Semidihedral group:SD16

Definition
The semidihedral group $$SD_{16}$$ (also denoted $$QD_{16}$$) is the semidihedral group (also called quasidihedral group) of order $$16$$. Specifically, it has the following presentation:

$$SD_{16} := \langle a,x \mid a^8 = x^2 = e, xax^{-1} = a^3 \rangle$$.

The group can also be defined as the general semilinear group of degree one over the field of nine elements.

Arithmetic functions
Note that the order can also be computed using the formula for the order of $$\Gamma L(1,p^r)$$ where $$p = 3, r = 2$$ and $$q = p^r = 9$$: The order of $$\Gamma L(1,q)$$ with $$q = p^r$$ is $$r(q - 1)$$, which becomes $$2(9 - 1) = 2(8) = 16$$.

Subgroups
There are a couple of interesting facts about this group:


 * Every subgroup of this group is an automorph-conjugate subgroup.
 * All the maximal subgroups are characteristic subgroups -- in fact, they are all isomorph-free subgroups.

Description by presentation
The group can be defined using a presentation as follows:

gap> F := FreeGroup(2);; gap> G := F/[F.1^8,F.2^2,F.2*F.1*F.2*F.1^(-3)];  gap> IdGroup(G); [ 16, 8 ]