Nontrivial semidirect product of Z4 and Z16

Definition
This group is defined as the external semidirect product of cyclic group:Z4 and cyclic group:Z16 where the generator of the latter acts by the inverse map on the former.

It is given by the presentation:

$$G := \langle a,b \mid a^4 = b^{16} = e, bab^{-1} = a^{-1} \rangle$$

Description by presentation
gap> F := FreeGroup(2);  gap> G := F/[F.1^4,F.2^(16),F.2*F.1*F.2^(-1)*F.1];  gap> IdGroup(G); [ 64, 44 ]