Semidirect product of Z16 and Z4 of semidihedral type

Definition
This group is defined as the external semidirect product where the base (acted upon) group is defining ingredient::cyclic group:Z16 and the acting group is cyclic group:Z4, and the generator of the acting group acts via the $$7^{th}$$ power map.

Equivalently, it is given by the presentation:

$$\! G := \langle a,b \mid a^{16} = b^4 = e, bab^{-1} = a^7 \rangle$$

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^(-7)];  gap> IdGroup(G); [ 64, 48 ]