Direct product of SmallGroup(32,4) and Z2

Definition
This group is defined as the external direct product of the following two groups: defining ingredient::semidirect product of Z8 and Z4 of M-type (GAP ID:(32,4)) and defining ingredient::cyclic group:Z2. Explicitly, it is given by the presentation:

$$G := \langle a,b,c \mid a^8 = b^4 = c^2 = e, ac = ca, bc = cb, bab^{-1} = a^9 \rangle$$