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.

Examples

 * Any cyclic group is metacyclic.
 * The direct product or semidirect product of two cyclic groups is metacyclic. These include the dihedral groups and the quasidihedral groups.
 * The dicyclic groups are metacyclic. (Note that a dicyclic group is not necessarily a semidirect product of two cyclic groups.)
 * Every finite group of squarefree order is metacyclic.
 * More generally every Z-group is metacyclic. A Z-group is a group whose Sylow subgroups are cyclic.