Inner holomorph of D8

Definition
This group can be defined in the following equivalent ways:


 * 1) It is the inner holomorph of the dihedral group of order eight. In other words, it is the semidirect product of the dihedral group by its inner automorphism group, which is isomorphic to a Klein four-group.
 * 2) It is the central product of the dihedral group of order eight with itself, with the common center identified.
 * 3) it is the inner holomorph of the quaternion group. In other words, it is the semidirect product of the quaternion group by its inner automorphism group, which is isomorphic to a Klein four-group.
 * 4) It is the central product of the quaternion group of order eight with itself, with the common center identified.
 * 5) It is the extraspecial group of order $$2^5$$ and '+' type.
 * 6) It is the subgroup of upper-triangular unipotent matrix group:U(4,2) given by the matrices with only corner entries, i.e., matrices over field:F2 of the form:

$$\begin{pmatrix} 1 & * & * & * \\ 0 & 1 & 0 & * \\ 0 & 0 & 1 & * \\ 0 & 0 & 0 & 1 \\\end{pmatrix}$$

The group can also be given by the presentation:

$$\langle x,y,z,a,b \mid x^2 = y^2 = z^2 = a^2 = b^2 = e, xy = yx, xz = zx, yz = zy, ax = xa, bx = xb, aya^{-1} = xy, az = za, by = yb, bzb^{-1} = yz \rangle$$

Long descriptions
Based on the inner holomorph idea, the group can be described as follows using GAP's DihedralGroup, AutomorphismGroup, InnerAutomorphismsAutomorphismGroup and SemidirectProduct functions:

gap> H := DihedralGroup(8);; gap> A := AutomorphismGroup(H);; gap> I := InnerAutomorphismsAutomorphismGroup(A);; gap> G := SemidirectProduct(I,H); 