# Classification of metacyclic p-groups

From Groupprops

This article gives a classification statement for certain kinds of groups of prime power order, subject to additional constraints.

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This page classifies all finite -groups that are metacyclic: there is a cyclic normal subgroup with a cyclic quotient group.

## Related facts

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

## First step of the classification: classifying for fixed normal subgroup and fixed quotient=

Let be natural numbers. We first determine all congruence classes of extensions (up to automorphism of the normal subgroup and of the quotient) with a cyclic normal subgroup of order and a cyclic quotient group of order .

(Note that at this step of the classification, we may get different groups that are isomorphic as groups, but not equivalent as extensions with the specified normal subgroup and specified quotient.