Semidirect product of Z16 and Z4 of dihedral type

Definition
This group is defined as the external semidirect product with base normal subgroup equal to cyclic group:Z16 and quotient group isomorphic to cyclic group:Z4, where the generator of the latter acts by the inverse map on the former.

Equivalently, it is given by the presentation:

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

Description by presentation
gap> F := FreeGroup(2);  gap> G := F/[F.1^(16),F.2^4,F.2*F.1*F.2^(-1)*F.1];  gap> IdGroup(G); [ 64, 47 ]