metacyclic group

Definition

A metacyclic group is a group $G$ that possesses a normal subgroup $N$ such that $N$ and $G/N$ are both cyclic.

•

All cyclic groups, and direct products of two cyclic groups.
•

All dihedral groups (including the infinite dihedral group).
•

All finite groups whose Sylow subgroups are cyclic (and so, in particular, all finite groups of squarefree (http://planetmath.org/SquareFreeNumber) order).

Subgroups (http://planetmath.org/Subgroup) and quotients (http://planetmath.org/QuotientGroup) of metacyclic groups are also metacyclic.

Metacyclic groups are obviously supersolvable, with Hirsch length at most $2$ .