The IsDihedral( G ) command determines whether the finite group G is isomorphic to a dihedral group of some (even) order, without using an (expensive) isomorphism test. It returns the value true if G is isomorphic to ${\mathrm{D}}_n$, for some positive integer $n$, and returns false otherwise.

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

$G≔\mathrm{Group}\left(\left[\mathrm{Perm}\left(\left[\left[1\,2\,3\,4\,5\,6\,7\right]\,\left[8\,9\,10\,11\,12\,13\,14\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[2\,7\right]\,\left[3\,6\right]\,\left[4\,5\right]\,\left[9\,14\right]\,\left[10\,13\right]\,\left[11\,12\right]\right]\right)\right]\right)$

$G≔⟨\left(1\,2\,3\,4\,5\,6\,7\right)\left(8\,9\,10\,11\,12\,13\,14\right)\,\left(2\,7\right)\left(3\,6\right)\left(4\,5\right)\left(9\,14\right)\left(10\,13\right)\left(11\,12\right)⟩$

$\mathrm{GroupOrder}\left(G\right)$

$14$

$\mathrm{IsDihedral}\left(G\right)$

$\mathrm{true}$

$\mathrm{AreIsomorphic}\left(G\,\mathrm{DihedralGroup}\left(7\right)\right)$

$\mathrm{true}$

$\mathrm{IsDihedral}\left(\mathrm{QuaternionGroup}\left(\right)\right)$

$\mathrm{false}$

$G≔\mathrm{McLaughlinGroup}\left(\right)\:$

$a≔\mathrm{RandomInvolution}\left(G\right)\:$

$b≔\mathrm{RandomInvolution}\left(G\right)\:$

$\mathrm{IsDihedral}\left(\mathrm{Group}\left(\left[a\,b\right]\right)\right)$

$\mathrm{true}$

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

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