Classification of finite p-groups with self-centralizing cyclic normal subgroup

Statement
A finite $$p$$-group (a group of prime power order) $$P$$, with a self-centralizing cyclic normal subgroup $$H$$, could be either of the following:


 * If $$p$$ is an odd prime, it is a faithful semidirect product of cyclic p-groups: it is the semidirect product of $$H$$ with a cyclic subgroup of the automorphism group of $$H$$.
 * If $$p = 2$$, there are a number of possibilities:

Related facts

 * Classification of finite p-groups with cyclic maximal subgroup
 * Classification of metacyclic p-groups