Semidirect product of Z8 and Z8 of dihedral type

Definition
This group is defined by the presentation:

$$\! G := \langle a,b \mid a^8 = b^8 = e, bab^{-1} = a^{-1} \rangle$$

Description by presentation
gap> F := FreeGroup(2);  gap> G := F/[F.1^8,F.2^8,F.2*F.1*F.2^(-1)*F.1];  gap> IdGroup(G); [ 64, 16 ]