The IsDicyclic( G ) command determines whether the finite group G is isomorphic to a dicyclic group of some order, without using an (expensive) isomorphism test. It returns the value true if G is isomorphic to a dicyclic group and returns false otherwise.

The IsQuaternionGroup( G ) command returns the value true if the group G is isomorphic to a (generalized) quaternion group, and returns false otherwise.

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

$G≔\mathrm{Group}\left(\left[\mathrm{Perm}\left(\left[\left[2\,5\,10\,16\,24\,14\,27\,4\right]\,\left[7\,19\,17\,26\,22\,9\,15\,13\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[2\,13\,24\,26\right]\,\left[4\,7\,16\,22\right]\,\left[5\,15\,14\,17\right]\,\left[9\,27\,19\,10\right]\right]\right)\right]\right)$

$G≔⟨\left(2\,5\,10\,16\,24\,14\,27\,4\right)\left(7\,19\,17\,26\,22\,9\,15\,13\right)\,\left(2\,13\,24\,26\right)\left(4\,7\,16\,22\right)\left(5\,15\,14\,17\right)\left(9\,27\,19\,10\right)⟩$

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

$16$

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

$\mathrm{true}$

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

$\mathrm{true}$

$G≔\mathrm{Group}\left(\left[\mathrm{Perm}\left(\left[\left[2\,5\,10\,4\right]\,\left[7\,19\,17\,15\,13\right]\,\left[14\,27\,16\,24\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[2\,27\,10\,24\right]\,\left[4\,16\,5\,14\right]\,\left[13\,19\right]\,\left[15\,17\right]\right]\right)\right]\right)$

$G≔⟨\left(2\,5\,10\,4\right)\left(7\,19\,17\,15\,13\right)\left(14\,27\,16\,24\right)\,\left(2\,27\,10\,24\right)\left(4\,16\,5\,14\right)\left(13\,19\right)\left(15\,17\right)⟩$

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

$\mathrm{true}$

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

$\mathrm{false}$

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

$40$

$\mathrm{IsDicyclic}\left(\mathrm{DihedralGroup}\left(18\right)\right)$

$\mathrm{false}$

$\mathrm{map}\left(\mathrm{IsDicyclic}\,\mathrm{AllSmallGroups}\left(24\right)\right)$

$\left[\mathrm{false}\,\mathrm{false}\,\mathrm{false}\,\mathrm{true}\,\mathrm{false}\,\mathrm{false}\,\mathrm{false}\,\mathrm{false}\,\mathrm{false}\,\mathrm{false}\,\mathrm{false}\,\mathrm{false}\,\mathrm{false}\,\mathrm{false}\,\mathrm{false}\right]$