior center of finite non-abelian metacyclic p-groups, p is an odd prime, for some small order groups using Groups, Algorithms and Programming (GAP) software. We also determine which of these groups are capable.