# Full octahedral group

A compound of cube and octahedron with full octahedral symmetry
The Cayley table of Oh doubles this one of Td (the symmetric group S4)

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

There are 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.

The Cayley table of Oh repeats the pattern of the Cayley table of S4. If, as in this article, the S4 based notation is used, the result of a concatenation of elements of Oh can be derived from the corresponding concatenation of elements of S4: With ${\displaystyle a,b,c\in S_{4}}$ and ${\displaystyle a',b',c'}$ being their respective inversions ${\displaystyle a\circ b=c}$ implies ${\displaystyle a\circ b'=a'\circ b=c'}$ and ${\displaystyle a'\circ b'=c}$.

Conjugacy classes
Elements of O Inversions of elements of O
identity 0 inversion 0'
3 × rotation by 180° about a 4-fold axis 7, 16, 23 3 × reflection in a plane perpendicular to a 4-fold axis 7', 16', 23'
8 × rotation by 120° about a 3-fold axis 3, 4, 8, 11, 12, 15, 19, 20 8 × rotoreflection by 60° 3', 4', 8', 11', 12', 15', 19', 20'
6 × rotation by 180° about a 2-fold axis 1', 2', 5', 6', 14', 21' 6 × reflection in a plane perpendicular to a 2-fold axis 1, 2, 5, 6, 14, 21
6 × rotation by 90° about a 4-fold axis 9', 10', 13', 17', 18', 22' 6 × rotoreflection by 90° 9, 10, 13, 17, 18, 22

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

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

A rotoreflection is a combination of rotation and reflection.

## Overview

### Truncated cuboctahedron

The vertices of the truncated cuboctahedron correspond to the elements of this group. Each of its faces of its dual, the disdyakis dodecahedron, is a fundamental domain.

### 8×6 matrix

S4 based identifiers
The JF compound used to illustrate the permutations

### Hexagon corresponding to top matrix row

3D diagrams

The files below illustrate the subgroup C3v or [3] that corresponds to the top matrix row. It contains the six permutations of the cube that leave the main diagonal fixed.

 permutohedron coordinates Cayley graph generated by and Cayley graph generated by and example solid

### 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.

cube

## 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:

 [...]+ rotate [...] reflect [...+,...+] cross [...+, ...] mixed

### Hasse diagrams

All 25 bundles of similar subgroups

### List of all subgroups

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

Oh S4 × C2 [4,3]

#### 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.)

### Different appearances of the same group

#### 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, with C4v or [4] being the most intuitive among them.
There are 12 individual subgroups (see above). Shown below are the ones where the square is seen from above the cube, i.e. from a point on the positive z-axis.

In the box above those of the four subgroups are the corresponding permutations of the square. It can be seen that the 2×2 transformation matrices of the square are the top left submatrices of the 3×3 matrices in the same column. The pattern of their bottom right entries is shown in a 4×2 matrix below 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.

### Cuboctahedral example solids and contained hexagons

 shown above cuboctahedral contained hexagon S4 blue red A4 × C2 A4 S4 green orange S3 blue C6 C3 S3 green

## 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. They work only in one direction, but bidict can be used to get back to the pairs.

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