# Metacyclic group

In group theory, a metacyclic group is an extension of a cyclic group by a cyclic group. That is, it is a group G for which there is a short exact sequence

$1 \rightarrow K \rightarrow G \rightarrow H \rightarrow 1,\,$

where H and K are cyclic. Equivalently, a metacyclic group is a group G having a cyclic normal subgroup N, such that the quotient G/N is also cyclic.

## Properties

Metacyclic groups are both supersolvable and metabelian.