Full octahedral group

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Created by Watchduck
One of the 48 permutations of the cube: The positive (anti-clockwise) rotation around the y-axis
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 and being their respective inversions implies and .


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 ,
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.


Overview[edit]

8×6 matrix[edit]

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 row[edit]

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 columns[edit]

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 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

Subgroups[edit]

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]

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

List of all subgroups[edit]

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

The group itself[edit]

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


Subgroups of order 24[edit]



Subgroups of order 16[edit]


Subgroups of order 12[edit]



Subgroups of order 8[edit]



Subgroups of order 6[edit]



Subgroups of order 4[edit]



Subgroups of order 3[edit]


Subgroups of order 2[edit]


The trivial group[edit]


Different appearances of the same group[edit]

Symmetry group of the cuboid[edit]

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 (clockwise) around the z-axis is shown below.

Symmetry group of the square[edit]

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. Shown below are the ones where the square is seen from above the cube, i.e. from a point on the positive z-axis.

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 triangle[edit]

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. 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 hexagons[edit]

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


Code[edit]

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 Python dictionaries are bijections from pairs to S4 based identifiers (n' as n+24), to permutations of 8 elements (i.e. cube vertices), and to the 3×3 transformation matrices.

Python dicts 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).