# 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.
Conjugacy classes :
| Elements of O | Elements of O | Elements of O | Inversions of elements of O | Inversions of 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 |
Examples :
| $(0,0)=0$ | $(3,0)=7$ | $(0,3)=3$ | $(7,1)=1'$ | $(4,5)=9'$ |
| neut | inv2 | rot3 | rot2 | rot1 |
| inv3 | ref1 | rotref3 | ref2 | rotref1 |
| $(7,0)=0'$ | $(4,0)=7'$ | $(7,3)=3'$ | $(0,1)=1$ | $(3,5)=9$ |
| 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 $S_{2}\wr S_{3}\simeq S_{2}^{3}\rtimes S_{3}$, and a natural way to identify its elements is as pairs $(m,n)$ with $m\in [0,2^{3})$ and $n\in [0,3!)$. But as it is also the direct product $S_{4}\times S_{2}$, one can simply identify the elements of tetrahedral subgrup T$_d$ as $a\in [0,4!)$ and their inversions as $a'$.
So e.g. the identity $(0,0)$ is represented as $0$ and the inversion $(7,0)$ as $0'$. $(3,1)$ is represented as $6$ and $(4,1)$ as $6'$.
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 :
* The reflection $7'$applied on the 120° rotation $4$gives the 60° rotoreflection $8'$.
* $7'\circ 4=8'$
* The reflection $7'$applied on the 90° rotation $22'$gives the 90° rotoreflection $17$.
* $7'\circ 22'=17$

## Cayley table

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

### S$_4$

The easier one is that in S$_4$ based notation: It simply doubles the Cayley table of S$_4$. With $a,b,c\in S_{4}$ and $a',b',c'$ being their respective inversions, $a\circ b=c$ implies $a'\circ b'=c$ and $a\circ b'=a'\circ b=c'$.
Example: $12'\circ 21=18'$

### pairs

The notation with pairs is probably more useful, but also more complicated:
    $(pm,pn)=(am,an)\circ (bm,bn)$
The calculation of $pm$ involves the cube vertex permutation shown in the images.  E.g. $\operatorname {cvp} (2,3)=(2,6,3,7,~0,4,1,5)$.
    $pm=\operatorname {cvp} (am,an)[bm]$
$pn$ simply follows from the Cayley table of S$_3$:
    $pn=an\circ bn$
Example: $(pm,pn)=(2,3)\circ (6,2)$
    $pm=\operatorname {cvp} (2,3)[6]=1$
    $pn=3\circ 2=5$
example :
| $12'$ | $\circ$ | $21$ | $=$ | $18'$ |
| $(2,3)$ | $\circ$ | $(6,2)$ | $=$ | $(1,5)$ |

## 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
* The right-hand coordinate system used in this article
truncated cuboctahedron :
number matrices :
S$_4$ based identifiers :
| 0 | 1 | 2 | 3 | 4 | 5 |
| 23' | 22' | 21' | 20' | 19' | 18' |
| 16' | 17' | 13' | 12' | 15' | 14' |
| 7 | 6 | 10 | 11 | 8 | 9 |
| 7' | 6' | 10' | 11' | 8' | 9' |
| 16 | 17 | 13 | 12 | 15 | 14 |
| 23 | 22 | 21 | 20 | 19 | 18 |
| 0' | 1' | 2' | 3' | 4' | 5' |
Reverse colexicographic ranks :
| 0 | 722 | 288 | 2210 | 1032 | 2354 |
| 5167 | 6490 | 5455 | 7978 | 6936 | 8258 |
| 11536 | 10213 | 12096 | 14018 | 10615 | 13738 |
| 16703 | 15981 | 17263 | 19786 | 16519 | 19642 |
| 23616 | 24338 | 23056 | 20533 | 23800 | 20677 |
| 28783 | 30106 | 28223 | 26301 | 29704 | 26581 |
| 35152 | 33829 | 34864 | 32341 | 33383 | 32061 |
| 40319 | 39597 | 40031 | 38109 | 39287 | 37965 |
Inversion numbers :
| 0 | 2 | 4 | 6 | 6 | 8 |
| 4 | 6 | 8 | 10 | 10 | 12 |
| 8 | 6 | 12 | 14 | 10 | 12 |
| 12 | 10 | 16 | 18 | 14 | 16 |
| 16 | 18 | 12 | 10 | 14 | 12 |
| 20 | 22 | 16 | 14 | 18 | 16 |
| 24 | 22 | 20 | 18 | 18 | 16 |
| 28 | 26 | 24 | 22 | 22 | 20 |
Left inversion counts :
| 00000000 | 00100010 | 00002200 | 00102030 | 00003210 | 00103130 |
| 01010101 | 00210021 | 01012301 | 00212041 | 00004321 | 00104241 |
| 00220022 | 01020102 | 00004422 | 00104252 | 01012402 | 00212052 |
| 01230123 | 01130113 | 01014523 | 00214263 | 01013513 | 00213163 |
| 00004444 | 00104454 | 00220044 | 01020304 | 00221054 | 01021404 |
| 01014545 | 00214465 | 01230145 | 01130315 | 00222165 | 01022515 |
| 00224466 | 01024546 | 00222266 | 01022526 | 01230246 | 01130326 |
| 01234567 | 01134557 | 01232367 | 01132537 | 01231357 | 01131437 |
It can be seen that the reverse colexicographic ranks of two inversions (in the sense of point reflections) add up to 40319 - the rank of the inversion (in the sense of point reflection) itself.
This means that their inversion sets (sets of pairs out of their natural order) are complements. (Beware the two different meanings of inversion here.) :

### Hexagon corresponding to top matrix row

3D diagrams :
| permutohedron coordinates | permutohedron | Cayley graph generated by and | Cayley graph generated by and |
| example solid | permutohedron | Cayley graph generated by and | Cayley graph generated by and |
2D equivalents :
| permutohedron | permutohedron | Cayley graph generated by two reflections | Cayley graph generated by reflection and rotation |

### 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 :
truncated cuboctahedron :
cube :
truncated cuboctahedron :
cube :
truncated cuboctahedron :
cube :
truncated cuboctahedron :
cube :
truncated cuboctahedron :
cube :
truncated cuboctahedron :

### Conjugacy classes

The full octahedral group has  A000712(3) = 10 conjugacy classes.
Two permutations $(m,n)$ and $(k,n)$ are complementary to each other, if $m+k=7$. 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) | inv2 (3 × 1) | inv2 (3 × 1) | ref1 (3 × 1) | ref1 (3 × 1) | ref1 (3 × 1) |
| 12 | 20 | 24 | 16 | 8 | 4 |
rot3 (4 × 2) :
| 6 | 6 | 14 | 14 | 18 | 18 | 18 | 18 |
rotref3 (4 × 2) :
| 22 | 22 | 14 | 14 | 10 | 10 | 10 | 10 |
rot2 (6 × 1) :
| 26 | 18 | 20 | 12 | 24 | 8 |
ref2 (6 × 1) :
| 2 | 10 | 8 | 16 | 4 | 20 |
| rot1 (3 × 2) | rot1 (3 × 2) | rot1 (3 × 2) | rot1 (3 × 2) | rot1 (3 × 2) | rot1 (3 × 2) | rotref1 (3 × 2) | rotref1 (3 × 2) | rotref1 (3 × 2) | rotref1 (3 × 2) | rotref1 (3 × 2) | rotref1 (3 × 2) |
| 6 | 6 | 12 | 12 | 12 | 12 | 22 | 22 | 16 | 16 | 16 | 16 |
Conjugacy classes of square permutations :
| neut | inv2 | ref1 | ref1 | ref2 | ref2 | rot1 | rot1 |
| 0 | 6 | 2 | 4 | 1 | 5 | 3 | 3 |
* Examples for tesseract and penteract
* The respective numbers of conjugacy classes for 4 and 5 dimensions are 20 and 36.
* The following dictionaries show an example pair and the corresponding permutation from each conjugacy class.
* (See here for the whole tesseract group.)
* 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

O$_h$ 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 |

### List of all subgroups

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

#### Subgroups of order 12

D$_{3d}$ Dih$_6$ [2$^+$,6] :
submatrix :

#### Subgroups of order 8

(Below the C$_2$$^3$ subgroups are shown in more detail.)
C$_{4v}$ Dih$_4$ green red [4] :
submatrix :
Permutations of the square for comparison :
| Permutations of the square for comparison | Cube permutations that leave the z-axis fixed |
(Below the Dih$_4$ subgroups are shown in more detail.)

#### Subgroups of order 6

(Below the S$_3$ subgroups are shown in more detail.)

#### The trivial group
### Different appearances of the same group
#### Symmetry group of the cuboid

The 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 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 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 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.

#### Symmetry group of the triangle

The symmetry group of the triangle appears in two essentially different ways as a subgroup of O$_h$, with C$_{3v}$ 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

| | S$_4$ blue red | A$_4$ × C$_2$ | A$_4$ | S$_4$ green orange |
| shown above | | | | |
| cuboctahedral | | | | |
| contained hexagon | | | | |
| | S$_3$ blue | C$_6$ | C$_3$ | S$_3$ 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. (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 S$_4$ based numbers).