Generalized dihedral group for direct product of Z4 and Z4

Definition
This group is defined as the generalized dihedral group corresponding to the abelian group given as the direct product of Z4 and Z4, i.e., it is the semidirect product of this group by cyclic group:Z2 where the non-identity element acts via the inverse map. It has the presentation:

$$\langle x,y,a \mid x^4 = y^4 = a^2 = e, xy = yx,axa^{-1} = x^{-1}, aya^{-1} = y^{-1} \rangle$$

Other descriptions
The group can be constructed using its presentation:

gap> F := FreeGroup(3);  gap> G := F/[F.1^4,F.2^4,F.3^2,F.1*F.2*F.1^(-1)*F.2^(-1),F.3*F.1*F.3^(-1)*F.1,F.3*F.2*F.3^(-1)*F.2]; 