Direct product of S4 and Z2

Definition
This group is defined in a number of equivalent ways:


 * 1) It is the full octahedral group: the group of all rigid symmetries (both orientation-preserving and orientation-reversing) of the regular octahedron.
 * 2) It is the full cube group: the group of all rigid symmetries (both orientation-preserving and orientation-reversing) of the cube.
 * 3) It is the external direct product of the symmetric group of degree four and the cyclic group of order two.
 * 4) It is the automorphism group of the general linear group:GL(2,3), i.e., the general linear group of degree two over the field of three elements.
 * 5) It is the wreath product of the cyclic group of order two and the symmetric group of degree three. In other words, it is the group $$(\Z_2 \times \Z_2 \times \Z_2) \rtimes S_3$$ where $$S_3$$ acts by coordinate permutations.
 * 6) It is the signed symmetric group of degree three. (Note: This is a reinterpretation of the previous definition as a wreath product).
 * 7) It is the projective general linear group of degree two over the ring $$\mathbb{Z}/4\mathbb{Z}$$, i.e., it is the group $$PGL(2,\mathbb{Z}/4\mathbb{Z})$$. Note that this is not the same as the group $$PGL(2,4)$$, which is the projective general linear group of degree two over field:F4 (and which is isomorphic to alternating group:A5, a group of order 60).