# Direct product of S4 and Z2

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## 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).

## GAP implementation

### Group ID

This finite group has order 48 and has ID 48 among the groups of order 48 in GAP's SmallGroup library. For context, there are groups of order 48. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(48,48)

For instance, we can use the following assignment in GAP to create the group and name it $G$:

gap> G := SmallGroup(48,48);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [48,48]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.

### Alternative descriptions

Description Functions used Mathematical comment
DirectProduct(SymmetricGroup(4),CyclicGroup(2)) DirectProduct, SymmetricGroup, CyclicGroup
AutomorphismGroup(GL(2,3)) AutomorphismGroup and GeneralLinearGroup
WreathProduct(CyclicGroup(2),SymmetricGroup(3)) WreathProduct, CyclicGroup, SymmetricGroup