SmallGroup(32,35)

Definition
This group is a quaternion group-like variant of the generalized dihedral group of direct product of Z4 and Z4. It is given by the presentation:

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

The group can also be described (up to isomorphism) as a subgroup of the direct product of Q8 and Q8 that is not isomorphic to the direct product of Q8 and Z4.

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