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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

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

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.

References

A. L. Shmel'kin (2001), "Metacyclic group", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Mathematics Encyclopedia