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 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.

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 and being their respective inversions implies and .


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.


OverviewEdit

Truncated cuboctahedronEdit

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.

Full octahedral group elements in truncated cuboctahedron; numbers.svg Full octahedral group elements in truncated cuboctahedron; JF.png Polyhedron great rhombi 6-8 max.png Polyhedron great rhombi 6-8 dual max.png


8×6 matrixEdit

Finite permutation number 0.svg Finite permutation number 1.svg Finite permutation number 2.svg Finite permutation number 3.svg Finite permutation number 4.svg Finite permutation number 5.svg
Cube vertex number 0.svg Cube permutation 0 0.svg Cube permutation 0 1.svg Cube permutation 0 2.svg Cube permutation 0 3.svg Cube permutation 0 4.svg Cube permutation 0 5.svg
Cube vertex number 1.svg Cube permutation 1 0.svg Cube permutation 1 1.svg Cube permutation 1 2.svg Cube permutation 1 3.svg Cube permutation 1 4.svg Cube permutation 1 5.svg
Cube vertex number 2.svg Cube permutation 2 0.svg Cube permutation 2 1.svg Cube permutation 2 2.svg Cube permutation 2 3.svg Cube permutation 2 4.svg Cube permutation 2 5.svg
Cube vertex number 3.svg Cube permutation 3 0.svg Cube permutation 3 1.svg Cube permutation 3 2.svg Cube permutation 3 3.svg Cube permutation 3 4.svg Cube permutation 3 5.svg
Cube vertex number 4.svg Cube permutation 4 0.svg Cube permutation 4 1.svg Cube permutation 4 2.svg Cube permutation 4 3.svg Cube permutation 4 4.svg Cube permutation 4 5.svg
Cube vertex number 5.svg Cube permutation 5 0.svg Cube permutation 5 1.svg Cube permutation 5 2.svg Cube permutation 5 3.svg Cube permutation 5 4.svg Cube permutation 5 5.svg
Cube vertex number 6.svg Cube permutation 6 0.svg Cube permutation 6 1.svg Cube permutation 6 2.svg Cube permutation 6 3.svg Cube permutation 6 4.svg Cube permutation 6 5.svg
Cube vertex number 7.svg Cube permutation 7 0.svg Cube permutation 7 1.svg Cube permutation 7 2.svg Cube permutation 7 3.svg Cube permutation 7 4.svg Cube permutation 7 5.svg
Finite permutation number 0.svg Finite permutation number 1.svg Finite permutation number 2.svg Finite permutation number 3.svg Finite permutation number 4.svg Finite permutation number 5.svg
Cube vertex number 0.svg Cube permutation 0 0 JF ortho.png Cube permutation 0 1 JF ortho.png Cube permutation 0 2 JF ortho.png Cube permutation 0 3 JF ortho.png Cube permutation 0 4 JF ortho.png Cube permutation 0 5 JF ortho.png
Cube vertex number 1.svg Cube permutation 1 0 JF ortho.png Cube permutation 1 1 JF ortho.png Cube permutation 1 2 JF ortho.png Cube permutation 1 3 JF ortho.png Cube permutation 1 4 JF ortho.png Cube permutation 1 5 JF ortho.png
Cube vertex number 2.svg Cube permutation 2 0 JF ortho.png Cube permutation 2 1 JF ortho.png Cube permutation 2 2 JF ortho.png Cube permutation 2 3 JF ortho.png Cube permutation 2 4 JF ortho.png Cube permutation 2 5 JF ortho.png
Cube vertex number 3.svg Cube permutation 3 0 JF ortho.png Cube permutation 3 1 JF ortho.png Cube permutation 3 2 JF ortho.png Cube permutation 3 3 JF ortho.png Cube permutation 3 4 JF ortho.png Cube permutation 3 5 JF ortho.png
Cube vertex number 4.svg Cube permutation 4 0 JF ortho.png Cube permutation 4 1 JF ortho.png Cube permutation 4 2 JF ortho.png Cube permutation 4 3 JF ortho.png Cube permutation 4 4 JF ortho.png Cube permutation 4 5 JF ortho.png
Cube vertex number 5.svg Cube permutation 5 0 JF ortho.png Cube permutation 5 1 JF ortho.png Cube permutation 5 2 JF ortho.png Cube permutation 5 3 JF ortho.png Cube permutation 5 4 JF ortho.png Cube permutation 5 5 JF ortho.png
Cube vertex number 6.svg Cube permutation 6 0 JF ortho.png Cube permutation 6 1 JF ortho.png Cube permutation 6 2 JF ortho.png Cube permutation 6 3 JF ortho.png Cube permutation 6 4 JF ortho.png Cube permutation 6 5 JF ortho.png
Cube vertex number 7.svg Cube permutation 7 0 JF ortho.png Cube permutation 7 1 JF ortho.png Cube permutation 7 2 JF ortho.png Cube permutation 7 3 JF ortho.png Cube permutation 7 4 JF ortho.png Cube permutation 7 5 JF ortho.png
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 rowEdit

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 Finite permutation number 1.svg and Finite permutation number 2.svg
Cayley graph generated by Finite permutation number 1.svg and Finite permutation number 4.svg
example solid

Cubes corresponding to matrix columnsEdit

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.

Finite permutation number 0.svg
cube
Cube permutation 6 0.svg Cube permutation 7 0.svg
Cube permutation 4 0.svg Cube permutation 5 0.svg
Cube permutation 2 0.svg Cube permutation 3 0.svg
Cube permutation 0 0.svg Cube permutation 1 0.svg
Cube permutation 6 0 JF.png Cube permutation 7 0 JF.png
Cube permutation 4 0 JF.png Cube permutation 5 0 JF.png
Cube permutation 2 0 JF.png Cube permutation 3 0 JF.png
Cube permutation 0 0 JF.png Cube permutation 1 0 JF.png

Conjugacy classesEdit

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)
Full octahedral group; set partition neut.svg Full octahedral group; set partition inv3.svg Full octahedral group; set partition inv2 0.svg Full octahedral group; set partition ref1 0.svg Full octahedral group; set partition rot3 0.svg Full octahedral group; set partition rotref3 0.svg Full octahedral group; set partition rot2 0b.svg Full octahedral group; set partition ref2 0b.svg Full octahedral group; set partition rot1 0.svg Full octahedral group; set partition rotref1 0.svg

SubgroupsEdit

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 diagramsEdit

All 25 bundles of similar subgroups
Full octahedral group; subgroups Hasse diagram.svg

List of all subgroupsEdit

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

The group itselfEdit

Oh S4 × C2 [4,3]
Subgroup of Oh; S4xC2; example solid.png Full octahedral group; cycle graph.svg
Subgroup of Oh; S4xC2; matrix.svg Subgroup of Oh; S4xC2; solid.png

Subgroups of order 24Edit


Subgroups of order 16Edit

Subgroups of order 12Edit


Subgroups of order 8Edit

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


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

Subgroups of order 6Edit

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

Subgroups of order 4Edit


Subgroups of order 3Edit

Subgroups of order 2Edit

The trivial groupEdit

Different appearances of the same groupEdit

Symmetry group of the cuboidEdit

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 squareEdit

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.

Blank300.png
Square permutation fix.svg
Square permutation horz.svg
Square permutation vert.svg
Square permutation cross.svg
Square permutation desc.svg
Square permutation left.svg
Square permutation right.svg
Square permutation asc.svg

Symmetry group of the triangleEdit

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.

Blank300.png
Triangle permutation fix.svg
Triangle permutation ref left.svg
Triangle permutation ref right.svg
Triangle permutation rot left.svg
Triangle permutation rot right.svg
Triangle permutation ref horz.svg

Cuboctahedral example solids and contained hexagonsEdit

S4 blue red A4 × C2 A4 S4 green orange
shown above Subgroup of Oh; S4 blue red; example solid.png Subgroup of Oh; A4xC2; example solid.png Subgroup of Oh; A4; example solid.png Subgroup of Oh; S4 green orange; example solid.png
cuboctahedral Subgroup of Oh; A4xC2; example solid (cuboctahedron).png Subgroup of Oh; A4; example solid (cuboctahedron).png Subgroup of Oh; S4 green orange; example solid (cuboctahedron).png
contained hexagon Subgroup of Oh; S3 blue 03; example solid.png Subgroup of Oh; C6 03; example solid.png Subgroup of Oh; C3 03; example solid.png Subgroup of Oh; S3 green 03; example solid.png
S3 blue C6 C3 S3 green


CodeEdit

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).