Holomorph of Z8

Definition
This group (which we shall call $$G$$ throughout) can be defined in either of these ways:


 * It is the holomorph of the cyclic group on eight elements. In other words, it is the direct product of the cyclic group on eight elements, with its automorphism group.
 * It is the holomorph of the ring $$\Z/8\Z$$. In other words, it is the member of family::general affine group $$GA(1,\Z/8\Z)$$.

The group has the following presentation (with $$e$$ denoting the identity element):

$$\! G := \langle a,x,y \mid a^8 = x^2 = y^2 = e, xax^{-1} = a^{-1}, yay^{-1} = a^5 \rangle$$

Other definitions
The group can be defined using GAP's AutomorphismGroup and SemidirectProduct functions. Here is a full code snippet:

gap> C := CyclicGroup(8);  gap> SemidirectProduct(AutomorphismGroup(C),C); 

This can be compressed by coding a function Holomorph for computing the holomorph of a group. With this function coded, we can use:

Holomorph(CyclicGroup(8))