Holomorph of D8

Definition
This group is defined in the following eqiuvalent ways:


 * It is the holomorph of the dihedral group of order eight, i.e., the semidirect product of the dihedral group of order eight and its automorphism group (which is also isomorphic to the dihedral group of order eight).
 * It is the $$2$$-Sylow subgroup of the holomorph of the quaternion group.

Alternative descriptions
The group can be constructed using the GAP commands DihedralGroup, AutomorphismGroup, and SemidirectProduct:

gap> G := DihedralGroup(8);  gap> A := AutomorphismGroup(G);  gap> S := SemidirectProduct(A,G); 

$$S$$ is the group we need.

It can also be constructed using a hand-coded GAP function: Holomorph, with which it becomes:

Holomorph(DihedralGroup(8))