Full octahedral group

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A compound of cube and octahedron with full octahedral symmetry
Cycle graph of Oh
Cayley table of the symmetric group S4
Cayley table of the symmetric group S3 with 3×3 permutation matrices

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 Sloane'sA000165(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.

Conjugacy classes
Elements of O Inversions of elements of O
identity neut 0 inversion inv3 0'
3 × rotation by 180° about a 4-fold axis inv2 7, 16, 23 3 × reflection in a plane perpendicular to a 4-fold axis ref1 7', 16', 23'
8 × rotation by 120° about a 3-fold axis rot3 3, 4, 8, 11, 12, 15, 19, 20 8 × rotoreflection by 60° rotref3 3', 4', 8', 11', 12', 15', 19', 20'
6 × rotation by 180° about a 2-fold axis rot2 1', 2', 5', 6', 14', 21' 6 × reflection in a plane perpendicular to a 2-fold axis ref2 1, 2, 5, 6, 14, 21
6 × rotation by 90° about a 4-fold axis rot1 9', 10', 13', 17', 18', 22' 6 × rotoreflection by 90° rotref1 9, 10, 13, 17, 18, 22

As the hyperoctahedral group of dimension 3 the full octahedral group is the wreath product ,
and a natural way to identify its elements is as pairs with and .
But as it is also the direct product , one can simply identify the elements of tetrahedral subgrup Td as and their inversions as .

So e.g. the identity is represented as and the inversion as .
is represented as and as .

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[edit | edit source]

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

S4[edit | edit source]

The easier one is that in S4 based notation: It simply doubles the Cayley table of S4.
With and being their respective inversions, implies and .


pairs[edit | edit source]

The notation with pairs is probably more useful, but also more complicated:

The calculation of involves the cube vertex permutation shown in the images.   E.g. .

simply follows from the Cayley table of S3:


Overview[edit | edit source]

Truncated cuboctahedron[edit | edit source]

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[edit | edit source]

S4 based identifiers
The JF compound used to illustrate the permutations
The right-hand coordinate system used in this article

Hexagon corresponding to top matrix row[edit | edit source]

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[edit | edit source]

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[edit | edit source]

The full octahedral group has Sloane'sA000712(3) = 10 conjugacy classes.

Two permutations and are complementary to each other, if .
Complementary permutations sum up to a vector of 7s, and their inversion sets are complements,
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.

neut (1) inv3 (1) inv2 (3) ref1 (3) rot3 (8) rotref3 (8) rot2 (6) ref2 (6) rot1 (6) rotref1 (6)

Subgroups[edit | edit source]

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[edit | edit source]

All 25 bundles of similar subgroups

List of all subgroups[edit | edit source]

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

The group itself[edit | edit source]

Oh S4 × C2 [4,3]
truncated cuboctahedron

Subgroups of order 24[edit | edit source]

Subgroups of order 16[edit | edit source]

Subgroups of order 12[edit | edit source]

Subgroups of order 8[edit | edit source]

(Below the C23 subgroups are shown in more detail.)

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

Subgroups of order 6[edit | edit source]

(Below the S3 subgroups are shown in more detail.)

Subgroups of order 4[edit | edit source]

Subgroups of order 3[edit | edit source]

Subgroups of order 2[edit | edit source]

The trivial group[edit | edit source]

Different appearances of the same group[edit | edit source]

Symmetry group of the cuboid[edit | edit source]

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[edit | edit source]

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.