Semidirect product of Z16 and Z4 via cube map

Definition
This group is defined as an external semidirect product where the base normal subgroup is cyclic group:Z16 and the acting quotient group is cyclic group:Z4 and the latter acts on the former via the cube map:

$$G := \langle a,b \mid a^{16} = b^4 = e, bab^{-1} = a^3 \rangle$$

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^(-3)];  gap> IdGroup(G); [ 64, 46 ]