I am interested in obtaining a permutation representation of the Pauli Group $G_1 = \langle X, Y, Z \rangle$. I think this would be easy enough as a "Regular representation" but then I learn that the Pauli group is the central product of $C_4$ and $D_4$. The definition of "central product" seems to reside in the darker recesses of group theory and I don't understand the terse notation I have found for it.

If someone could provide the Mathematica code for the central product of two groups such that I can see what it is step by step I would be very happy.