Metacyclic group

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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

$1 \rightarrow C_m \rightarrow G \rightarrow C_n \rightarrow 1.$

Equivalently, a metacyclic group is a group G having a cyclic normal subgroup N, such that the quotient G/N is also cyclic.

[edit] Properties

Metacyclic groups are both supersolvable and metabelian.

An interesting property of metacyclic groups is that every complex irreducible representation of every metacyclic group can be easily constructed from a diagram.

[edit] Examples

Any cyclic group is metacyclic.
The direct product or semidirect product of two cyclic groups is metacyclic. These include the dihedral groups, the quasidihedral groups, and the dicyclic 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.