Metacyclic group

In group theory, a metacyclic group is an extension of a cyclic group by a cyclic group. Equivalently, a metacyclic group is a group ${\displaystyle G}$ having a cyclic normal subgroup ${\displaystyle N}$, such that the quotient ${\displaystyle G/N}$ is also cyclic.

A group ${\displaystyle G}$ is metacyclic is there is a normal subgroup ${\displaystyle N}$ such that the sequence below is exact:^[1]

${\displaystyle 1\rightarrow N\rightarrow G\rightarrow G/N\rightarrow 1,\,}$