Full octahedral group



The full octahedral group Oh is the of dimension 3. This article mainly looks at it as the symmetry group of the cube.

There are (3) = 48 permutation of the cube. Half of them are its rotations, forming the subgroup O (the symmetric group S4), and the other half are their inversions. The inversion is the permutation that exchanges opposite vertices of the cube. It is not to be confused with the inversion of a permutation.

As the hyperoctahedral group of dimension 3 the full octahedral group is the $$S_2 \wr S_3 \simeq S_2^3 \rtimes S_3$$, and a natural way to identify its elements is as pairs $$(m, n)$$ with $$m \in [0, 2^3)$$ and $$n \in [0, 3!)$$. But as it is also the direct product $$S_4 \times S_2$$, one can simply identify the elements of tetrahedral subgrup Td as $$a \in [0, 4!)$$ and their inversions as $$a'$$.

So e.g. the identity $$(0, 0)$$ is represented as $$0$$ and the inversion $$(7, 0)$$ as $$0'$$. $$(3, 1)$$ is represented as $$6$$ and $$(4, 1)$$ as $$6'$$.

A rotoreflection is a combination of rotation and reflection. While a rotation leaves its axis and a reflection leaves its plane unchanged, a rotoreflection leaves only the center unchanged.

Cayley table
As there are two ways to denote the elements of this group, there are two ways to write the Cayley table.

S4
The easier one is that in S4 based notation: It simply doubles the Cayley table of S4. With $$a, b, c \in S_4$$ and $$a', b', c'$$ being their respective inversions, $$a \circ b = c$$ implies $$a' \circ b' = c$$ and $$a \circ b' = a' \circ b = c'$$.

Example: $$12' \circ 21 = 18'$$

pairs
The notation with pairs is probably more useful, but also more complicated:
 * $$(pm, pn) = (am, an) \circ (bm, bn)$$

The calculation of $$pm$$ involves the cube vertex permutation shown in the images. E.g. $$\mathrm{cvp}(2, 3) = (2, 6, 3, 7, ~ 0, 4, 1, 5)$$.


 * $$pm = \mathrm{cvp}(am, an)[bm]$$

$$pn$$ simply follows from the Cayley table of S3:
 * $$pn = an \circ bn$$

Example: $$(pm, pn) = (2, 3) \circ (6, 2)$$
 * $$pm = \mathrm{cvp}(2, 3)[6] = 1$$
 * $$pn = 3 \circ 2 = 5$$

Truncated cuboctahedron
The vertices of the correspond to the elements of this group. Each of its faces of its dual, the, is a.

Cubes corresponding to matrix columns
Each of the six cubes in the following collapsible boxes shows one of the basic permutations from the top row of the matrix in the bottom left position. In the other seven positions are the products of applying the reflections along coordinate axes on these basic permutations.

Conjugacy classes
The full octahedral group has (3) = 10 conjugacy classes.

Two permutations $$(m, n)$$ and $$(k, n)$$ are complementary to each other, if $$m + k = 7$$. Complementary permutations sum up to a vector of 7s, and their inversion sets are , so their inversion numbers sum up to 28. (Compare one of the number matrices above.)

The conjugacy classes below are always shown in complementary pairs (like inv2/ref1 or rot2/ref2). The numbers over the triangles are the inversion numbers of the corresponding permutations. It can be seen that corresponding numbers add up to 28.

Subgroups
Oh has 98 individual subgroups, which are all shown in the list below. (A Python dictionary of them can be found here.)

They naturally divide in 33 bundles of similar subgroups, whose elements belong to the same conjugacy classes. In this article these bundles are given naive names based on some of the colors assigned to their elements (like Dih4 green orange). Each of them has a collapsible box below, containing representations of the individual subgroups.

These belong to 25 bigger bundles, which can be identified with a label in Schoenflies or Coxeter notation (like D2d or [2+,4]). Each of them has a vertex in the big Hasse diagram below.

Four different kinds of Coxeter notation can be distinguished, based on where they contain plus signs:

List of all subgroups

 * For the same list including all permutations of the respective example solids, see Full octahedral group/List of all subgroups.

Subgroups of order 12
{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;" ! bgcolor="#f0f0e0" colspan="5"| D3d  Dih6   [2+,6]
 * rowspan="2" style="width:240px;"| Subgroup of Oh; Dih6 03; example solid.png
 * Subgroup of Oh; Dih6 03; cycle graph.svg
 * Subgroup of Oh; Dih6 11; cycle graph.svg
 * Subgroup of Oh; Dih6 15; cycle graph.svg
 * Subgroup of Oh; Dih6 08; cycle graph.svg
 * Subgroup of Oh; Dih6 03; matrix.svg Subgroup of Oh; Dih6 03; solid.png
 * Subgroup of Oh; Dih6 11; matrix.svg Subgroup of Oh; Dih6 11; solid.png
 * Subgroup of Oh; Dih6 15; matrix.svg Subgroup of Oh; Dih6 15; solid.png
 * Subgroup of Oh; Dih6 08; matrix.svg Subgroup of Oh; Dih6 08; solid.png
 * colspan="5"|
 * Subgroup of Oh; Dih6 08; matrix.svg Subgroup of Oh; Dih6 08; solid.png
 * colspan="5"|
 * colspan="5"|


 * }

Subgroups of order 8
(Below the C23 subgroups are shown in more detail.)

(Below the Dih4 subgroups are shown in more detail.)

Subgroups of order 6
(Below the S3 subgroups are shown in more detail.)

Symmetry group of the cuboid
The symmetry group of the cuboid C23 appears in two essentially different ways as a subgroup of Oh. The one where the cuboid is the cube itself is the most intuitive one. In the other one the cuboid is the original cube rotated by 45° around an axis. The one where it is rotated around the z-axis is shown below. There are 4 individual subgroups (see above).

Symmetry group of the square
The symmetry group of the square appears in four essentially different ways as a subgroup of Oh. (C4v or [4] being the most intuitive among them.) There are 12 individual Dih4 subgroups (see above). Shown below are the four where the square is seen "from above", i.e. a point on the positive z-axis.

In the white box above the colored boxes of the four subgroups are the permutations of the square. Their 2×2 transformation matrices are the top left submatrices of the four 3×3 matrices in the same column. So the last non-zero entry of the 3×3 matrix determines the permutation in this column. (So each column has only two different permutations.) The pattern of these eight last entries identifies the subgroup. It is shown on the left in the little 4×2 matrix under the example solid.

Symmetry group of the triangle
The symmetry group of the triangle appears in two essentially different ways as a subgroup of Oh, with C3v or [3] being the most intuitive among them. There are 8 individual subgroups (see above). Shown below are the ones where the triangle is seen from a point on the negative main diagonal of the coordinate system.

Code
The Python code used to create many of the illustrations in this article can be found here: https://github.com/watchduck/full_octahedral_group

The following code shows bijections from pairs to other representations:

These are Python dictionaries without the surrounding braces. (Dicts work only in one direction, but bidict can be used to get back to the pairs.)

A dictionary of these permutations and their properties (including conjugacy class and inverse) can be found here.

A dictionary of all the subgroups can be found here (as a bijection from naive names to tuples of S4 based numbers).