Full octahedral group
The full octahedral group O_{h} is the hyperoctahedral group of dimension 3. This article mainly looks at it as the symmetry group of the cube.
There are A000165(3) = 48 permutation of the cube. Half of them are its rotations, forming the subgroup O (the symmetric group S_{4}), 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.
Elements of O  Inversions of elements of O  

identity  neut  0  inversion  inv3  0' 
3 × rotation by 180° about a 4fold axis  inv2  7, 16, 23  3 × reflection in a plane perpendicular to a 4fold axis  ref1  7', 16', 23' 
8 × rotation by 120° about a 3fold 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 2fold axis  rot2  1', 2', 5', 6', 14', 21'  6 × reflection in a plane perpendicular to a 2fold axis  ref2  1, 2, 5, 6, 14, 21 
6 × rotation by 90° about a 4fold axis  rot1  9', 10', 13', 17', 18', 22'  6 × rotoreflection by 90°  rotref1  9, 10, 13, 17, 18, 22 
Examples  

neut  inv2  rot3  rot2  rot1 
inv3  ref1  rotref3  ref2  rotref1 
See full list below: 8×6 matrix or Conjugacy classes 
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 T_{d} 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.
Illustration of rotoreflections  

 

Cayley table
editAs there are two ways to denote the elements of this group, there are two ways to write the Cayley table.
S_{4}
editThe easier one is that in S_{4} based notation: It simply doubles the Cayley table of S_{4}.
With and being their respective inversions, implies and .
Example:
pairs
editThe 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 S_{3}:
Example:
example  


Overview
editTruncated cuboctahedron
editThe 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

Hexagon corresponding to top matrix row
edit3D diagrams  

The files below illustrate the subgroup C_{3v} or [3] that corresponds to the top matrix row. It contains the six permutations of the cube that leave the main diagonal fixed. 
2D equivalents  

The rest of this article uses left action, i.e. means first , then , 
details  

 
This is left action again, so the 3×3 permutation matrices shown here are the transposes of those in the small permutohedron in the box above  which only makes a difference for and . 
Cubes corresponding to matrix columns
editEach 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
editThe full octahedral group has A000712(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) 

neut (1 × 1)  inv3 (1 × 1) 

0 
28 
inv2 (3 × 1)  ref1 (3 × 1)  

12 
20 
24 
16 
8 
4 
rot3 (4 × 2)  

6 
14 
18 
18  
rotref3 (4 × 2)  

22 
14 
10 
10  
rot2 (6 × 1)  

26 
18 
20 
12 
24 
8 
ref2 (6 × 1)  

2 
10 
8 
16 
4 
20 
rot1 (3 × 2)  rotref1 (3 × 2)  

6 
12 
12 
22 
16 
16  
Conjugacy classes of square permutations  

neut  inv2  ref1  ref2  rot1  
0 
6 
2 
4 
1 
5 
3  
Examples for tesseract and penteract 

The respective numbers of conjugacy classes for 4 and 5 dimensions are 20 and 36. examples4 = {
(0, 0): (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15),
(0, 1): (0, 1, 2, 3, 8, 9, 10, 11, 4, 5, 6, 7, 12, 13, 14, 15),
(0, 3): (0, 1, 4, 5, 8, 9, 12, 13, 2, 3, 6, 7, 10, 11, 14, 15),
(0, 7): (0, 2, 1, 3, 8, 10, 9, 11, 4, 6, 5, 7, 12, 14, 13, 15),
(0, 9): (0, 2, 4, 6, 8, 10, 12, 14, 1, 3, 5, 7, 9, 11, 13, 15),
(1, 0): (1, 0, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10, 13, 12, 15, 14),
(1, 1): (1, 3, 0, 2, 5, 7, 4, 6, 9, 11, 8, 10, 13, 15, 12, 14),
(1, 2): (1, 0, 3, 2, 9, 8, 11, 10, 5, 4, 7, 6, 13, 12, 15, 14),
(1, 3): (1, 3, 5, 7, 0, 2, 4, 6, 9, 11, 13, 15, 8, 10, 12, 14),
(1, 7): (1, 3, 0, 2, 9, 11, 8, 10, 5, 7, 4, 6, 13, 15, 12, 14),
(1, 8): (1, 0, 5, 4, 9, 8, 13, 12, 3, 2, 7, 6, 11, 10, 15, 14),
(1, 9): (1, 3, 5, 7, 9, 11, 13, 15, 0, 2, 4, 6, 8, 10, 12, 14),
(3, 0): (3, 2, 1, 0, 7, 6, 5, 4, 11, 10, 9, 8, 15, 14, 13, 12),
(3, 2): (3, 2, 7, 6, 1, 0, 5, 4, 11, 10, 15, 14, 9, 8, 13, 12),
(3, 6): (3, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4, 15, 14, 13, 12),
(3, 8): (3, 2, 7, 6, 11, 10, 15, 14, 1, 0, 5, 4, 9, 8, 13, 12),
(3, 16): (3, 7, 11, 15, 2, 6, 10, 14, 1, 5, 9, 13, 0, 4, 8, 12),
(7, 0): (7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8),
(7, 6): (7, 6, 5, 4, 15, 14, 13, 12, 3, 2, 1, 0, 11, 10, 9, 8),
(15, 0): (15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0)
}
examples5 = {
(0, 0): (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31),
(0, 1): (0, 1, 2, 3, 4, 5, 6, 7, 16, 17, 18, 19, 20, 21, 22, 23, 8, 9, 10, 11, 12, 13, 14, 15, 24, 25, 26, 27, 28, 29, 30, 31),
(0, 3): (0, 1, 2, 3, 8, 9, 10, 11, 16, 17, 18, 19, 24, 25, 26, 27, 4, 5, 6, 7, 12, 13, 14, 15, 20, 21, 22, 23, 28, 29, 30, 31),
(0, 7): (0, 1, 4, 5, 2, 3, 6, 7, 16, 17, 20, 21, 18, 19, 22, 23, 8, 9, 12, 13, 10, 11, 14, 15, 24, 25, 28, 29, 26, 27, 30, 31),
(0, 9): (0, 1, 4, 5, 8, 9, 12, 13, 16, 17, 20, 21, 24, 25, 28, 29, 2, 3, 6, 7, 10, 11, 14, 15, 18, 19, 22, 23, 26, 27, 30, 31),
(0, 27): (0, 2, 1, 3, 8, 10, 9, 11, 16, 18, 17, 19, 24, 26, 25, 27, 4, 6, 5, 7, 12, 14, 13, 15, 20, 22, 21, 23, 28, 30, 29, 31),
(0, 33): (0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31),
(1, 0): (1, 0, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10, 13, 12, 15, 14, 17, 16, 19, 18, 21, 20, 23, 22, 25, 24, 27, 26, 29, 28, 31, 30),
(1, 1): (1, 3, 0, 2, 5, 7, 4, 6, 9, 11, 8, 10, 13, 15, 12, 14, 17, 19, 16, 18, 21, 23, 20, 22, 25, 27, 24, 26, 29, 31, 28, 30),
(1, 2): (1, 0, 3, 2, 5, 4, 7, 6, 17, 16, 19, 18, 21, 20, 23, 22, 9, 8, 11, 10, 13, 12, 15, 14, 25, 24, 27, 26, 29, 28, 31, 30),
(1, 3): (1, 3, 5, 7, 0, 2, 4, 6, 9, 11, 13, 15, 8, 10, 12, 14, 17, 19, 21, 23, 16, 18, 20, 22, 25, 27, 29, 31, 24, 26, 28, 30),
(1, 7): (1, 3, 0, 2, 5, 7, 4, 6, 17, 19, 16, 18, 21, 23, 20, 22, 9, 11, 8, 10, 13, 15, 12, 14, 25, 27, 24, 26, 29, 31, 28, 30),
(1, 8): (1, 0, 3, 2, 9, 8, 11, 10, 17, 16, 19, 18, 25, 24, 27, 26, 5, 4, 7, 6, 13, 12, 15, 14, 21, 20, 23, 22, 29, 28, 31, 30),
(1, 9): (1, 3, 5, 7, 9, 11, 13, 15, 0, 2, 4, 6, 8, 10, 12, 14, 17, 19, 21, 23, 25, 27, 29, 31, 16, 18, 20, 22, 24, 26, 28, 30),
(1, 26): (1, 0, 5, 4, 3, 2, 7, 6, 17, 16, 21, 20, 19, 18, 23, 22, 9, 8, 13, 12, 11, 10, 15, 14, 25, 24, 29, 28, 27, 26, 31, 30),
(1, 27): (1, 3, 5, 7, 0, 2, 4, 6, 17, 19, 21, 23, 16, 18, 20, 22, 9, 11, 13, 15, 8, 10, 12, 14, 25, 27, 29, 31, 24, 26, 28, 30),
(1, 31): (1, 3, 0, 2, 9, 11, 8, 10, 17, 19, 16, 18, 25, 27, 24, 26, 5, 7, 4, 6, 13, 15, 12, 14, 21, 23, 20, 22, 29, 31, 28, 30),
(1, 32): (1, 0, 5, 4, 9, 8, 13, 12, 17, 16, 21, 20, 25, 24, 29, 28, 3, 2, 7, 6, 11, 10, 15, 14, 19, 18, 23, 22, 27, 26, 31, 30),
(1, 33): (1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30),
(3, 0): (3, 2, 1, 0, 7, 6, 5, 4, 11, 10, 9, 8, 15, 14, 13, 12, 19, 18, 17, 16, 23, 22, 21, 20, 27, 26, 25, 24, 31, 30, 29, 28),
(3, 2): (3, 2, 7, 6, 1, 0, 5, 4, 11, 10, 15, 14, 9, 8, 13, 12, 19, 18, 23, 22, 17, 16, 21, 20, 27, 26, 31, 30, 25, 24, 29, 28),
(3, 6): (3, 2, 1, 0, 7, 6, 5, 4, 19, 18, 17, 16, 23, 22, 21, 20, 11, 10, 9, 8, 15, 14, 13, 12, 27, 26, 25, 24, 31, 30, 29, 28),
(3, 8): (3, 2, 7, 6, 11, 10, 15, 14, 1, 0, 5, 4, 9, 8, 13, 12, 19, 18, 23, 22, 27, 26, 31, 30, 17, 16, 21, 20, 25, 24, 29, 28),
(3, 16): (3, 7, 11, 15, 2, 6, 10, 14, 1, 5, 9, 13, 0, 4, 8, 12, 19, 23, 27, 31, 18, 22, 26, 30, 17, 21, 25, 29, 16, 20, 24, 28),
(3, 26): (3, 2, 7, 6, 1, 0, 5, 4, 19, 18, 23, 22, 17, 16, 21, 20, 11, 10, 15, 14, 9, 8, 13, 12, 27, 26, 31, 30, 25, 24, 29, 28),
(3, 30): (3, 2, 1, 0, 11, 10, 9, 8, 19, 18, 17, 16, 27, 26, 25, 24, 7, 6, 5, 4, 15, 14, 13, 12, 23, 22, 21, 20, 31, 30, 29, 28),
(3, 32): (3, 2, 7, 6, 11, 10, 15, 14, 19, 18, 23, 22, 27, 26, 31, 30, 1, 0, 5, 4, 9, 8, 13, 12, 17, 16, 21, 20, 25, 24, 29, 28),
(3, 40): (3, 7, 11, 15, 2, 6, 10, 14, 19, 23, 27, 31, 18, 22, 26, 30, 1, 5, 9, 13, 0, 4, 8, 12, 17, 21, 25, 29, 16, 20, 24, 28),
(7, 0): (7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24),
(7, 6): (7, 6, 5, 4, 15, 14, 13, 12, 3, 2, 1, 0, 11, 10, 9, 8, 23, 22, 21, 20, 31, 30, 29, 28, 19, 18, 17, 16, 27, 26, 25, 24),
(7, 24): (7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 31, 30, 29, 28, 27, 26, 25, 24),
(7, 30): (7, 6, 5, 4, 15, 14, 13, 12, 23, 22, 21, 20, 31, 30, 29, 28, 3, 2, 1, 0, 11, 10, 9, 8, 19, 18, 17, 16, 27, 26, 25, 24),
(7, 60): (7, 6, 15, 14, 23, 22, 31, 30, 5, 4, 13, 12, 21, 20, 29, 28, 3, 2, 11, 10, 19, 18, 27, 26, 1, 0, 9, 8, 17, 16, 25, 24),
(15, 0): (15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16),
(15, 24): (15, 14, 13, 12, 11, 10, 9, 8, 31, 30, 29, 28, 27, 26, 25, 24, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16),
(31, 0): (31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0)
}

Subgroups
editO_{h} 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 Dih_{4} 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 D_{2d} 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
editAll 25 bundles of similar subgroups 

tetrahedral, rotational and reflective subgroups  

chiral tetrahedral and pyritohedral subgroups  

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
editO_{h} S_{4} × C_{2} [4,3]  

Subgroups of order 24
editT_{d} S_{4} green orange [3,3]  

O S_{4} blue red [4,3]^{+}  

T_{h} A_{4} × C_{2} [3^{+},4]  

Subgroups of order 16
editD_{4h} Dih_{4} × C_{2} [2,4]  

Subgroups of order 12
editT A_{4} [3,3]^{+}  


D_{3d} Dih_{6} [2^{+},6]  


Subgroups of order 8
editD_{2h} C_{2}^{3} white [2,2]  

D_{2h} C_{2}^{3} green [2,2]  

(Below the C_{2}^{3} subgroups are shown in more detail.)
C_{4h} C_{4} × C_{2} [4^{+},2]  

C_{4v} Dih_{4} green red [4]  


D_{2d} Dih_{4} blue orange [2^{+},4]  

D_{2d} Dih_{4} green orange [2^{+},4]  

     
D_{4} Dih_{4} blue red [2,4]^{+}  

(Below the Dih_{4} subgroups are shown in more detail.)
Subgroups of order 6
editC_{3v} S_{3} green [3]  

D_{3} S_{3} blue [2,3]^{+}  

(Below the S_{3} subgroups are shown in more detail.)
S_{6} C_{6} [2^{+},6^{+}]  

Subgroups of order 4
editS_{4} C_{4} orange [2^{+},4^{+}]  

C_{4} C_{4} red [4]^{+}  

C_{2h} = D_{1d} V inv white [2^{+},2]  

C_{2h} = D_{1d} V inv green [2^{+},2]  

C_{2v} V green blue yellow [2]  

C_{2v} V yellow white [2]  

C_{2v} V green white [2]  

D_{2} V blue white [2,2]^{+}  

D_{2} V white [2,2]^{+}  

Subgroups of order 3
editC_{3} C_{3} [3]^{+}  

Subgroups of order 2
editS_{2} C_{2} inv [2^{+},2^{+}]  

C_{s} = C_{1v} C_{2} yellow [ ]  

C_{s} = C_{1v} C_{2} green [ ]  

C_{2} C_{2} white [2]^{+}  

C_{2} C_{2} blue [2]^{+}  

The trivial group
editC_{1} C_{1} [ ]^{+}  

Different appearances of the same group
editSymmetry group of the cuboid
editThe symmetry group of the cuboid C_{2}^{3} appears in two essentially different ways as a subgroup of O_{h}.
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 zaxis is shown below.
There are 4 individual subgroups (see above).
D_{2h} C_{2}^{3} white [2,2]  

D_{2h} C_{2}^{3} green [2,2]  

Symmetry group of the square
editThe symmetry group of the square appears in four essentially different ways as a subgroup of O_{h}. (C_{4v} or [4] being the most intuitive among them.)
There are 12 individual Dih_{4} subgroups (see above). Shown below are the four where the square is seen "from above", i.e. a point on the positive zaxis.
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 nonzero 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.
Square permutations for comparison  

C_{4v} Dih_{4} green red [4]  
