Let $G$ be a finite of prime-power order. We say that $G$ is special if either $G$ is elementary abelian, or if the center, derived subgroup, and Frattini subgroup of $G$ all coincide and is elementary abelian. If, in addition, these coindicent subgroups of $G$ have prime order, then $G$ is said to be extraspecial. Note that non-trivial abelian groups are not extraspecial, since their centers and derived subgroups cannot be equal.

The IsSpecial( G ) command returns true if the permutation group G is a special $p$-group, for some prime number $p$.

The IsExtraspecial( G ) command returns true if the permutation group G is an extraspecial $p$-group, for some prime number $p$.

Both commands return false if the group G is not a $p$-group for any prime number $p$.

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

$\mathrm{IsSpecial}\left(\mathrm{Alt}\left(4\right)\right)$

$\mathrm{false}$

$\mathrm{IsSpecial}\left(\mathrm{CyclicGroup}\left(3\right)\right)$

$\mathrm{true}$

$\mathrm{IsExtraspecial}\left(\mathrm{CyclicGroup}\left(3\right)\right)$

$\mathrm{false}$

$\mathrm{IsSpecial}\left(\mathrm{CyclicGroup}\left(4\right)\right)$

$\mathrm{false}$

$\mathrm{IsSpecial}\left(\mathrm{ElementaryGroup}\left(11\,4\right)\right)$

$\mathrm{true}$

$\mathrm{IsExtraspecial}\left(\mathrm{ElementaryGroup}\left(11\,4\right)\right)$

$\mathrm{false}$

$\mathrm{IsSpecial}\left(\mathrm{DihedralGroup}\left(4\right)\right)$

$\mathrm{true}$

$\mathrm{IsSpecial}\left(\mathrm{DihedralGroup}\left(16\right)\right)$

$\mathrm{false}$

$\mathrm{GroupOrder}\left(\mathrm{Center}\left(\mathrm{DihedralGroup}\left(16\right)\right)\right)$

$2$

$\mathrm{GroupOrder}\left(\mathrm{DerivedSubgroup}\left(\mathrm{DihedralGroup}\left(16\right)\right)\right)$

$8$

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

$\mathrm{true}$

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

$\mathrm{true}$

$\mathrm{map}\left(\mathrm{GroupOrder}\,\left[\mathrm{Center}\,\mathrm{DerivedSubgroup}\,\mathrm{FrattiniSubgroup}\right]\left(\mathrm{QuaternionGroup}\left(\right)\right)\right)$

$\left[2\,2\,2\right]$

$\mathrm{IsSpecial}\left(\mathrm{SmallGroup}\left(1331\,5\right)\right)$

$\mathrm{true}$

$\mathrm{IsSpecial}\left(\mathrm{QuaternionGroup}\left(5\right)\right)$

$\mathrm{false}$

$\mathrm{IsExtraspecial}\left(\mathrm{QuaternionGroup}\left(5\right)\right)$

$\mathrm{false}$

$\mathrm{GroupOrder}\left(\mathrm{Center}\left(\mathrm{QuaternionGroup}\left(5\right)\right)\right)$

$2$

$\mathrm{GroupOrder}\left(\mathrm{DerivedSubgroup}\left(\mathrm{QuaternionGroup}\left(5\right)\right)\right)$

$8$

$\mathrm{IsExtraspecial}\left(\mathrm{SmallGroup}\left(1331\,5\right)\right)$

$\mathrm{false}$

$\mathrm{IsSpecial}\left(\mathrm{TrivialGroup}\left(\right)\right)$

$\mathrm{true}$

$\mathrm{IsExtraspecial}\left(\mathrm{TrivialGroup}\left(\right)\right)$

$\mathrm{false}$

$G≔\mathrm{PSL}\left(2\,9\right)\:$

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

$C≔\mathrm{Centralizer}\left(g\,G\right)$

$C≔⟨\left(2\,5\right)\left(3\,4\right)\,\left(1\,6\right)\left(3\,4\right)\,\left(2\,3\right)\left(4\,5\right)⟩$

$\mathrm{IsExtraspecial}\left(C\right)$

$\mathrm{true}$

$\mathrm{AreIsomorphic}\left(C\,\mathrm{DihedralGroup}\left(4\right)\right)$

$\mathrm{true}$