A group $G$ is metacyclic if it is an extension of a cyclic group by another cyclic group. The extension need not be proper; that is, a cyclic group is metacyclic.

The IsMetacyclic( G ) command attempts to determine whether the group G is metacyclic.  It returns true if G is metacyclic and returns false otherwise.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$\mathrm{IsMetacyclic}\left(\mathrm{Symm}\left(3\right)\right)$

$\mathrm{true}$

$\mathrm{IsMetacyclic}\left(\mathrm{Symm}\left(4\right)\right)$

$\mathrm{false}$

$\mathrm{IsMetacyclic}\left(\mathrm{MetacyclicGroup}\left(3\,2\,3\right)\right)$

$\mathrm{true}$

$\mathrm{IsMetacyclic}\left(\mathrm{FrobeniusGroup}\left(186\,1\right)\right)$

$\mathrm{true}$

$\mathrm{IsMetacyclic}\left(\mathrm{CyclicGroup}\left(9\right)\right)$

$\mathrm{true}$

The GroupTheory[IsMetacyclic] command was introduced in Maple 2019.

For more information on Maple 2019 changes, see Updates in Maple 2019.