SmallGroup(32,24)

Definition
This group is defined by the following presentation:

$$\langle a,b,c \mid a^4 = b^4 = c^2 = e, ab = ba, ac = ca, cbc^{-1} = a^2b \rangle$$

It can alternatively be defined in the following equivalent ways:


 * It is the central product of SmallGroup(16,3) and cyclic group:Z4 over a common cyclic central subgroup of order two.
 * It is the central product of nontrivial semidirect product of Z4 and Z4 (ID: (16,4)) and cyclic group:Z4 over a common cyclic central subgroup of order two.

Description by presentation
Here is the GAP code to define this group using a presentation:

gap> F := FreeGroup(3);  gap> G := F/[F.1^4,F.2^4, F.1*F.2*F.1^(-1)*F.2^(-1), F.3^2, F.1*F.3*F.1^(-1)*F.3^(-1),F.3*F.2*F.3^(-1)*F.2^(-1)*F.1^2 ]; 