View source for Full octahedral group

[[File:Polyhedron pair 6-8.png|thumb|A {{w|compound of cube and octahedron}} with full octahedral symmetry]]
[[File:Full octahedral group; cycle graph.svg|thumb|[[w:Cycle graph (algebra)|Cycle graph]] of O<sub>h</sub>]]
[[File:Symmetric group 4; Cayley table (left); numbers.svg|thumb|Cayley table of the [[Symmetric group S4|symmetric group S<sub>4</sub>]]]]
[[File:Symmetric group 3; Cayley table; matrices (left).svg|thumb|Cayley table of the symmetric group S<sub>3</sub> with 3×3 permutation matrices]]

The '''[[w:Octahedral symmetry|full octahedral group]] O<sub>h</sub>''' is the {{w2|hyperoctahedral group}} of dimension 3. This article mainly looks at it as the symmetry group of the cube.

There are {{oeis|A000165}}(3) = 48 permutation of the cube. Half of them are its rotations, forming the subgroup O (the [[symmetric group S4|symmetric group S<sub>4</sub>]]), and the other half are their inversions.<br>
The [[w:Point reflection|inversion]] is the permutation that exchanges opposite vertices of the cube. <small>It is not to be confused with the [[Inversion (discrete mathematics)|inversion of a permutation]].</small>

{| class=wikitable
|+ {{w2|Conjugacy class}}es
!colspan="3" style="border-right: 3px solid black;"|Elements of O
!colspan="3"|Inversions of elements of O
|-
|identity
|style="background:#fff;"|'''neut'''
|style="background:#fff; border-right: 3px solid black;"|'''0'''
| inversion
|style="background:#ffff7f;"|'''inv3'''
|style="background:#ffff7f;"|'''0<nowiki>'</nowiki>'''
|-
|3 &times; rotation by 180° about a 4-fold axis
|style="background:#fff;"|'''inv2'''
|style="background:#fff; border-right: 3px solid black;"|'''7, 16, 23'''
|3 &times; reflection in a plane perpendicular to a 4-fold axis
|style="background:#ffff7f;"|'''ref1'''
|style="background:#ffff7f;"|'''7', 16', 23<nowiki>'</nowiki>'''
|-
|8 &times; rotation by 120° about a 3-fold axis
|style="background:#fff;"|rot3
|style="background:#fff; border-right: 3px solid black;"|3, 4, 8, 11, 12, 15,  19, 20
|8 &times; rotoreflection by 60°
|style="background:#ffff7f;"|rotref3
|style="background:#ffff7f;"|3', 4', 8', 11', 12', 15',  19', 20'
|-
|6 &times; rotation by 180° about a 2-fold axis
|style="background:#3375ff;"|rot2
|style="background:#3375ff; border-right: 3px solid black;"|1', 2', 5', 6', 14', 21'
|6 &times; reflection in a plane perpendicular to a 2-fold axis
|style="background:#33d42a;"|ref2
|style="background:#33d42a;"|1, 2, 5, 6, 14, 21
|-
|6 &times; rotation by 90° about a 4-fold axis
|style="background:#ef2500;"|rot1
|style="background:#ef2500; border-right: 3px solid black;"|9', 10', 13', 17', 18', 22'
|6 &times; rotoreflection by 90°
|style="background:#ffa200;"|rotref1
|style="background:#ffa200;"|9, 10, 13, 17, 18, 22
|}

{| class="wikitable collapsible collapsed" style="text-align: center;"
!colspan="5"| Examples
|-
| [[File:Cube permutation 0 0.svg|180px]]<br><math>(0,0) = 0</math>
| [[File:Cube permutation 3 0.svg|180px]]<br><math>(3,0) = 7</math>
| [[File:Cube permutation 0 3.svg|180px]]<br><math>(0,3) = 3</math>
| [[File:Cube permutation 7 1.svg|180px]]<br><math>(7,1) = 1'</math>
| [[File:Cube permutation 4 5.svg|180px]]<br><math>(4,5) = 9'</math>
|-
|style="background:#fff;"|'''neut'''
|style="background:#fff;"|'''inv2'''
|style="background:#fff;"|rot3
|style="background:#3375ff;"|rot2
|style="background:#ef2500;"|rot1
|- style="border-top: 3px solid black"
|style="background:#ffff7f;"|'''inv3'''
|style="background:#ffff7f;"|'''ref1'''
|style="background:#ffff7f;"|rotref3
|style="background:#33d42a;"|ref2
|style="background:#ffa200;"|rotref1
|-
| <math>(7,0) = 0'</math><br>[[File:Cube permutation 7 0.svg|180px]]
| <math>(4,0) = 7'</math><br>[[File:Cube permutation 4 0.svg|180px]]
| <math>(7,3) = 3'</math><br>[[File:Cube permutation 7 3.svg|180px]]
| <math>(0,1) = 1</math><br>[[File:Cube permutation 0 1.svg|180px]]
| <math>(3,5) = 9</math><br>[[File:Cube permutation 3 5.svg|180px]]
|-
|colspan="5"| See full list below: [[#8×6 matrix|8×6 matrix]] or [[#Conjugacy classes|Conjugacy classes]]
|}

As the [[w:hyperoctahedral group|hyperoctahedral group]] of dimension 3 the full octahedral group is the {{w2|wreath product}} <math>S_2 \wr S_3 \simeq S_2^3 \rtimes S_3</math>,<br>and a natural way to identify its elements is as pairs <math>(m, n)</math> with <math>m \in [0, 2^3)</math> and <math>n \in [0, 3!)</math>.<br>But as it is also the [[w:Direct product of groups|direct product]] <math>S_4 \times S_2</math>, one can simply identify the elements of tetrahedral subgrup ''T<sub>d</sub>'' as <math>a \in [0, 4!)</math> and their inversions as <math>a'</math>.

So e.g. the identity <math>(0, 0)</math> is represented as <math>0</math> and the inversion <math>(7, 0)</math> as <math>0'</math>.<br>
<math>(3, 1)</math> is represented as <math>6</math> and <math>(4, 1)</math> as <math>6'</math>.

A [[w:Improper rotation|rotoreflection]] is a combination of rotation and reflection.<br>
<small>While a rotation leaves its axis and a reflection leaves its plane unchanged, a rotoreflection leaves only the center unchanged.</small>

{| class="wikitable collapsible collapsed" style="text-align: center;"
!colspan="5"| Illustration of rotoreflections
|-
|
{{multiple image
 | align = left
 | image1 = Cube permutation 4 0.svg
 | width1 =  230
 | caption1 = The reflection <math>7'</math>
 | image2 = Cube permutation 0 4.svg
 | width2 =  230
 | caption2 = applied on the 120° rotation <math>4</math>
 | image3 = Cube permutation 4 4.svg
 | width3 =  230
 | caption3 = gives the 60° rotoreflection <math>8'</math>.
}}
<br><math>7' \circ 4 = 8'</math>
|-
|
{{multiple image
 | align = left
 | image1 = Cube permutation 4 0.svg
 | width1 =  230
 | caption1 = The reflection <math>7'</math>
 | image2 = Cube permutation 1 1.svg
 | width2 =  230
 | caption2 = applied on the 90° rotation <math>22'</math>
 | image3 = Cube permutation 5 1.svg
 | width3 =  230
 | caption3 = gives the 90° rotoreflection <math>17</math>.
}}
<br><math>7' \circ 22' = 17</math>
|}


==Cayley table==

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

===S<sub>4</sub>===

The easier one is that in [[symmetric group S4|S<sub>4</sub>]] based notation: It simply doubles the Cayley table of S<sub>4</sub>.<br>
With <math>a, b, c \in S_4</math> and <math>a', b', c'</math> being their respective inversions, <math>a \circ b = c</math> implies <math>a' \circ b' = c</math> and <math>a \circ b' = a' \circ b = c'</math>.

Example: <math>12' \circ 21 = 18'</math>

===pairs===

The notation with pairs is probably more useful, but also more complicated:
: <math>(pm, pn) = (am, an) \circ (bm, bn)</math>

The calculation of <math>pm</math> involves the cube vertex permutation shown in the images. &nbsp;
E.g. <math>\operatorname{cvp}(2, 3) = (2, 6, 3, 7, ~ 0, 4, 1, 5)</math>.

: <math>pm = \operatorname{cvp}(am, an)[bm]</math>

<math>pn</math> simply follows from the Cayley table of S<sub>3</sub>:
: <math>pn = an \circ bn</math>

Example: <math>(pm, pn) = (2, 3) \circ (6, 2)</math>
: <math>pm = \operatorname{cvp}(2, 3)[6] = 1</math>
: <math>pn = 3 \circ 2 = 5</math><br>

{| class="wikitable collapsible collapsed" style="text-align: center;"
!example
|-
|
{|
| [[File:Cube permutation 2 3.svg|230px|border]]
|
| [[File:Cube permutation 6 2.svg|230px|border]]
| 
| [[File:Cube permutation 1 5.svg|230px|border]]
|-
| <math>12'</math>
|rowspan="2"| <math>\circ</math>
| <math>21</math>
|rowspan="2"| <math>=</math>
| <math>18'</math>
|-
| <math>(2, 3)</math>
| <math>(6, 2)</math>
| <math>(1, 5)</math>
|}
|}

==Overview==


===Truncated cuboctahedron===

The vertices of the {{w|truncated cuboctahedron}} correspond to the elements of this group. Each of its faces of its dual, the {{w|disdyakis dodecahedron}}, is a {{w|fundamental domain}}.

{| style="width: 100%;"
| [[File:Full octahedral group elements in truncated cuboctahedron; numbers.svg|280px]]
| [[File:Full octahedral group elements in truncated cuboctahedron; JF.png|320px]]
| [[File:Polyhedron great rhombi 6-8 max.png|280px]]
| [[File:Polyhedron great rhombi 6-8 dual max.png|280px]]
|}


===8×6 matrix===

{| class="collapsible open" style="width: 100%; text-align: center; border: 1px solid #ddd;"
! bgcolor="#eee"|
|-
|
{{Full octahedral group; image matrices}}
<!--START NESTED BOX 1-->
{| class="collapsible collapsed" style="width: 100%; border: 1px solid #999; border-bottom:none;"
!bgcolor="white"| truncated cuboctahedron
|-
|{{Full octahedral group; image matrices; truncated octahedron}}
|}<!--END NESTED BOX 1-->
<!--START NESTED BOX 2-->
{| class="collapsible collapsed" style="width: 100%; border: 1px solid #999; background-color: white;"
!bgcolor="white"| number matrices
|-
|{{Full octahedral group; text matrices}}
|-
|style="text-align:left;"|
It can be seen that the reverse colexicographic ranks of two inversions <small>(in the sense of [[w:Point reflection|point reflections]])</small> add up to 40319 - the rank of ''the inversion'' <small>(in the sense of point reflection)</small> itself.<br>
This means that their [[Inversion (discrete mathematics)|inversion sets]] <small>(sets of pairs out of their natural order)</small> are complements. (Beware the two different meanings of ''inversion'' here.)
|}<!--END NESTED BOX 2-->
|}

===Hexagon corresponding to top matrix row===

{| class="collapsible open" style="width: 100%; border: 1px solid #ddd; border-bottom:0;"
! bgcolor="#eee"| 3D diagrams
|-
|
The files below illustrate the subgroup '''C<sub>3v</sub>''' or '''[3]''' that corresponds to the top matrix row. It contains the six permutations of the cube that leave the main diagonal fixed.

{| style="width:100%; text-align:center;"
| [[File:Permutohedron order 3 in cube, 1-based.png|thumb|center|200px|permutohedron coordinates]]
|rowspan="2"| [[File:Permutohedron of S3; cube permutations.png|thumb|center|400px|[[w:permutohedron|permutohedron]] ]]
|rowspan="2"| [[File:Cayley graph of S3 with cube permutations; generators a, b.png|thumb|center|400px|[[w:Cayley graph|Cayley graph]] generated by [[File:Finite permutation number 1.svg|20px]] and [[File:Finite permutation number 2.svg|20px]]]]
|rowspan="2"| [[File:Cayley graph of S3 with cube permutations; generators a, r.png|thumb|center|400px|Cayley graph generated by [[File:Finite permutation number 1.svg|20px]] and [[File:Finite permutation number 4.svg|20px]]]]
|-
| [[File:Subgroup of Oh; S3 green 03; example solid.png|thumb|center|150px|example solid]]
|}
|}

{| class="collapsible collapsed" style="width: 100%; border: 1px solid #ddd; border-bottom:0;"
! bgcolor="#eee"| 2D equivalents
|-
|
The rest of this article uses left [[w:group action|action]], i.e. <math>f \cdot g</math> means ''first <math>g</math>, then <math>f</math>'',<br>
but these two-dimensional diagrams use right action, i.e. <math>f \cdot g</math> means ''first <math>f</math>, then <math>g</math>''.<br>
So <math>{\color{RoyalBlue}a} ~ \widehat{=}</math> [[File:Finite permutation number 1.svg|20px]] and <math>{\color{RubineRed}r} ~ \widehat{=}</math> [[File:Finite permutation number 4.svg|20px]], but <math>{\color{RoyalBlue}a} \cdot {\color{RubineRed}r} ~ \widehat{=}</math> [[File:Finite permutation number 4.svg|20px]]<math>\cdot</math>[[File:Finite permutation number 1.svg|20px]]<math>=</math>[[File:Finite permutation number 5.svg|20px]] and <math>{\color{RubineRed}r} \cdot {\color{RoyalBlue}a} ~ \widehat{=}</math> [[File:Finite permutation number 1.svg|20px]]<math>\cdot</math>[[File:Finite permutation number 4.svg|20px]]<math>=</math>[[File:Finite permutation number 2.svg|20px]].

{| style="width:100%; text-align:center;"
| [[File:Permutohedron of S3 with permutation results as coordinates; numbers.svg|thumb|center|200px|permutohedron]]
| [[File:Permutohedron of S3 with permutation results as coordinates; triangles.svg|thumb|center|400px|permutohedron]]
| [[File:Cayley graph of S3 with triangles; generators a, b.svg|thumb|center|400px|Cayley graph generated by two reflections]]
| [[File:Cayley graph of S3 with triangles; generators a, r.svg|thumb|center|400px|Cayley graph generated by reflection and rotation]]
|}
|}

{| class="collapsible collapsed" style="width: 100%; border: 1px solid #ddd;"
!colspan="2" bgcolor="#eee"| details
|-
|style="width:260px;"| [[File:Subgroup of Oh; S3 green 03; example solid (triangle).png|thumb|center|200px|permuted triangle in the cube (alternative example solid)]]
| {{Full octahedral group; hexagon}}
|-
|colspan="2"| This is left action again, so the 3×3 permutation matrices shown here are the [[w:Transpose|transposes]] of those in the small permutohedron in the box above - which only makes a difference for [[File:Finite permutation number 3.svg|20px]] and [[File:Finite permutation number 4.svg|20px]].
|}

===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.<br>
In the other seven positions are the products of applying the reflections along coordinate axes on these basic permutations.

{{Full octahedral group; cube of cube perms|0}}
{{Full octahedral group; cube of cube perms|1}}
{{Full octahedral group; cube of cube perms|2}}
{{Full octahedral group; cube of cube perms|3}}
{{Full octahedral group; cube of cube perms|4}}
{{Full octahedral group; cube of cube perms|5}}

===Conjugacy classes===

The full octahedral group has {{oeis|A000712}}(3) = 10 conjugacy classes.

Two permutations <math>(m, n)</math> and <math>(k, n)</math> are complementary to each other, if <math>m + k = 7</math>.<br>
Complementary permutations sum up to a vector of 7s, and their [[inversion (discrete mathematics)|inversion sets]] are {{w|Complement (set theory)|complements}},<br>
so their inversion numbers sum up to 28.  <small>(Compare one of the number matrices [[#8×6 matrix|above]].)</small>

The conjugacy classes below are always shown in complementary pairs (like inv2/ref1 or rot2/ref2).<br>
The numbers over the triangles are the inversion numbers of the corresponding permutations. It can be seen that corresponding numbers add up to 28.

{| class="wikitable" style="width: 100%; text-align: center;"
|-
!style="background: #ffffff;"| neut <small>(1)</small>
!style="background: #ffff7f; border-right: 3px solid #a2a9b1;"| inv3 <small>(1)</small>
!style="background: #ffffff;"| inv2 <small>(3)</small>
!style="background: #ffff7f; border-right: 3px solid #a2a9b1;"| ref1 <small>(3)</small>
!style="background: #ffffff;"| rot3 <small>(8)</small>
!style="background: #ffff7f; border-right: 3px solid #a2a9b1;"| rotref3 <small>(8)</small>
!style="background: #3375ff;"| rot2 <small>(6)</small>
!style="background: #33d42a; border-right: 3px solid #a2a9b1;"| ref2 <small>(6)</small>
!style="background: #ef2500;"| rot1 <small>(6)</small>
!style="background: #ffa200;"| rotref1 <small>(6)</small>
|-
| [[File:Full octahedral group; set partition neut.svg|120px]]
|style="border-right: 3px solid #a2a9b1;"| [[File:Full octahedral group; set partition inv3.svg|120px]]
| [[File:Full octahedral group; set partition inv2 0.svg|120px]]
|style="border-right: 3px solid #a2a9b1;"| [[File:Full octahedral group; set partition ref1 0.svg|120px]]
| [[File:Full octahedral group; set partition rot3 0.svg|120px]]
|style="border-right: 3px solid #a2a9b1;"| [[File:Full octahedral group; set partition rotref3 0.svg|120px]]
| [[File:Full octahedral group; set partition rot2 0b.svg|120px]]
|style="border-right: 3px solid #a2a9b1;"| [[File:Full octahedral group; set partition ref2 0b.svg|120px]]
| [[File:Full octahedral group; set partition rot1 0.svg|120px]]
| [[File:Full octahedral group; set partition rotref1 0.svg|120px]]
|}

{| class="wikitable collapsible collapsed" style="width: 100%; text-align: center;"
|-
!style="width: 50%; background: #fff; border-right: 3px solid black;"| neut <small>(1 × 1)</small>
!style="width: 50%; background: #ffff7f;"| inv3 <small>(1 × 1)</small>
|-
|style="border-right: 3px solid black;"| 0<br>[[File:Full octahedral group; set partition neut.svg|180px]]
| 28<br>[[File:Full octahedral group; set partition inv3.svg|180px]]
|-
|style="border-right: 3px solid black;"| [[File:Cube permutation 0 0.svg|100px|border]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Cube permutation 7 0.svg|100px|border]]<br>[[File:Cube vertex number 7.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
|}

{| class="wikitable collapsible collapsed" style="width: 100%; text-align: center;"
|-
!colspan="3" style="width: 50%; background: #fff; border-right: 3px solid black;"| inv2 <small>(3 × 1)</small>
!colspan="3" style="width: 50%; background: #ffff7f;"| ref1 <small>(3 × 1)</small>
|-
| 12<br>[[File:Full octahedral group; set partition inv2 0.svg|180px]]
| 20<br>[[File:Full octahedral group; set partition inv2 1.svg|180px]]
|style="border-right: 3px solid black;"| 24<br>[[File:Full octahedral group; set partition inv2 2.svg|180px]]
| 16<br>[[File:Full octahedral group; set partition ref1 2.svg|180px]]
| 8<br>[[File:Full octahedral group; set partition ref1 1.svg|180px]]
| 4<br>[[File:Full octahedral group; set partition ref1 0.svg|180px]]
|-
| [[File:Cube permutation 3 0.svg|100px|border]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Cube permutation 5 0.svg|100px|border]]<br>[[File:Cube vertex number 5.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
|style="border-right: 3px solid black;"| [[File:Cube permutation 6 0.svg|100px|border]]<br>[[File:Cube vertex number 6.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Cube permutation 4 0.svg|100px|border]]<br>[[File:Cube vertex number 4.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Cube permutation 2 0.svg|100px|border]]<br>[[File:Cube vertex number 2.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Cube permutation 1 0.svg|100px|border]]<br>[[File:Cube vertex number 1.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
|}

{| class="wikitable collapsible collapsed" style="width: 100%; text-align: center;"
|-
!colspan="8" style="width: 50%; background: #fff;"| rot3 <small>(4 × 2)</small>
|-
|colspan="2" style="border-right: 3px solid #a2a9b1;"| 6<br>[[File:Full octahedral group; set partition rot3 0.svg|180px]]
|colspan="2" style="border-right: 3px solid #a2a9b1;"| 14<br>[[File:Full octahedral group; set partition rot3 1.svg|180px]]
|colspan="2" style="border-right: 3px solid #a2a9b1;"| 18<br>[[File:Full octahedral group; set partition rot3 2.svg|180px]]
|colspan="2"| 18<br>[[File:Full octahedral group; set partition rot3 3.svg|180px]]
|-
| [[File:Cube permutation 0 3.svg|100px|border]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 3.svg|20px]]
|style="border-right: 3px solid #a2a9b1;"| [[File:Cube permutation 0 4.svg|100px|border]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 4.svg|20px]]
| [[File:Cube permutation 5 3.svg|100px|border]]<br>[[File:Cube vertex number 5.svg|20px]] [[File:Finite permutation number 3.svg|20px]]
|style="border-right: 3px solid #a2a9b1;"| [[File:Cube permutation 3 4.svg|100px|border]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 4.svg|20px]]
| [[File:Cube permutation 3 3.svg|100px|border]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 3.svg|20px]]
|style="border-right: 3px solid #a2a9b1;"| [[File:Cube permutation 6 4.svg|100px|border]]<br>[[File:Cube vertex number 6.svg|20px]] [[File:Finite permutation number 4.svg|20px]]
| [[File:Cube permutation 6 3.svg|100px|border]]<br>[[File:Cube vertex number 6.svg|20px]] [[File:Finite permutation number 3.svg|20px]]
| [[File:Cube permutation 5 4.svg|100px|border]]<br>[[File:Cube vertex number 5.svg|20px]] [[File:Finite permutation number 4.svg|20px]]
|}

{| class="wikitable collapsible collapsed" style="width: 100%; text-align: center;"
|-
!colspan="8" style="width: 50%; background: #ffff7f;"| rotref3 <small>(4 × 2)</small>
|-
|colspan="2" style="border-right: 3px solid #a2a9b1;"| 22<br>[[File:Full octahedral group; set partition rotref3 0.svg|180px]]
|colspan="2" style="border-right: 3px solid #a2a9b1;"| 14<br>[[File:Full octahedral group; set partition rotref3 1.svg|180px]]
|colspan="2" style="border-right: 3px solid #a2a9b1;"| 10<br>[[File:Full octahedral group; set partition rotref3 2.svg|180px]]
|colspan="2"| 10<br>[[File:Full octahedral group; set partition rotref3 3.svg|180px]]
|-
| [[File:Cube permutation 7 3.svg|100px|border]]<br>[[File:Cube vertex number 7.svg|20px]] [[File:Finite permutation number 3.svg|20px]]
|style="border-right: 3px solid #a2a9b1;"| [[File:Cube permutation 7 4.svg|100px|border]]<br>[[File:Cube vertex number 7.svg|20px]] [[File:Finite permutation number 4.svg|20px]]
| [[File:Cube permutation 2 3.svg|100px|border]]<br>[[File:Cube vertex number 2.svg|20px]] [[File:Finite permutation number 3.svg|20px]]
|style="border-right: 3px solid #a2a9b1;"| [[File:Cube permutation 4 4.svg|100px|border]]<br>[[File:Cube vertex number 4.svg|20px]] [[File:Finite permutation number 4.svg|20px]]
| [[File:Cube permutation 4 3.svg|100px|border]]<br>[[File:Cube vertex number 4.svg|20px]] [[File:Finite permutation number 3.svg|20px]]
|style="border-right: 3px solid #a2a9b1;"| [[File:Cube permutation 1 4.svg|100px|border]]<br>[[File:Cube vertex number 1.svg|20px]] [[File:Finite permutation number 4.svg|20px]]
| [[File:Cube permutation 1 3.svg|100px|border]]<br>[[File:Cube vertex number 1.svg|20px]] [[File:Finite permutation number 3.svg|20px]]
| [[File:Cube permutation 2 4.svg|100px|border]]<br>[[File:Cube vertex number 2.svg|20px]] [[File:Finite permutation number 4.svg|20px]]
|}

{| class="wikitable collapsible collapsed" style="width: 100%; text-align: center;"
|-
!colspan="6" style="width: 50%; background: #3375ff; border-right: 3px solid black;"| rot2 <small>(6 × 1)</small>
|-
| 26<br>[[File:Full octahedral group; set partition rot2 2b.svg|180px]]
|style="border-right: 3px solid #a2a9b1;"| 18<br>[[File:Full octahedral group; set partition rot2 2a.svg|180px]]
| 20<br>[[File:Full octahedral group; set partition rot2 1b.svg|180px]]
|style="border-right: 3px solid #a2a9b1;"| 12<br>[[File:Full octahedral group; set partition rot2 1a.svg|180px]]
| 24<br>[[File:Full octahedral group; set partition rot2 0b.svg|180px]]
| 8<br>[[File:Full octahedral group; set partition rot2 0a.svg|180px]]
|-
| [[File:Cube permutation 7 1.svg|100px|border]]<br>[[File:Cube vertex number 7.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
|style="border-right: 3px solid #a2a9b1;"| [[File:Cube permutation 4 1.svg|100px|border]]<br>[[File:Cube vertex number 4.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
| [[File:Cube permutation 7 5.svg|100px|border]]<br>[[File:Cube vertex number 7.svg|20px]] [[File:Finite permutation number 5.svg|20px]]
|style="border-right: 3px solid #a2a9b1;"| [[File:Cube permutation 2 5.svg|100px|border]]<br>[[File:Cube vertex number 2.svg|20px]] [[File:Finite permutation number 5.svg|20px]]
| [[File:Cube permutation 7 2.svg|100px|border]]<br>[[File:Cube vertex number 7.svg|20px]] [[File:Finite permutation number 2.svg|20px]]
| [[File:Cube permutation 1 2.svg|100px|border]]<br>[[File:Cube vertex number 1.svg|20px]] [[File:Finite permutation number 2.svg|20px]]
|}

{| class="wikitable collapsible collapsed" style="width: 100%; text-align: center;"
|-
!colspan="6" style="width: 50%; background: #33d42a;"| ref2 <small>(6 × 1)</small>
|-
| 2<br>[[File:Full octahedral group; set partition ref2 0a.svg|180px]]
|style="border-right: 3px solid #a2a9b1;"| 10<br>[[File:Full octahedral group; set partition ref2 0b.svg|180px]]
| 8<br>[[File:Full octahedral group; set partition ref2 1a.svg|180px]]
|style="border-right: 3px solid #a2a9b1;"| 16<br>[[File:Full octahedral group; set partition ref2 1b.svg|180px]]
| 4<br>[[File:Full octahedral group; set partition ref2 2a.svg|180px]]
| 20<br>[[File:Full octahedral group; set partition ref2 2b.svg|180px]]
|-
| [[File:Cube permutation 0 1.svg|100px|border]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
|style="border-right: 3px solid #a2a9b1;"| [[File:Cube permutation 3 1.svg|100px|border]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
| [[File:Cube permutation 0 5.svg|100px|border]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 5.svg|20px]]
|style="border-right: 3px solid #a2a9b1;"| [[File:Cube permutation 5 5.svg|100px|border]]<br>[[File:Cube vertex number 5.svg|20px]] [[File:Finite permutation number 5.svg|20px]]
| [[File:Cube permutation 0 2.svg|100px|border]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 2.svg|20px]]
| [[File:Cube permutation 6 2.svg|100px|border]]<br>[[File:Cube vertex number 6.svg|20px]] [[File:Finite permutation number 2.svg|20px]]
|}

{| class="wikitable collapsible collapsed" style="width: 100%; text-align: center;"
|-
!colspan="6" style="width: 50%; background: #ef2500; border-right: 3px solid black;"| rot1 <small>(3 × 2)</small>
!colspan="6" style="width: 50%; background: #ffa200;"| rotref1 <small>(3 × 2)</small>
|-
|colspan="2"| 6<br>[[File:Full octahedral group; set partition rot1 0.svg|180px]]
|colspan="2"| 12<br>[[File:Full octahedral group; set partition rot1 1.svg|180px]]
|colspan="2" style="border-right: 3px solid black;"| 12<br>[[File:Full octahedral group; set partition rot1 2.svg|180px]]
|colspan="2"| 22<br>[[File:Full octahedral group; set partition rotref1 0.svg|180px]]
|colspan="2"| 16<br>[[File:Full octahedral group; set partition rotref1 1.svg|180px]]
|colspan="2"| 16<br>[[File:Full octahedral group; set partition rotref1 2.svg|180px]]
|-
| [[File:Cube permutation 2 1.svg|100px|border]]<br>[[File:Cube vertex number 2.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
| [[File:Cube permutation 1 1.svg|100px|border]]<br>[[File:Cube vertex number 1.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
| [[File:Cube permutation 1 5.svg|100px|border]]<br>[[File:Cube vertex number 1.svg|20px]] [[File:Finite permutation number 5.svg|20px]]
| [[File:Cube permutation 4 5.svg|100px|border]]<br>[[File:Cube vertex number 4.svg|20px]] [[File:Finite permutation number 5.svg|20px]]
| [[File:Cube permutation 4 2.svg|100px|border]]<br>[[File:Cube vertex number 4.svg|20px]] [[File:Finite permutation number 2.svg|20px]]
|style="border-right: 3px solid black;"| [[File:Cube permutation 2 2.svg|100px|border]]<br>[[File:Cube vertex number 2.svg|20px]] [[File:Finite permutation number 2.svg|20px]]
| [[File:Cube permutation 5 1.svg|100px|border]]<br>[[File:Cube vertex number 5.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
| [[File:Cube permutation 6 1.svg|100px|border]]<br>[[File:Cube vertex number 6.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
| [[File:Cube permutation 6 5.svg|100px|border]]<br>[[File:Cube vertex number 6.svg|20px]] [[File:Finite permutation number 5.svg|20px]]
| [[File:Cube permutation 3 5.svg|100px|border]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 5.svg|20px]]
| [[File:Cube permutation 3 2.svg|100px|border]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 2.svg|20px]]
| [[File:Cube permutation 5 2.svg|100px|border]]<br>[[File:Cube vertex number 5.svg|20px]] [[File:Finite permutation number 2.svg|20px]]
|}

{| class="wikitable collapsible collapsed" style="width: 100%; text-align: center;"
|-
!colspan="8"| Conjugacy classes of square permutations
|-
!style="border-right: 3px solid #a2a9b1;"| neut
!style="border-right: 3px solid #a2a9b1;"| inv2
!style="border-right: 3px solid #a2a9b1;" colspan="2"| ref1
!style="border-right: 3px solid #a2a9b1;" colspan="2"| ref2
!colspan="2"| rot1
|-
|style="border-right: 3px solid #a2a9b1;"| 0<br>[[File:Dihedral group Dih 4; set partition neut.svg|120px]]
|style="border-right: 3px solid #a2a9b1;"| 6<br>[[File:Dihedral group Dih 4; set partition inv2.svg|120px]]
| 2<br>[[File:Dihedral group Dih 4; set partition ref1 0.svg|120px]]
|style="border-right: 3px solid #a2a9b1;"| 4<br>[[File:Dihedral group Dih 4; set partition ref1 1.svg|120px]]
| 1<br>[[File:Dihedral group Dih 4; set partition ref2 a.svg|120px]]
|style="border-right: 3px solid #a2a9b1;"| 5<br>[[File:Dihedral group Dih 4; set partition ref2 b.svg|120px]]
|colspan="2"| 3<br>[[File:Dihedral group Dih 4; set partition rot1.svg|120px]]
|-
|style="border-right: 3px solid #a2a9b1;"| [[File:Square permutation 0 0, colored.svg|100px|border]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
|style="border-right: 3px solid #a2a9b1;"| [[File:Square permutation 3 0, colored.svg|100px|border]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Square permutation 1 0, colored.svg|100px|border]]<br>[[File:Cube vertex number 1.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
|style="border-right: 3px solid #a2a9b1;"| [[File:Square permutation 2 0, colored.svg|100px|border]]<br>[[File:Cube vertex number 2.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Square permutation 0 1, colored.svg|100px|border]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
|style="border-right: 3px solid #a2a9b1;"| [[File:Square permutation 3 1, colored.svg|100px|border]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
| [[File:Square permutation 1 1, colored.svg|100px|border]]<br>[[File:Cube vertex number 1.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
| [[File:Square permutation 2 1, colored.svg|100px|border]]<br>[[File:Cube vertex number 2.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
|}

{| class="wikitable collapsible collapsed" style="width: 100%;"
|-
! Examples for tesseract and penteract
|-
|
The respective numbers of conjugacy classes for 4 and 5 dimensions are 20 and 36.<br>
The following dictionaries show an example pair and the corresponding permutation from each conjugacy class.<br>
(See [https://pastebin.com/VuKq29SV here] for the whole tesseract group.)

<syntaxhighlight lang="python">
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)
}
</syntaxhighlight>
|}

==Subgroups==
O<sub>h</sub> has 98 individual subgroups, which are all shown in the list below. <small>(A Python dictionary of them can be found [https://github.com/watchduck/full_octahedral_group/blob/master/projects/p03_subgroups/store_dicts.py here].)</small>

They naturally divide in 33 bundles of similar subgroups, whose elements belong to the same [[w:conjugacy class|conjugacy classes]].<br>
In this article these bundles are given naive names based on some of the colors assigned to their elements (like '''Dih<sub>4</sub> green orange''').<br>
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 [[w:Schoenflies notation|Schoenflies]] or [[w:Coxeter notation|Coxeter]] notation (like '''D<sub>2d</sub>''' or '''[2<sup>+</sup>,4]''').<br>
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:
{| class="wikitable"
|- style="background: #f0e0f0;"
| '''[...]<sup>+</sup>''' || rotate
|- style="background: #e0f0f0;"
| '''[...]''' || reflect
|- style="background: #e0e0e0;"
| '''[...<sup>+</sup>,...<sup>+</sup>]''' || cross
|- style="background: #f0f0e0;"
| '''[...<sup>+</sup>, ...]''' || mixed
|}


===Hasse diagrams===

{| class="collapsible open" style="width: 100%; text-align: center; border: 1px solid #ddd; border-bottom: 0;"
! bgcolor="#eee"| All 25 bundles of similar subgroups
|-
| [[File:Full octahedral group; subgroups Hasse diagram.svg|center|1000px]]
|}
{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #ddd; border-bottom: 0;"
! colspan="4" bgcolor="#eee"| tetrahedral, rotational and reflective subgroups
|-
| [[File:Symmetric group S4; lattice of subgroups Hasse diagram; 11 different cycle graphs.svg|thumb|center|300px|(Subgroups of S<sub>4</sub> as an abstract group)]]
| [[File:Full octahedral group; subgroups Hasse diagram; Td.svg|thumb|center|300px|Subgroups of S<sub>4</sub> as the tetrahedral group '''T<sub>d</sub>''' or '''[3,3]''']]
| [[File:Full octahedral group; subgroups Hasse diagram; rotational.svg|thumb|center|300px|Rotational subgroups<br>Subgroups of S<sub>4</sub> as the chiral octahedral group '''O''' or '''[4,3]<sup>+</sup>''']]
| [[File:Full octahedral group; subgroups Hasse diagram; reflective.svg|thumb|center|300px|Reflective subgroups<br>Coxeter notation is like in the file on the left without the plus signs. But there is no '''[2,3]''', and no green edge from '''[2,2]''' to '''[3,3]'''.]]
|}
{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #ddd;"
! colspan="2" bgcolor="#eee"| chiral tetrahedral and pyritohedral subgroups
|-
| [[File:Full octahedral group; subgroups Hasse diagram; T.svg|thumb|center|260px|Subgroups of [[w:Alternating group|A]]<sub>4</sub> as the chiral tetrahedral group '''T''' or '''[3,3]<sup>+</sup>''']]
| [[File:Full octahedral group; subgroups Hasse diagram; Th.svg|thumb|center|550px|Subgroups of A<sub>4</sub>×C<sub>2</sub> as the pyritohedral group '''T<sub>h</sub>''' or '''[3<sup>+</sup>,4]''']]
|}

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

====The group itself====

{| class="collapsible open" style="width: 100%; text-align: center; border: 1px solid #999;"
!colspan="3" bgcolor="#e0f0f0"|<span style="float:left; padding-left:50px;">O<sub>h</sub></span>   S<sub>4</sub> × C<sub>2</sub>   <span style="float:right; padding-right:50px;">[4,3]</span>
|-
|rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; S4xC2; example solid.png|230px]]
| [[File:Full octahedral group; cycle graph.svg|700px]]
|rowspan="2" style="width:240px;"| [[File:Polyhedron great rhombi 6-8 max.png|thumb|230px|{{w|truncated cuboctahedron}} ]]
|-
| [[File:Subgroup of Oh; S4xC2; matrix.svg|50px]] [[File:Subgroup of Oh; S4xC2; solid.png|80px]]
|}

====Subgroups of order 24====

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;"
!colspan="3" bgcolor="#e0f0f0"|<span style="float:left; padding-left:50px;">T<sub>d</sub></span>    S<sub>4</sub> green orange   <span style="float:right; padding-right:50px;">[3,3]</span>
|-
|rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; S4 green orange; example solid.png|230px]]
| [[File:Subgroup of Oh; S4 green orange; cycle graph.svg|700px]]
|rowspan="2" style="width:240px;"| [[File:Polyhedron great rhombi 6-8 subsolid tetrahedral maxmatch.png|thumb|230px|nonuniform {{w|truncated octahedron}} ]]
|-
| [[File:Subgroup of Oh; S4 green orange; matrix.svg|50px|]] [[File:Subgroup of Oh; S4 green orange; solid.png|80px]]
|}
{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;"
!colspan="3" bgcolor="#f0e0f0"|<span style="float:left; padding-left:50px;">O</span>    S<sub>4</sub> blue red   <span style="float:right; padding-right:50px;">[4,3]<sup>+</sup></span>
|-
|rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; S4 blue red; example solid.png|230px]]
| [[File:Subgroup of Oh; S4 blue red; cycle graph.svg|700px]]
|rowspan="2" style="width:240px;"| [[File:Polyhedron great rhombi 6-8 subsolid snub right maxmatch.png|thumb|230px|nonuniform {{w|snub cube}} ]]
|-
| [[File:Subgroup of Oh; S4 blue red; matrix.svg|50px|]] [[File:Subgroup of Oh; S4 blue red; solid.png|80px]]
|}


{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;"
!colspan="3" bgcolor="#f0f0e0"|<span style="float:left; padding-left:50px;">T<sub>h</sub></span>   A<sub>4</sub> × C<sub>2</sub>   <span style="float:right; padding-right:50px;">[3<sup>+</sup>,4]</span>
|-
|rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; A4xC2; example solid.png|230px]]
| [[File:Subgroup of Oh; A4xC2; cycle graph.svg|550px]]
|rowspan="2" style="width:240px;"| [[File:Polyhedron great rhombi 6-8 subsolid pyritohedral maxmatch.png|thumb|230px|cantic snub octahedron]]
|-
| [[File:Subgroup of Oh; A4xC2; matrix.svg|50px|]] [[File:Subgroup of Oh; A4xC2; solid.png|80px]]
|}

====Subgroups of order 16====

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;"
! bgcolor="#e0f0f0" colspan="4"|<span style="float:left; padding-left:50px;">D<sub>4h</sub></span>   Dih<sub>4</sub> × C<sub>2</sub>   <span style="float:right; padding-right:50px;">[2,4]</span>
|-
|rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; Dih4xC2 07; example solid.png|230px]]
| [[File:Subgroup of Oh; Dih4xC2 07; cycle graph.svg|400px|]]
| [[File:Subgroup of Oh; Dih4xC2 16; cycle graph.svg|400px|]]
| [[File:Subgroup of Oh; Dih4xC2 23; cycle graph.svg|400px|]]
|-
| [[File:Subgroup of Oh; Dih4xC2 07; matrix.svg|50px|]] [[File:Subgroup of Oh; Dih4xC2 07; solid.png|80px]]
| [[File:Subgroup of Oh; Dih4xC2 16; matrix.svg|50px|]] [[File:Subgroup of Oh; Dih4xC2 16; solid.png|80px]]
| [[File:Subgroup of Oh; Dih4xC2 23; matrix.svg|50px|]] [[File:Subgroup of Oh; Dih4xC2 23; solid.png|80px]]
|}

====Subgroups of order 12====

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;"
!colspan="3" bgcolor="#f0e0f0"|<span style="float:left; padding-left:50px;">T</span>    A<sub>4</sub>   <span style="float:right; padding-right:50px;">[3,3]<sup>+</sup></span>
|-
|rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; A4; example solid.png|230px]]
| [[File:Subgroup of Oh; A4; cycle graph.svg|450px]]
|rowspan="2" style="width:240px;"| [[File:Polyhedron great rhombi 6-8 subsolid 20 maxmatch.png|thumb|230px|nonuniform {{w|snub octahedron}} (i.e. {{w|Icosahedron#Pyritohedral_symmetry|icosahedron}}) ]]
|-
| [[File:Subgroup of Oh; A4; matrix.svg|50px|]] [[File:Subgroup of Oh; A4; solid.png|80px]]
|-
|colspan="3"|<!--START NESTED BOX-->
{| class="collapsible collapsed" style="width: 100%; border: 1px solid #999;"
!bgcolor="white"| submatrix
|-
|{{Full octahedral group; submatrix A4}}
|}<!--END NESTED BOX-->
|}


{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;"
! bgcolor="#f0f0e0" colspan="5"|<span style="float:left; padding-left:50px;">D<sub>3d</sub></span>   Dih<sub>6</sub>   <span style="float:right; padding-right:50px;">[2<sup>+</sup>,6]</span>
|-
|rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; Dih6 03; example solid.png|230px]]
| [[File:Subgroup of Oh; Dih6 03; cycle graph.svg|200px|]]
| [[File:Subgroup of Oh; Dih6 11; cycle graph.svg|200px|]]
| [[File:Subgroup of Oh; Dih6 15; cycle graph.svg|200px|]]
| [[File:Subgroup of Oh; Dih6 08; cycle graph.svg|200px|]]
|-
| [[File:Subgroup of Oh; Dih6 03; matrix.svg|50px|]] [[File:Subgroup of Oh; Dih6 03; solid.png|80px]]
| [[File:Subgroup of Oh; Dih6 11; matrix.svg|50px|]] [[File:Subgroup of Oh; Dih6 11; solid.png|80px]]
| [[File:Subgroup of Oh; Dih6 15; matrix.svg|50px|]] [[File:Subgroup of Oh; Dih6 15; solid.png|80px]]
| [[File:Subgroup of Oh; Dih6 08; matrix.svg|50px|]] [[File:Subgroup of Oh; Dih6 08; solid.png|80px]]
|-
|colspan="5"|<!--START NESTED BOX-->
{| class="collapsible collapsed" style="width: 100%; border: 1px solid #999;"
!bgcolor="white"| submatrix
|-
|{{Full octahedral group; submatrix Dih6 03}}
|}<!--END NESTED BOX-->
|}

====Subgroups of order 8====

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;"
!colspan="2" bgcolor="#e0f0f0"|<span style="float:left; padding-left:50px;">D<sub>2h</sub></span>   C<sub>2</sub><sup>3</sup> white   <span style="float:right; padding-right:50px;">[2,2]</span>
|-
|rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; C2^3 white; example solid.png|230px]]
| [[File:Subgroup of Oh; C2^3 white; cycle graph.svg|230px]]
|-
| [[File:Subgroup of Oh; C2^3 white; matrix.svg|50px|]] [[File:Subgroup of Oh; C2^3 white; solid.png|80px|]]
|}
{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;"
! bgcolor="#e0f0f0" colspan="4"|<span style="float:left; padding-left:50px;">D<sub>2h</sub></span>   C<sub>2</sub><sup>3</sup> green   <span style="float:right; padding-right:50px;">[2,2]</span>
|-
|rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; C2^3 green 07; example solid.png|230px]]
| [[File:Subgroup of Oh; C2^3 green 07; cycle graph.svg|220px|]]
| [[File:Subgroup of Oh; C2^3 green 16; cycle graph.svg|220px|]]
| [[File:Subgroup of Oh; C2^3 green 23; cycle graph.svg|220px|]]
|-
| [[File:Subgroup of Oh; C2^3 green 07; matrix.svg|50px|]] [[File:Subgroup of Oh; C2^3 green 07; solid.png|80px|]]
| [[File:Subgroup of Oh; C2^3 green 16; matrix.svg|50px|]] [[File:Subgroup of Oh; C2^3 green 16; solid.png|80px|]]
| [[File:Subgroup of Oh; C2^3 green 23; matrix.svg|50px|]] [[File:Subgroup of Oh; C2^3 green 23; solid.png|80px|]]
|}

<small>([[#Symmetry group of the cuboid|Below]] the C<sub>2</sub><sup>3</sup> subgroups are shown in more detail.)</small>

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;"
! bgcolor="#f0f0e0" colspan="4"|<span style="float:left; padding-left:50px;">C<sub>4h</sub></span>   C<sub>4</sub> × C<sub>2</sub>   <span style="float:right; padding-right:50px;">[4<sup>+</sup>,2]</span>
|-
|rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; C4xC2 07; example solid.png|230px]]
| [[File:Subgroup of Oh; C4xC2 07; cycle graph.svg|350px|]]
| [[File:Subgroup of Oh; C4xC2 16; cycle graph.svg|350px|]]
| [[File:Subgroup of Oh; C4xC2 23; cycle graph.svg|350px|]]
|-
| [[File:Subgroup of Oh; C4xC2 07; matrix.svg|50px|]] [[File:Subgroup of Oh; C4xC2 07; solid.png|80px|]]
| [[File:Subgroup of Oh; C4xC2 16; matrix.svg|50px|]] [[File:Subgroup of Oh; C4xC2 16; solid.png|80px|]]
| [[File:Subgroup of Oh; C4xC2 23; matrix.svg|50px|]] [[File:Subgroup of Oh; C4xC2 23; solid.png|80px|]]
|}


{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;"
! bgcolor="#e0f0f0" colspan="4"|<span style="float:left; padding-left:50px;">C<sub>4v</sub></span>   Dih<sub>4</sub> green red   <span style="float:right; padding-right:50px;">[4]</span>
|-
|rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; Dih4 green red 07; example solid.png|230px]]
| [[File:Subgroup of Oh; Dih4 green red 07; cycle graph.svg|200px|]]
| [[File:Subgroup of Oh; Dih4 green red 16; cycle graph.svg|200px|]]
| [[File:Subgroup of Oh; Dih4 green red 23; cycle graph.svg|200px|]]
|-
| [[File:Subgroup of Oh; Dih4 green red 07; matrix.svg|50px|]] [[File:Subgroup of Oh; Dih4 green red 07; solid.png|80px|]]
| [[File:Subgroup of Oh; Dih4 green red 16; matrix.svg|50px|]] [[File:Subgroup of Oh; Dih4 green red 16; solid.png|80px|]]
| [[File:Subgroup of Oh; Dih4 green red 23; matrix.svg|50px|]] [[File:Subgroup of Oh; Dih4 green red 23; solid.png|80px|]]
|-
|colspan="4"|<!--START NESTED BOX-->
{| class="collapsible collapsed" style="width: 100%; border: 1px solid #999;"
!bgcolor="white"| submatrix
|-
|{{Full octahedral group; submatrix Dih4 green red 07}}
|}<!--END NESTED BOX-->
|}

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;"
! bgcolor="#f0f0e0" colspan="4"|<span style="float:left; padding-left:50px;">D<sub>2d</sub></span>   Dih<sub>4</sub> blue orange   <span style="float:right; padding-right:50px;">[2<sup>+</sup>,4]</span>
|-
|rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; Dih4 blue orange 07; example solid.png|230px]]
| [[File:Subgroup of Oh; Dih4 blue orange 07; cycle graph.svg|200px|]]
| [[File:Subgroup of Oh; Dih4 blue orange 16; cycle graph.svg|200px|]]
| [[File:Subgroup of Oh; Dih4 blue orange 23; cycle graph.svg|200px|]]
|-
| [[File:Subgroup of Oh; Dih4 blue orange 07; matrix.svg|50px|]] [[File:Subgroup of Oh; Dih4 blue orange 07; solid.png|80px|]]
| [[File:Subgroup of Oh; Dih4 blue orange 16; matrix.svg|50px|]] [[File:Subgroup of Oh; Dih4 blue orange 16; solid.png|80px|]]
| [[File:Subgroup of Oh; Dih4 blue orange 23; matrix.svg|50px|]] [[File:Subgroup of Oh; Dih4 blue orange 23; solid.png|80px|]]
|}

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;"
! bgcolor="#f0f0e0" colspan="4"|<span style="float:left; padding-left:50px;">D<sub>2d</sub></span>   Dih<sub>4</sub> green orange   <span style="float:right; padding-right:50px;">[2<sup>+</sup>,4]</span>
|-
|rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; Dih4 green orange 07; example solid.png|230px]]
| [[File:Subgroup of Oh; Dih4 green orange 07; cycle graph.svg|200px|]]
| [[File:Subgroup of Oh; Dih4 green orange 16; cycle graph.svg|200px|]]
| [[File:Subgroup of Oh; Dih4 green orange 23; cycle graph.svg|200px|]]
|-
| [[File:Subgroup of Oh; Dih4 green orange 07; matrix.svg|50px|]]| [[File:Subgroup of Oh; Dih4 green orange 07; solid.png|80px|]]
| [[File:Subgroup of Oh; Dih4 green orange 16; matrix.svg|50px|]]| [[File:Subgroup of Oh; Dih4 green orange 16; solid.png|80px|]]
| [[File:Subgroup of Oh; Dih4 green orange 23; matrix.svg|50px|]]| [[File:Subgroup of Oh; Dih4 green orange 23; solid.png|80px|]]
|}

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;"
! bgcolor="#f0e0f0" colspan="4"|<span style="float:left; padding-left:50px;">D<sub>4</sub></span>   Dih<sub>4</sub> blue red   <span style="float:right; padding-right:50px;">[2,4]<sup>+</sup></span>
|-
|rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; Dih4 blue red 07; example solid.png|230px]]
| [[File:Subgroup of Oh; Dih4 blue red 07; cycle graph.svg|200px|]]
| [[File:Subgroup of Oh; Dih4 blue red 16; cycle graph.svg|200px|]]
| [[File:Subgroup of Oh; Dih4 blue red 23; cycle graph.svg|200px|]]
|-
| [[File:Subgroup of Oh; Dih4 blue red 07; matrix.svg|50px|]] [[File:Subgroup of Oh; Dih4 blue red 07; solid.png|80px|]]
| [[File:Subgroup of Oh; Dih4 blue red 16; matrix.svg|50px|]] [[File:Subgroup of Oh; Dih4 blue red 16; solid.png|80px|]]
| [[File:Subgroup of Oh; Dih4 blue red 23; matrix.svg|50px|]] [[File:Subgroup of Oh; Dih4 blue red 23; solid.png|80px|]]
|}

<small>([[#Symmetry group of the square|Below]] the Dih<sub>4</sub> subgroups are shown in more detail.)</small>

====Subgroups of order 6====

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;"
! bgcolor="#e0f0f0" colspan="5"|<span style="float:left; padding-left:50px;">C<sub>3v</sub></span>   S<sub>3</sub> green   <span style="float:right; padding-right:50px;">[3]</span>
|-
|rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; S3 green 03; example solid.png|230px]]
| [[File:Subgroup of Oh; S3 green 03; cycle graph.svg|170px|]]
| [[File:Subgroup of Oh; S3 green 11; cycle graph.svg|170px|]]
| [[File:Subgroup of Oh; S3 green 15; cycle graph.svg|170px|]]
| [[File:Subgroup of Oh; S3 green 08; cycle graph.svg|170px|]]
|-
| [[File:Subgroup of Oh; S3 green 03; matrix.svg|50px|]] [[File:Subgroup of Oh; S3 green 03; solid.png|80px]]
| [[File:Subgroup of Oh; S3 green 11; matrix.svg|50px|]] [[File:Subgroup of Oh; S3 green 11; solid.png|80px]]
| [[File:Subgroup of Oh; S3 green 15; matrix.svg|50px|]] [[File:Subgroup of Oh; S3 green 15; solid.png|80px]]
| [[File:Subgroup of Oh; S3 green 08; matrix.svg|50px|]] [[File:Subgroup of Oh; S3 green 08; solid.png|80px]]
|}
{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;"
! bgcolor="#f0e0f0" colspan="5"|<span style="float:left; padding-left:50px;">D<sub>3</sub></span>   S<sub>3</sub> blue   <span style="float:right; padding-right:50px;">[2,3]<sup>+</sup></span>
|-
|rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; S3 blue 03; example solid.png|230px]]
| [[File:Subgroup of Oh; S3 blue 03; cycle graph.svg|170px|]]
| [[File:Subgroup of Oh; S3 blue 11; cycle graph.svg|170px|]]
| [[File:Subgroup of Oh; S3 blue 15; cycle graph.svg|170px|]]
| [[File:Subgroup of Oh; S3 blue 08; cycle graph.svg|170px|]]
|-
| [[File:Subgroup of Oh; S3 blue 03; matrix.svg|50px|]] [[File:Subgroup of Oh; S3 blue 03; solid.png|80px]]
| [[File:Subgroup of Oh; S3 blue 11; matrix.svg|50px|]] [[File:Subgroup of Oh; S3 blue 11; solid.png|80px]]
| [[File:Subgroup of Oh; S3 blue 15; matrix.svg|50px|]] [[File:Subgroup of Oh; S3 blue 15; solid.png|80px]]
| [[File:Subgroup of Oh; S3 blue 08; matrix.svg|50px|]] [[File:Subgroup of Oh; S3 blue 08; solid.png|80px]]
|}

<small>([[#Symmetry group of the triangle|Below]] the S<sub>3</sub> subgroups are shown in more detail.)</small>

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;"
! bgcolor="#e0e0e0" colspan="5"|<span style="float:left; padding-left:50px;">S<sub>6</sub></span>   C<sub>6</sub>   <span style="float:right; padding-right:50px;">[2<sup>+</sup>,6<sup>+</sup>]</span>
|-
|rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; C6 03; example solid.png|230px]]
| [[File:Subgroup of Oh; C6 03; cycle graph.svg|200px|]]
| [[File:Subgroup of Oh; C6 11; cycle graph.svg|200px|]]
| [[File:Subgroup of Oh; C6 15; cycle graph.svg|200px|]]
| [[File:Subgroup of Oh; C6 08; cycle graph.svg|200px|]]
|-
| [[File:Subgroup of Oh; C6 03; matrix.svg|50px|]] [[File:Subgroup of Oh; C6 03; solid.png|80px]]
| [[File:Subgroup of Oh; C6 11; matrix.svg|50px|]] [[File:Subgroup of Oh; C6 11; solid.png|80px]]
| [[File:Subgroup of Oh; C6 15; matrix.svg|50px|]] [[File:Subgroup of Oh; C6 15; solid.png|80px]]
| [[File:Subgroup of Oh; C6 08; matrix.svg|50px|]] [[File:Subgroup of Oh; C6 08; solid.png|80px]]
|}

====Subgroups of order 4====

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;"
! bgcolor="#e0e0e0" colspan="4"|<span style="float:left; padding-left:50px;">S<sub>4</sub></span>   C<sub>4</sub> orange   <span style="float:right; padding-right:50px;">[2<sup>+</sup>,4<sup>+</sup>]</span>
|-
|rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; C4 orange 07; example solid.png|230px]]
| [[File:Subgroup of Oh; C4 orange 07; cycle graph.svg|180px|]]
| [[File:Subgroup of Oh; C4 orange 16; cycle graph.svg|180px|]]
| [[File:Subgroup of Oh; C4 orange 23; cycle graph.svg|180px|]]
|-
| [[File:Subgroup of Oh; C4 orange 07; matrix.svg|50px|]] [[File:Subgroup of Oh; C4 orange 07; solid.png|80px]]
| [[File:Subgroup of Oh; C4 orange 16; matrix.svg|50px|]] [[File:Subgroup of Oh; C4 orange 16; solid.png|80px]]
| [[File:Subgroup of Oh; C4 orange 23; matrix.svg|50px|]] [[File:Subgroup of Oh; C4 orange 23; solid.png|80px]]
|}
{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;"
! bgcolor="#f0e0f0" colspan="4"|<span style="float:left; padding-left:50px;">C<sub>4</sub></span>   C<sub>4</sub> red   <span style="float:right; padding-right:50px;">[4]<sup>+</sup></span>
|-
|rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; C4 red 07; example solid.png|230px]]
| [[File:Subgroup of Oh; C4 red 07; cycle graph.svg|180px|]]
| [[File:Subgroup of Oh; C4 red 16; cycle graph.svg|180px|]]
| [[File:Subgroup of Oh; C4 red 23; cycle graph.svg|180px|]]
|-
| [[File:Subgroup of Oh; C4 red 07; matrix.svg|50px|]] [[File:Subgroup of Oh; C4 red 07; solid.png|80px]]
| [[File:Subgroup of Oh; C4 red 16; matrix.svg|50px|]] [[File:Subgroup of Oh; C4 red 16; solid.png|80px]]
| [[File:Subgroup of Oh; C4 red 23; matrix.svg|50px|]] [[File:Subgroup of Oh; C4 red 23; solid.png|80px]]
|}


{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;"
! bgcolor="#f0f0e0" colspan="4"|<span style="float:left; padding-left:50px;">C<sub>2h</sub> = D<sub>1d</sub></span>   V inv white   <span style="float:right; padding-right:50px;">[2<sup>+</sup>,2]</span>
|-
|rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; V inv white 07; example solid.png|230px]]
| [[File:Subgroup of Oh; V inv white 07; cycle graph.svg|80px|]]
| [[File:Subgroup of Oh; V inv white 16; cycle graph.svg|80px|]]
| [[File:Subgroup of Oh; V inv white 23; cycle graph.svg|80px|]]
|-
| [[File:Subgroup of Oh; V inv white 07; matrix.svg|50px|]] [[File:Subgroup of Oh; V inv white 07; solid.png|80px]]
| [[File:Subgroup of Oh; V inv white 16; matrix.svg|50px|]] [[File:Subgroup of Oh; V inv white 16; solid.png|80px]]
| [[File:Subgroup of Oh; V inv white 23; matrix.svg|50px|]] [[File:Subgroup of Oh; V inv white 23; solid.png|80px]]
|}
{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;"
! bgcolor="#f0f0e0" colspan="7"|<span style="float:left; padding-left:50px;">C<sub>2h</sub> = D<sub>1d</sub></span>   V inv green   <span style="float:right; padding-right:50px;">[2<sup>+</sup>,2]</span>
|-
|rowspan="3" style="width:240px;"| [[File:Subgroup of Oh; V inv green 01; example solid.png|230px]]
| [[File:Subgroup of Oh; V inv green 01; cycle graph.svg|80px|]]
| [[File:Subgroup of Oh; V inv green 06; cycle graph.svg|80px|]]
| [[File:Subgroup of Oh; V inv green 05; cycle graph.svg|80px|]]
| [[File:Subgroup of Oh; V inv green 14; cycle graph.svg|80px|]]
| [[File:Subgroup of Oh; V inv green 02; cycle graph.svg|80px|]]
| [[File:Subgroup of Oh; V inv green 21; cycle graph.svg|80px|]]
|-
| [[File:Subgroup of Oh; V inv green 01; matrix.svg|50px|]]
| [[File:Subgroup of Oh; V inv green 06; matrix.svg|50px|]]
| [[File:Subgroup of Oh; V inv green 05; matrix.svg|50px|]]
| [[File:Subgroup of Oh; V inv green 14; matrix.svg|50px|]]
| [[File:Subgroup of Oh; V inv green 02; matrix.svg|50px|]]
| [[File:Subgroup of Oh; V inv green 21; matrix.svg|50px|]]
|-
| [[File:Subgroup of Oh; V inv green 01; solid.png|80px]]
| [[File:Subgroup of Oh; V inv green 06; solid.png|80px]]
| [[File:Subgroup of Oh; V inv green 05; solid.png|80px]]
| [[File:Subgroup of Oh; V inv green 14; solid.png|80px]]
| [[File:Subgroup of Oh; V inv green 02; solid.png|80px]]
| [[File:Subgroup of Oh; V inv green 21; solid.png|80px]]
|}
{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;"
! bgcolor="#e0f0f0" colspan="7"|<span style="float:left; padding-left:50px;">C<sub>2v</sub></span>   V green blue yellow   <span style="float:right; padding-right:50px;">[2]</span>
|-
|rowspan="3" style="width:240px;"| [[File:Subgroup of Oh; V gby 01; example solid.png|230px]]
| [[File:Subgroup of Oh; V gby 01; cycle graph.svg|180px|]]
| [[File:Subgroup of Oh; V gby 06; cycle graph.svg|180px|]]
| [[File:Subgroup of Oh; V gby 05; cycle graph.svg|180px|]]
| [[File:Subgroup of Oh; V gby 14; cycle graph.svg|180px|]]
| [[File:Subgroup of Oh; V gby 02; cycle graph.svg|180px|]]
| [[File:Subgroup of Oh; V gby 21; cycle graph.svg|180px|]]
|-
| [[File:Subgroup of Oh; V gby 01; matrix.svg|50px|]]
| [[File:Subgroup of Oh; V gby 06; matrix.svg|50px|]]
| [[File:Subgroup of Oh; V gby 05; matrix.svg|50px|]]
| [[File:Subgroup of Oh; V gby 14; matrix.svg|50px|]]
| [[File:Subgroup of Oh; V gby 02; matrix.svg|50px|]]
| [[File:Subgroup of Oh; V gby 21; matrix.svg|50px|]]
|-
| [[File:Subgroup of Oh; V gby 01; solid.png|80px]]
| [[File:Subgroup of Oh; V gby 06; solid.png|80px]]
| [[File:Subgroup of Oh; V gby 05; solid.png|80px]]
| [[File:Subgroup of Oh; V gby 14; solid.png|80px]]
| [[File:Subgroup of Oh; V gby 02; solid.png|80px]]
| [[File:Subgroup of Oh; V gby 21; solid.png|80px]]
|}
{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;"
! bgcolor="#e0f0f0" colspan="4"|<span style="float:left; padding-left:50px;">C<sub>2v</sub></span>   V yellow white   <span style="float:right; padding-right:50px;">[2]</span>
|-
|rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; V yellow white 07; example solid.png|230px]]
| [[File:Subgroup of Oh; V yellow white 07; cycle graph.svg|150px|]]
| [[File:Subgroup of Oh; V yellow white 16; cycle graph.svg|150px|]]
| [[File:Subgroup of Oh; V yellow white 23; cycle graph.svg|150px|]]
|-
| [[File:Subgroup of Oh; V yellow white 07; matrix.svg|50px|]] [[File:Subgroup of Oh; V yellow white 07; solid.png|80px]]
| [[File:Subgroup of Oh; V yellow white 16; matrix.svg|50px|]] [[File:Subgroup of Oh; V yellow white 16; solid.png|80px]]
| [[File:Subgroup of Oh; V yellow white 23; matrix.svg|50px|]] [[File:Subgroup of Oh; V yellow white 23; solid.png|80px]]
|}
{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;"
! bgcolor="#e0f0f0" colspan="4"|<span style="float:left; padding-left:50px;">C<sub>2v</sub></span>   V green white   <span style="float:right; padding-right:50px;">[2]</span>
|-
|rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; V green white 07; example solid.png|230px]]
| [[File:Subgroup of Oh; V green white 07; cycle graph.svg|180px|]]
| [[File:Subgroup of Oh; V green white 16; cycle graph.svg|180px|]]
| [[File:Subgroup of Oh; V green white 23; cycle graph.svg|180px|]]
|-
| [[File:Subgroup of Oh; V green white 07; matrix.svg|50px|]] [[File:Subgroup of Oh; V green white 07; solid.png|80px]]
| [[File:Subgroup of Oh; V green white 16; matrix.svg|50px|]] [[File:Subgroup of Oh; V green white 16; solid.png|80px]]
| [[File:Subgroup of Oh; V green white 23; matrix.svg|50px|]] [[File:Subgroup of Oh; V green white 23; solid.png|80px]]
|}
{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;"
! bgcolor="#f0e0f0" colspan="4"|<span style="float:left; padding-left:50px;">D<sub>2</sub></span>   V blue white   <span style="float:right; padding-right:50px;">[2,2]<sup>+</sup></span>
|-
|rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; V blue white; example solid.png|230px]]
| [[File:Subgroup of Oh; V blue white 07; cycle graph.svg|180px|]]
| [[File:Subgroup of Oh; V blue white 16; cycle graph.svg|180px|]]
| [[File:Subgroup of Oh; V blue white 23; cycle graph.svg|180px|]]
|-
| [[File:Subgroup of Oh; V blue white 07; matrix.svg|50px|]] [[File:Subgroup of Oh; V blue white 07; solid.png|80px]]
| [[File:Subgroup of Oh; V blue white 16; matrix.svg|50px|]] [[File:Subgroup of Oh; V blue white 16; solid.png|80px]]
| [[File:Subgroup of Oh; V blue white 23; matrix.svg|50px|]] [[File:Subgroup of Oh; V blue white 23; solid.png|80px]]
|}
{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;"
! bgcolor="#f0e0f0" colspan="2"|<span style="float:left; padding-left:50px;">D<sub>2</sub></span>   V white   <span style="float:right; padding-right:50px;">[2,2]<sup>+</sup></span>
|-
|rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; V white; example solid.png|230px]]
| [[File:Subgroup of Oh; V white; cycle graph.svg|220px|]]
|-
| [[File:Subgroup of Oh; V white; matrix.svg|50px|]] [[File:Subgroup of Oh; V white; solid.png|80px]]
|}

====Subgroups of order 3====

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;"
! bgcolor="#f0e0f0" colspan="5"|<span style="float:left; padding-left:50px;">C<sub>3</sub></span>   C<sub>3</sub>   <span style="float:right; padding-right:50px;">[3]<sup>+</sup></span>
|-
|rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; C3 03; example solid.png|230px]]
| [[File:Subgroup of Oh; C3 03; cycle graph.svg|170px|]]
| [[File:Subgroup of Oh; C3 11; cycle graph.svg|170px|]]
| [[File:Subgroup of Oh; C3 15; cycle graph.svg|170px|]]
| [[File:Subgroup of Oh; C3 08; cycle graph.svg|170px|]]
|-
| [[File:Subgroup of Oh; C3 03; matrix.svg|50px|]] [[File:Subgroup of Oh; C3 03; solid.png|80px]]
| [[File:Subgroup of Oh; C3 11; matrix.svg|50px|]] [[File:Subgroup of Oh; C3 11; solid.png|80px]]
| [[File:Subgroup of Oh; C3 15; matrix.svg|50px|]] [[File:Subgroup of Oh; C3 15; solid.png|80px]]
| [[File:Subgroup of Oh; C3 08; matrix.svg|50px|]] [[File:Subgroup of Oh; C3 08; solid.png|80px]]
|}

====Subgroups of order 2====

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;"
!colspan="2" bgcolor="#e0e0e0"|<span style="float:left; padding-left:50px;">S<sub>2</sub></span>   C<sub>2</sub> inv   <span style="float:right; padding-right:50px;">[2<sup>+</sup>,2<sup>+</sup>]</span>
|-
|rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; C2 inv; example solid.png|230px]]
| [[File:Subgroup of Oh; C2 inv; cycle graph.svg|70px|]]
|-
| [[File:Subgroup of Oh; C2 inv; matrix.svg|50px|]] [[File:Subgroup of Oh; C2 inv; solid.png|80px]]
|}
{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;"
! bgcolor="#e0f0f0" colspan="4"|<span style="float:left; padding-left:50px;">C<sub>s</sub> = C<sub>1v</sub></span>   C<sub>2</sub> yellow   <span style="float:right; padding-right:50px;">[ ]</span>
|-
|rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; C2 yellow 07; example solid.png|230px]]
| [[File:Subgroup of Oh; C2 yellow 07; cycle graph.svg|70px|]]
| [[File:Subgroup of Oh; C2 yellow 16; cycle graph.svg|70px|]]
| [[File:Subgroup of Oh; C2 yellow 23; cycle graph.svg|70px|]]
|-
| [[File:Subgroup of Oh; C2 yellow 07; matrix.svg|50px|]] [[File:Subgroup of Oh; C2 yellow 07; solid.png|80px]]
| [[File:Subgroup of Oh; C2 yellow 16; matrix.svg|50px|]] [[File:Subgroup of Oh; C2 yellow 16; solid.png|80px]]
| [[File:Subgroup of Oh; C2 yellow 23; matrix.svg|50px|]] [[File:Subgroup of Oh; C2 yellow 23; solid.png|80px]]
|}
{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;"
! bgcolor="#e0f0f0" colspan="7"|<span style="float:left; padding-left:50px;">C<sub>s</sub> = C<sub>1v</sub></span>   C<sub>2</sub> green   <span style="float:right; padding-right:50px;">[ ]</span>
|-
|rowspan="3" style="width:240px;"| [[File:Subgroup of Oh; C2 green 01; example solid.png|230px]]
| [[File:Subgroup of Oh; C2 green 01; cycle graph.svg|70px|]]
| [[File:Subgroup of Oh; C2 green 06; cycle graph.svg|70px|]]
| [[File:Subgroup of Oh; C2 green 05; cycle graph.svg|70px|]]
| [[File:Subgroup of Oh; C2 green 14; cycle graph.svg|70px|]]
| [[File:Subgroup of Oh; C2 green 02; cycle graph.svg|70px|]]
| [[File:Subgroup of Oh; C2 green 21; cycle graph.svg|70px|]]
|-
| [[File:Subgroup of Oh; C2 green 01; matrix.svg|50px|]]
| [[File:Subgroup of Oh; C2 green 06; matrix.svg|50px|]]
| [[File:Subgroup of Oh; C2 green 05; matrix.svg|50px|]]
| [[File:Subgroup of Oh; C2 green 14; matrix.svg|50px|]]
| [[File:Subgroup of Oh; C2 green 02; matrix.svg|50px|]]
| [[File:Subgroup of Oh; C2 green 21; matrix.svg|50px|]]
|-
| [[File:Subgroup of Oh; C2 green 01; solid.png|80px]]
| [[File:Subgroup of Oh; C2 green 06; solid.png|80px]]
| [[File:Subgroup of Oh; C2 green 05; solid.png|80px]]
| [[File:Subgroup of Oh; C2 green 14; solid.png|80px]]
| [[File:Subgroup of Oh; C2 green 02; solid.png|80px]]
| [[File:Subgroup of Oh; C2 green 21; solid.png|80px]]
|}
{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;"
! bgcolor="#f0e0f0" colspan="4"|<span style="float:left; padding-left:50px;">C<sub>2</sub></span>   C<sub>2</sub> white   <span style="float:right; padding-right:50px;">[2]<sup>+</sup></span>
|-
|rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; C2 white 07; example solid.png|230px]]
| [[File:Subgroup of Oh; C2 white 07; cycle graph.svg|70px|]]
| [[File:Subgroup of Oh; C2 white 16; cycle graph.svg|70px|]]
| [[File:Subgroup of Oh; C2 white 23; cycle graph.svg|70px|]]
|-
| [[File:Subgroup of Oh; C2 white 07; matrix.svg|50px|]] [[File:Subgroup of Oh; C2 white 07; solid.png|80px]]
| [[File:Subgroup of Oh; C2 white 16; matrix.svg|50px|]] [[File:Subgroup of Oh; C2 white 16; solid.png|80px]]
| [[File:Subgroup of Oh; C2 white 23; matrix.svg|50px|]] [[File:Subgroup of Oh; C2 white 23; solid.png|80px]]
|}
{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;"
! bgcolor="#f0e0f0" colspan="7"|<span style="float:left; padding-left:50px;">C<sub>2</sub></span>   C<sub>2</sub> blue   <span style="float:right; padding-right:50px;">[2]<sup>+</sup></span>
|-
|rowspan="3" style="width:240px;"| [[File:Subgroup of Oh; C2 blue 01; example solid.png|230px]]
| [[File:Subgroup of Oh; C2 blue 01; cycle graph.svg|70px|]]
| [[File:Subgroup of Oh; C2 blue 06; cycle graph.svg|70px|]]
| [[File:Subgroup of Oh; C2 blue 05; cycle graph.svg|70px|]]
| [[File:Subgroup of Oh; C2 blue 14; cycle graph.svg|70px|]]
| [[File:Subgroup of Oh; C2 blue 02; cycle graph.svg|70px|]]
| [[File:Subgroup of Oh; C2 blue 21; cycle graph.svg|70px|]]
|-
| [[File:Subgroup of Oh; C2 blue 01; matrix.svg|50px|]]
| [[File:Subgroup of Oh; C2 blue 06; matrix.svg|50px|]]
| [[File:Subgroup of Oh; C2 blue 05; matrix.svg|50px|]]
| [[File:Subgroup of Oh; C2 blue 14; matrix.svg|50px|]]
| [[File:Subgroup of Oh; C2 blue 02; matrix.svg|50px|]]
| [[File:Subgroup of Oh; C2 blue 21; matrix.svg|50px|]]
|-
| [[File:Subgroup of Oh; C2 blue 01; solid.png|80px]]
| [[File:Subgroup of Oh; C2 blue 06; solid.png|80px]]
| [[File:Subgroup of Oh; C2 blue 05; solid.png|80px]]
| [[File:Subgroup of Oh; C2 blue 14; solid.png|80px]]
| [[File:Subgroup of Oh; C2 blue 02; solid.png|80px]]
| [[File:Subgroup of Oh; C2 blue 21; solid.png|80px]]
|}

====The trivial group====

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;"
! bgcolor="#f0e0f0" colspan="2"|<span style="float:left; padding-left:50px;">C<sub>1</sub></span>   C<sub>1</sub>   <span style="float:right; padding-right:50px;">[ ]<sup>+</sup></span>
|-
|rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; trivial; example solid.png|230px]]
|style="padding:30px;"| [[File:Circlemarker.svg|45px|]]
|-
| [[File:Subgroup of Oh; trivial; matrix.svg|50px|]] [[File:Subgroup of Oh; trivial; solid.png|80px]]
|}

===Different appearances of the same group===


====Symmetry group of the cuboid====

The symmetry group of the cuboid C<sub>2</sub><sup>3</sup> appears in two essentially different ways as a subgroup of O<sub>h</sub>.<br>
The one where the cuboid is the cube itself is the most intuitive one.<br>
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.<br>
There are 4 individual subgroups <small>(see [[#Subgroups of order 8|above]]).</small>

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;"
!colspan="9" bgcolor="#e0f0f0"|<span style="float:left; padding-left:50px;">D<sub>2h</sub></span>   C<sub>2</sub><sup>3</sup> white   <span style="float:right; padding-right:50px;">[2,2]</span>
|-
| [[File:Subgroup of Oh; C2^3 white; example solid.png|180px]]
| [[File:Cube permutation 0 0.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Cube permutation 1 0.svg|150px]]<br>[[File:Cube vertex number 1.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Cube permutation 2 0.svg|150px]]<br>[[File:Cube vertex number 2.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Cube permutation 3 0.svg|150px]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Cube permutation 4 0.svg|150px]]<br>[[File:Cube vertex number 4.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Cube permutation 5 0.svg|150px]]<br>[[File:Cube vertex number 5.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Cube permutation 6 0.svg|150px]]<br>[[File:Cube vertex number 6.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Cube permutation 7 0.svg|150px]]<br>[[File:Cube vertex number 7.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
|}

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;"
! bgcolor="#e0f0f0" colspan="9"|<span style="float:left; padding-left:50px;">D<sub>2h</sub></span>   C<sub>2</sub><sup>3</sup> green   <span style="float:right; padding-right:50px;">[2,2]</span>
|-
| [[File:Subgroup of Oh; C2^3 green 07; example solid.png|180px]]
| [[File:Cube permutation 0 0.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Cube permutation 0 1.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
| [[File:Cube permutation 3 1.svg|150px]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
| [[File:Cube permutation 3 0.svg|150px]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Cube permutation 4 0.svg|150px]]<br>[[File:Cube vertex number 4.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Cube permutation 4 1.svg|150px]]<br>[[File:Cube vertex number 4.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
| [[File:Cube permutation 7 1.svg|150px]]<br>[[File:Cube vertex number 7.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
| [[File:Cube permutation 7 0.svg|150px]]<br>[[File:Cube vertex number 7.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
|}

====Symmetry group of the square====

The [[w:Dihedral group of order 8|symmetry group of the square]] appears in four essentially different ways as a subgroup of O<sub>h</sub>. (C<sub>4v</sub> or [4] being the most intuitive among them.)<br>
There are 12 individual Dih<sub>4</sub> subgroups <small>(see [[#Subgroups of order 8|above]])</small>. 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.

{| style="width: 100%; text-align: center; border: 1px solid #fff;"
| [[File:Blank300.png|180px]]
| <div style="padding: 0 25px;">[[File:Square permutation fix.svg|100px]]</div>
| <div style="padding: 0 25px;">[[File:Square permutation horz.svg|100px]]</div>
| <div style="padding: 0 25px;">[[File:Square permutation vert.svg|100px]]</div>
| <div style="padding: 0 25px;">[[File:Square permutation cross.svg|100px]]</div>
| <div style="padding: 0 25px;">[[File:Square permutation desc.svg|100px]]</div>
| <div style="padding: 0 25px;">[[File:Square permutation left.svg|100px]]</div>
| <div style="padding: 0 25px;">[[File:Square permutation right.svg|100px]]</div>
| <div style="padding: 0 25px;">[[File:Square permutation asc.svg|100px]]</div>
|}

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;"
! bgcolor="white" colspan="9"|Square permutations for comparison
|-
| [[File:Dihedral group of order 8; Cayley table (element orders 1,2,2,4,4,2,2,2); subgroup of S4 (left).svg|180px]]
| [[File:Square permutation 0 0, colored.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Square permutation 1 0, colored.svg|150px]]<br>[[File:Cube vertex number 1.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Square permutation 2 0, colored.svg|150px]]<br>[[File:Cube vertex number 2.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Square permutation 3 0, colored.svg|150px]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Square permutation 0 1, colored.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
| [[File:Square permutation 1 1, colored.svg|150px]]<br>[[File:Cube vertex number 1.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
| [[File:Square permutation 2 1, colored.svg|150px]]<br>[[File:Cube vertex number 2.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
| [[File:Square permutation 3 1, colored.svg|150px]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
|}

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;"
! bgcolor="#e0f0f0" colspan="9"|<span style="float:left; padding-left:50px;">C<sub>4v</sub></span>   Dih<sub>4</sub> green red   <span style="float:right; padding-right:50px;">[4]</span>
|-
| [[File:Subgroup of Oh; Dih4 green red 07; example solid.png|180px]]<br>[[File:Square subgroup of the cube, pattern of bottom right digits in 3x3 matrices, 0.svg|40px]]
| [[File:Cube permutation 0 0.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Cube permutation 1 0.svg|150px]]<br>[[File:Cube vertex number 1.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Cube permutation 2 0.svg|150px]]<br>[[File:Cube vertex number 2.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Cube permutation 3 0.svg|150px]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Cube permutation 0 1.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
| [[File:Cube permutation 1 1.svg|150px]]<br>[[File:Cube vertex number 1.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
| [[File:Cube permutation 2 1.svg|150px]]<br>[[File:Cube vertex number 2.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
| [[File:Cube permutation 3 1.svg|150px]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
|}

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;"
! bgcolor="#f0f0e0" colspan="9"|<span style="float:left; padding-left:50px;">D<sub>2d</sub></span>   Dih<sub>4</sub> blue orange   <span style="float:right; padding-right:50px;">[2<sup>+</sup>,4]</span>
|-
| [[File:Subgroup of Oh; Dih4 blue orange 07; example solid.png|180px]]<br>[[File:Square subgroup of the cube, pattern of bottom right digits in 3x3 matrices, 1.svg|40px]]
| [[File:Cube permutation 0 0.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Cube permutation 1 0.svg|150px]]<br>[[File:Cube vertex number 1.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Cube permutation 2 0.svg|150px]]<br>[[File:Cube vertex number 2.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Cube permutation 3 0.svg|150px]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Cube permutation 4 1.svg|150px]]<br>[[File:Cube vertex number 4.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
| [[File:Cube permutation 5 1.svg|150px]]<br>[[File:Cube vertex number 5.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
| [[File:Cube permutation 6 1.svg|150px]]<br>[[File:Cube vertex number 6.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
| [[File:Cube permutation 7 1.svg|150px]]<br>[[File:Cube vertex number 7.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
|}

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;"
! bgcolor="#f0f0e0" colspan="9"|<span style="float:left; padding-left:50px;">D<sub>2d</sub></span>   Dih<sub>4</sub> green orange   <span style="float:right; padding-right:50px;">[2<sup>+</sup>,4]</span>
|-
| [[File:Subgroup of Oh; Dih4 green orange 07; example solid.png|180px]]<br>[[File:Square subgroup of the cube, pattern of bottom right digits in 3x3 matrices, 2.svg|40px]]
| [[File:Cube permutation 0 0.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Cube permutation 5 0.svg|150px]]<br>[[File:Cube vertex number 5.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Cube permutation 6 0.svg|150px]]<br>[[File:Cube vertex number 6.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Cube permutation 3 0.svg|150px]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Cube permutation 0 1.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
| [[File:Cube permutation 5 1.svg|150px]]<br>[[File:Cube vertex number 5.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
| [[File:Cube permutation 6 1.svg|150px]]<br>[[File:Cube vertex number 6.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
| [[File:Cube permutation 3 1.svg|150px]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
|}

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;"
! bgcolor="#f0e0f0" colspan="9"|<span style="float:left; padding-left:50px;">D<sub>4</sub></span>   Dih<sub>4</sub> blue red   <span style="float:right; padding-right:50px;">[2,4]<sup>+</sup></span>
|-
| [[File:Subgroup of Oh; Dih4 blue red 07; example solid.png|180px]]<br>[[File:Square subgroup of the cube, pattern of bottom right digits in 3x3 matrices, 3.svg|40px]]
| [[File:Cube permutation 0 0.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Cube permutation 5 0.svg|150px]]<br>[[File:Cube vertex number 5.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Cube permutation 6 0.svg|150px]]<br>[[File:Cube vertex number 6.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Cube permutation 3 0.svg|150px]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Cube permutation 4 1.svg|150px]]<br>[[File:Cube vertex number 4.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
| [[File:Cube permutation 1 1.svg|150px]]<br>[[File:Cube vertex number 1.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
| [[File:Cube permutation 2 1.svg|150px]]<br>[[File:Cube vertex number 2.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
| [[File:Cube permutation 7 1.svg|150px]]<br>[[File:Cube vertex number 7.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
|}

====Symmetry group of the triangle====

The [[w:Dihedral group of order 6|symmetry group of the triangle]] appears in two essentially different ways as a subgroup of O<sub>h</sub>, with C<sub>3v</sub> or [3] being the most intuitive among them.<br>
There are 8 individual subgroups <small>(see [[#Subgroups of order 6|above]])</small>. Shown below are the ones where the triangle is seen from a point on the negative main diagonal of the coordinate system.<br>

{| style="width: 100%; text-align: center; border: 1px solid #fff;"
| [[File:Blank300.png|180px]]
| <div style="padding: 0 20px;">[[File:Triangle permutation fix.svg|110px]]</div>
| <div style="padding: 0 20px;">[[File:Triangle permutation ref left.svg|110px]]</div>
| <div style="padding: 0 20px;">[[File:Triangle permutation ref right.svg|110px]]</div>
| <div style="padding: 0 20px;">[[File:Triangle permutation rot left.svg|110px]]</div>
| <div style="padding: 0 20px;">[[File:Triangle permutation rot right.svg|110px]]</div>
| <div style="padding: 0 20px;">[[File:Triangle permutation ref horz.svg|110px]]</div>
|}

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;"
! bgcolor="#e0f0f0" colspan="7"|<span style="float:left; padding-left:50px;">C<sub>3v</sub></span>   S<sub>3</sub> green   <span style="float:right; padding-right:50px;">[3]</span>
|-
| [[File:Subgroup of Oh; S3 green 03; example solid (triangle).png|180px]]
| [[File:Cube permutation 0 0.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Cube permutation 0 1.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
| [[File:Cube permutation 0 2.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 2.svg|20px]]
| [[File:Cube permutation 0 3.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 3.svg|20px]]
| [[File:Cube permutation 0 4.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 4.svg|20px]]
| [[File:Cube permutation 0 5.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 5.svg|20px]]
|}

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;"
! bgcolor="#f0e0f0" colspan="7"|<span style="float:left; padding-left:50px;">D<sub>3</sub></span>   S<sub>3</sub> blue   <span style="float:right; padding-right:50px;">[2,3]<sup>+</sup></span>
|-
| [[File:Subgroup of Oh; S3 blue 03; example solid (with numbers).png|180px]]
| [[File:Cube permutation 0 0.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 0.svg|20px]]
| [[File:Cube permutation 7 1.svg|150px]]<br>[[File:Cube vertex number 7.svg|20px]] [[File:Finite permutation number 1.svg|20px]]
| [[File:Cube permutation 7 2.svg|150px]]<br>[[File:Cube vertex number 7.svg|20px]] [[File:Finite permutation number 2.svg|20px]]
| [[File:Cube permutation 0 3.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 3.svg|20px]]
| [[File:Cube permutation 0 4.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 4.svg|20px]]
| [[File:Cube permutation 7 5.svg|150px]]<br>[[File:Cube vertex number 7.svg|20px]] [[File:Finite permutation number 5.svg|20px]]
|}

===Cuboctahedral example solids and contained hexagons===

{| class="collapsible open" style="width: 100%; text-align: center; border: 1px solid #ddd;"
! bgcolor="#eee"|
|-
|
{| class="wikitable" style="width: 100%; text-align: center;"
!
|style="background: #f0e0f0;"| '''S<sub>4</sub> blue red'''
|style="background: #f0f0e0;"| '''A<sub>4</sub> × C<sub>2</sub>'''
|style="background: #f0e0f0;"| '''A<sub>4</sub>'''
|style="background: #e0f0f0;"| '''S<sub>4</sub> green orange'''
|-
! shown above
|rowspan="2" | [[File:Subgroup of Oh; S4 blue red; example solid.png|200px]]
| [[File:Subgroup of Oh; A4xC2; example solid.png|200px]]
| [[File:Subgroup of Oh; A4; example solid.png|200px]]
| [[File:Subgroup of Oh; S4 green orange; example solid.png|200px]]
|-
! cuboctahedral
| [[File:Subgroup of Oh; A4xC2; example solid (cuboctahedron).png|200px]]
| [[File:Subgroup of Oh; A4; example solid (cuboctahedron).png|200px]]
| [[File:Subgroup of Oh; S4 green orange; example solid (cuboctahedron).png|200px]]
|-
! contained hexagon
| [[File:Subgroup of Oh; S3 blue 03; example solid.png|200px]]
| [[File:Subgroup of Oh; C6 03; example solid.png|200px]]
| [[File:Subgroup of Oh; C3 03; example solid.png|200px]]
| [[File:Subgroup of Oh; S3 green 03; example solid.png|200px]]
|-
!
|style="background: #f0e0f0;"| '''S<sub>3</sub> blue'''
|style="background: #e0e0e0;"| '''C<sub>6</sub>'''
|style="background: #f0e0f0;"| '''C<sub>3</sub>'''
|style="background: #e0f0f0;"| '''S<sub>3</sub> 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:

{| class="collapsible collapsed" style="width: 100%; border: 1px solid #ddd; border-bottom: none; background-color: white;"
!style="background-color: #eee; font-weight: normal;"| '''S<sub>4</sub>'''-based identifiers
|-
|
<math>n'</math> is represented as <math>n+24</math>.
 <span style="color:gray;">(0, 0):</span>  0, <span style="color:gray;">(0, 1):</span>  1, <span style="color:gray;">(0, 2):</span>  2, <span style="color:gray;">(0, 3):</span>  3, <span style="color:gray;">(0, 4):</span>  4, <span style="color:gray;">(0, 5):</span>  5,
 <span style="color:gray;">(1, 0):</span> 47, <span style="color:gray;">(1, 1):</span> 46, <span style="color:gray;">(1, 2):</span> 45, <span style="color:gray;">(1, 3):</span> 44, <span style="color:gray;">(1, 4):</span> 43, <span style="color:gray;">(1, 5):</span> 42,
 <span style="color:gray;">(2, 0):</span> 40, <span style="color:gray;">(2, 1):</span> 41, <span style="color:gray;">(2, 2):</span> 37, <span style="color:gray;">(2, 3):</span> 36, <span style="color:gray;">(2, 4):</span> 39, <span style="color:gray;">(2, 5):</span> 38,
 <span style="color:gray;">(3, 0):</span>  7, <span style="color:gray;">(3, 1):</span>  6, <span style="color:gray;">(3, 2):</span> 10, <span style="color:gray;">(3, 3):</span> 11, <span style="color:gray;">(3, 4):</span>  8, <span style="color:gray;">(3, 5):</span>  9,
 <span style="color:gray;">(4, 0):</span> 31, <span style="color:gray;">(4, 1):</span> 30, <span style="color:gray;">(4, 2):</span> 34, <span style="color:gray;">(4, 3):</span> 35, <span style="color:gray;">(4, 4):</span> 32, <span style="color:gray;">(4, 5):</span> 33,
 <span style="color:gray;">(5, 0):</span> 16, <span style="color:gray;">(5, 1):</span> 17, <span style="color:gray;">(5, 2):</span> 13, <span style="color:gray;">(5, 3):</span> 12, <span style="color:gray;">(5, 4):</span> 15, <span style="color:gray;">(5, 5):</span> 14,
 <span style="color:gray;">(6, 0):</span> 23, <span style="color:gray;">(6, 1):</span> 22, <span style="color:gray;">(6, 2):</span> 21, <span style="color:gray;">(6, 3):</span> 20, <span style="color:gray;">(6, 4):</span> 19, <span style="color:gray;">(6, 5):</span> 18,
 <span style="color:gray;">(7, 0):</span> 24, <span style="color:gray;">(7, 1):</span> 25, <span style="color:gray;">(7, 2):</span> 26, <span style="color:gray;">(7, 3):</span> 27, <span style="color:gray;">(7, 4):</span> 28, <span style="color:gray;">(7, 5):</span> 29
|}

{| class="collapsible collapsed" style="width: 100%; border: 1px solid #ddd; border-bottom: none; background-color: white;"
!style="background-color: #eee; font-weight: normal;"| Permutations of '''cube''' vertices
|-
|
The initial order is <code>(-1, -1, -1), (1, -1, -1), (-1, 1, -1), (1, 1, -1), (-1, -1, 1), (1, -1, 1), (-1, 1, 1), (1, 1, 1)</code>.

 <span style="white-space: nowrap;"><span style="color:gray;">(0, 0):</span> (0, 1, 2, 3, 4, 5, 6, 7), <span style="color:gray;">(0, 1):</span> (0, 2, 1, 3, 4, 6, 5, 7), <span style="color:gray;">(0, 2):</span> (0, 1, 4, 5, 2, 3, 6, 7), <span style="color:gray;">(0, 3):</span> (0, 4, 1, 5, 2, 6, 3, 7), <span style="color:gray;">(0, 4):</span> (0, 2, 4, 6, 1, 3, 5, 7), <span style="color:gray;">(0, 5):</span> (0, 4, 2, 6, 1, 5, 3, 7),</span>
 <span style="white-space: nowrap;"><span style="color:gray;">(1, 0):</span> (1, 0, 3, 2, 5, 4, 7, 6), <span style="color:gray;">(1, 1):</span> (1, 3, 0, 2, 5, 7, 4, 6), <span style="color:gray;">(1, 2):</span> (1, 0, 5, 4, 3, 2, 7, 6), <span style="color:gray;">(1, 3):</span> (1, 5, 0, 4, 3, 7, 2, 6), <span style="color:gray;">(1, 4):</span> (1, 3, 5, 7, 0, 2, 4, 6), <span style="color:gray;">(1, 5):</span> (1, 5, 3, 7, 0, 4, 2, 6),</span>
 <span style="white-space: nowrap;"><span style="color:gray;">(2, 0):</span> (2, 3, 0, 1, 6, 7, 4, 5), <span style="color:gray;">(2, 1):</span> (2, 0, 3, 1, 6, 4, 7, 5), <span style="color:gray;">(2, 2):</span> (2, 3, 6, 7, 0, 1, 4, 5), <span style="color:gray;">(2, 3):</span> (2, 6, 3, 7, 0, 4, 1, 5), <span style="color:gray;">(2, 4):</span> (2, 0, 6, 4, 3, 1, 7, 5), <span style="color:gray;">(2, 5):</span> (2, 6, 0, 4, 3, 7, 1, 5),</span>
 <span style="white-space: nowrap;"><span style="color:gray;">(3, 0):</span> (3, 2, 1, 0, 7, 6, 5, 4), <span style="color:gray;">(3, 1):</span> (3, 1, 2, 0, 7, 5, 6, 4), <span style="color:gray;">(3, 2):</span> (3, 2, 7, 6, 1, 0, 5, 4), <span style="color:gray;">(3, 3):</span> (3, 7, 2, 6, 1, 5, 0, 4), <span style="color:gray;">(3, 4):</span> (3, 1, 7, 5, 2, 0, 6, 4), <span style="color:gray;">(3, 5):</span> (3, 7, 1, 5, 2, 6, 0, 4),</span>
 <span style="white-space: nowrap;"><span style="color:gray;">(4, 0):</span> (4, 5, 6, 7, 0, 1, 2, 3), <span style="color:gray;">(4, 1):</span> (4, 6, 5, 7, 0, 2, 1, 3), <span style="color:gray;">(4, 2):</span> (4, 5, 0, 1, 6, 7, 2, 3), <span style="color:gray;">(4, 3):</span> (4, 0, 5, 1, 6, 2, 7, 3), <span style="color:gray;">(4, 4):</span> (4, 6, 0, 2, 5, 7, 1, 3), <span style="color:gray;">(4, 5):</span> (4, 0, 6, 2, 5, 1, 7, 3),</span>
 <span style="white-space: nowrap;"><span style="color:gray;">(5, 0):</span> (5, 4, 7, 6, 1, 0, 3, 2), <span style="color:gray;">(5, 1):</span> (5, 7, 4, 6, 1, 3, 0, 2), <span style="color:gray;">(5, 2):</span> (5, 4, 1, 0, 7, 6, 3, 2), <span style="color:gray;">(5, 3):</span> (5, 1, 4, 0, 7, 3, 6, 2), <span style="color:gray;">(5, 4):</span> (5, 7, 1, 3, 4, 6, 0, 2), <span style="color:gray;">(5, 5):</span> (5, 1, 7, 3, 4, 0, 6, 2),</span>
 <span style="white-space: nowrap;"><span style="color:gray;">(6, 0):</span> (6, 7, 4, 5, 2, 3, 0, 1), <span style="color:gray;">(6, 1):</span> (6, 4, 7, 5, 2, 0, 3, 1), <span style="color:gray;">(6, 2):</span> (6, 7, 2, 3, 4, 5, 0, 1), <span style="color:gray;">(6, 3):</span> (6, 2, 7, 3, 4, 0, 5, 1), <span style="color:gray;">(6, 4):</span> (6, 4, 2, 0, 7, 5, 3, 1), <span style="color:gray;">(6, 5):</span> (6, 2, 4, 0, 7, 3, 5, 1),</span>
 <span style="white-space: nowrap;"><span style="color:gray;">(7, 0):</span> (7, 6, 5, 4, 3, 2, 1, 0), <span style="color:gray;">(7, 1):</span> (7, 5, 6, 4, 3, 1, 2, 0), <span style="color:gray;">(7, 2):</span> (7, 6, 3, 2, 5, 4, 1, 0), <span style="color:gray;">(7, 3):</span> (7, 3, 6, 2, 5, 1, 4, 0), <span style="color:gray;">(7, 4):</span> (7, 5, 3, 1, 6, 4, 2, 0), <span style="color:gray;">(7, 5):</span> (7, 3, 5, 1, 6, 2, 4, 0)</span>
|}

{| class="collapsible collapsed" style="width: 100%; border: 1px solid #ddd; border-bottom: none; background-color: white;"
!style="background-color: #eee; font-weight: normal;"| Permutations of '''octahedron''' vertices
|-
|
The initial order is <code>(-1, 0, 0), (1, 0, 0), (0, -1, 0), (0, 1, 0), (0, 0, -1), (0, 0, 1)</code>.

 <span style="white-space: nowrap;"><span style="color:gray;">(0, 0):</span> (0, 1, 2, 3, 4, 5), <span style="color:gray;">(0, 1):</span> (2, 3, 0, 1, 4, 5), <span style="color:gray;">(0, 2):</span> (0, 1, 4, 5, 2, 3), <span style="color:gray;">(0, 3):</span> (4, 5, 0, 1, 2, 3), <span style="color:gray;">(0, 4):</span> (2, 3, 4, 5, 0, 1), <span style="color:gray;">(0, 5):</span> (4, 5, 2, 3, 0, 1),</span>
 <span style="white-space: nowrap;"><span style="color:gray;">(1, 0):</span> (1, 0, 2, 3, 4, 5), <span style="color:gray;">(1, 1):</span> (2, 3, 1, 0, 4, 5), <span style="color:gray;">(1, 2):</span> (1, 0, 4, 5, 2, 3), <span style="color:gray;">(1, 3):</span> (4, 5, 1, 0, 2, 3), <span style="color:gray;">(1, 4):</span> (2, 3, 4, 5, 1, 0), <span style="color:gray;">(1, 5):</span> (4, 5, 2, 3, 1, 0),</span>
 <span style="white-space: nowrap;"><span style="color:gray;">(2, 0):</span> (0, 1, 3, 2, 4, 5), <span style="color:gray;">(2, 1):</span> (3, 2, 0, 1, 4, 5), <span style="color:gray;">(2, 2):</span> (0, 1, 4, 5, 3, 2), <span style="color:gray;">(2, 3):</span> (4, 5, 0, 1, 3, 2), <span style="color:gray;">(2, 4):</span> (3, 2, 4, 5, 0, 1), <span style="color:gray;">(2, 5):</span> (4, 5, 3, 2, 0, 1),</span>
 <span style="white-space: nowrap;"><span style="color:gray;">(3, 0):</span> (1, 0, 3, 2, 4, 5), <span style="color:gray;">(3, 1):</span> (3, 2, 1, 0, 4, 5), <span style="color:gray;">(3, 2):</span> (1, 0, 4, 5, 3, 2), <span style="color:gray;">(3, 3):</span> (4, 5, 1, 0, 3, 2), <span style="color:gray;">(3, 4):</span> (3, 2, 4, 5, 1, 0), <span style="color:gray;">(3, 5):</span> (4, 5, 3, 2, 1, 0),</span>
 <span style="white-space: nowrap;"><span style="color:gray;">(4, 0):</span> (0, 1, 2, 3, 5, 4), <span style="color:gray;">(4, 1):</span> (2, 3, 0, 1, 5, 4), <span style="color:gray;">(4, 2):</span> (0, 1, 5, 4, 2, 3), <span style="color:gray;">(4, 3):</span> (5, 4, 0, 1, 2, 3), <span style="color:gray;">(4, 4):</span> (2, 3, 5, 4, 0, 1), <span style="color:gray;">(4, 5):</span> (5, 4, 2, 3, 0, 1),</span>
 <span style="white-space: nowrap;"><span style="color:gray;">(5, 0):</span> (1, 0, 2, 3, 5, 4), <span style="color:gray;">(5, 1):</span> (2, 3, 1, 0, 5, 4), <span style="color:gray;">(5, 2):</span> (1, 0, 5, 4, 2, 3), <span style="color:gray;">(5, 3):</span> (5, 4, 1, 0, 2, 3), <span style="color:gray;">(5, 4):</span> (2, 3, 5, 4, 1, 0), <span style="color:gray;">(5, 5):</span> (5, 4, 2, 3, 1, 0),</span>
 <span style="white-space: nowrap;"><span style="color:gray;">(6, 0):</span> (0, 1, 3, 2, 5, 4), <span style="color:gray;">(6, 1):</span> (3, 2, 0, 1, 5, 4), <span style="color:gray;">(6, 2):</span> (0, 1, 5, 4, 3, 2), <span style="color:gray;">(6, 3):</span> (5, 4, 0, 1, 3, 2), <span style="color:gray;">(6, 4):</span> (3, 2, 5, 4, 0, 1), <span style="color:gray;">(6, 5):</span> (5, 4, 3, 2, 0, 1),</span>
 <span style="white-space: nowrap;"><span style="color:gray;">(7, 0):</span> (1, 0, 3, 2, 5, 4), <span style="color:gray;">(7, 1):</span> (3, 2, 1, 0, 5, 4), <span style="color:gray;">(7, 2):</span> (1, 0, 5, 4, 3, 2), <span style="color:gray;">(7, 3):</span> (5, 4, 1, 0, 3, 2), <span style="color:gray;">(7, 4):</span> (3, 2, 5, 4, 1, 0), <span style="color:gray;">(7, 5):</span> (5, 4, 3, 2, 1, 0)</span>
|}

{| class="collapsible collapsed" style="width: 100%; border: 1px solid #ddd; border-bottom: none; background-color: white;"
!colspan="2" style="background-color: #eee; font-weight: normal;"| Pairs in '''reverse colex order'''
|-
|colspan="2"|
The [[Lexicographic_and_colexicographic_order#Permutations|reverse colexicographic order of permutations]] is the order of their respective {{w|Factorial number system|integer values}}.<br>
The list of permutations of octahedron vertices on the right can be generalized as the infinite list of {{w|cross-polytope}} vertices.<br>
<small>(Its first eight elements are the permutations of the square within a 4×2 matrix.)</small>
|- style="text-align: center;"
| '''cube'''
| '''octahedron'''
|- style="vertical-align: top;"
|
 (0, 0), (0, 2), (0, 1), (0, 4), (0, 3), (0, 5),
 (1, 0), (1, 2), (1, 1), (1, 4), (1, 3), (1, 5),
 (2, 1), (2, 4), (2, 0), (2, 2), (2, 5), (2, 3),
 (3, 1), (3, 4), (3, 0), (3, 2), (3, 5), (3, 3),
 (4, 3), (4, 5), (4, 2), (4, 0), (4, 4), (4, 1),
 (5, 3), (5, 5), (5, 2), (5, 0), (5, 4), (5, 1),
 (6, 5), (6, 3), (6, 4), (6, 1), (6, 2), (6, 0),
 (7, 5), (7, 3), (7, 4), (7, 1), (7, 2), (7, 0)
|
 (0, 0), (1, 0), (2, 0), (3, 0), (0, 1), (2, 1), (1, 1), (3, 1),
 (4, 0), (5, 0), (6, 0), (7, 0), (4, 1), (6, 1), (5, 1), (7, 1), 
 (0, 2), (1, 2), (4, 2), (5, 2), (0, 3), (4, 3), (1, 3), (5, 3), 
 (2, 2), (3, 2), (6, 2), (7, 2), (2, 3), (6, 3), (3, 3), (7, 3), 
 (0, 4), (2, 4), (4, 4), (6, 4), (0, 5), (4, 5), (2, 5), (6, 5), 
 (1, 4), (3, 4), (5, 4), (7, 4), (1, 5), (5, 5), (3, 5), (7, 5)
|}

{| class="collapsible collapsed" style="width: 100%; border: 1px solid #ddd; border-bottom: none; background-color: white;"
!style="background-color: #eee; font-weight: normal;"| '''3×3 matrices'''
|-
|
 <span style="white-space: nowrap;"><span style="color:gray;">(0, 0):</span> [[ 1, 0, 0], [0, 1, 0], [0, 0, 1]], <span style="color:gray;">(0, 1):</span> [[0, 1, 0], [ 1, 0, 0], [0, 0, 1]], <span style="color:gray;">(0, 2):</span> [[ 1, 0, 0], [0, 0, 1], [0, 1, 0]], <span style="color:gray;">(0, 3):</span> [[0, 1, 0], [0, 0, 1], [ 1, 0, 0]], <span style="color:gray;">(0, 4):</span> [[0, 0, 1], [ 1, 0, 0], [0, 1, 0]], <span style="color:gray;">(0, 5):</span> [[0, 0, 1], [0, 1, 0], [ 1, 0, 0]],</span>
 <span style="white-space: nowrap;"><span style="color:gray;">(1, 0):</span> [[-1, 0, 0], [0, 1, 0], [0, 0, 1]], <span style="color:gray;">(1, 1):</span> [[0,-1, 0], [ 1, 0, 0], [0, 0, 1]], <span style="color:gray;">(1, 2):</span> [[-1, 0, 0], [0, 0, 1], [0, 1, 0]], <span style="color:gray;">(1, 3):</span> [[0,-1, 0], [0, 0, 1], [ 1, 0, 0]], <span style="color:gray;">(1, 4):</span> [[0, 0,-1], [ 1, 0, 0], [0, 1, 0]], <span style="color:gray;">(1, 5):</span> [[0, 0,-1], [0, 1, 0], [ 1, 0, 0]],</span>
 <span style="white-space: nowrap;"><span style="color:gray;">(2, 0):</span> [[ 1, 0, 0], [0,-1, 0], [0, 0, 1]], <span style="color:gray;">(2, 1):</span> [[0, 1, 0], [-1, 0, 0], [0, 0, 1]], <span style="color:gray;">(2, 2):</span> [[ 1, 0, 0], [0, 0,-1], [0, 1, 0]], <span style="color:gray;">(2, 3):</span> [[0, 1, 0], [0, 0,-1], [ 1, 0, 0]], <span style="color:gray;">(2, 4):</span> [[0, 0, 1], [-1, 0, 0], [0, 1, 0]], <span style="color:gray;">(2, 5):</span> [[0, 0, 1], [0,-1, 0], [ 1, 0, 0]],</span>
 <span style="white-space: nowrap;"><span style="color:gray;">(3, 0):</span> [[-1, 0, 0], [0,-1, 0], [0, 0, 1]], <span style="color:gray;">(3, 1):</span> [[0,-1, 0], [-1, 0, 0], [0, 0, 1]], <span style="color:gray;">(3, 2):</span> [[-1, 0, 0], [0, 0,-1], [0, 1, 0]], <span style="color:gray;">(3, 3):</span> [[0,-1, 0], [0, 0,-1], [ 1, 0, 0]], <span style="color:gray;">(3, 4):</span> [[0, 0,-1], [-1, 0, 0], [0, 1, 0]], <span style="color:gray;">(3, 5):</span> [[0, 0,-1], [0,-1, 0], [ 1, 0, 0]],</span>
 <span style="white-space: nowrap;"><span style="color:gray;">(4, 0):</span> [[ 1, 0, 0], [0, 1, 0], [0, 0,-1]], <span style="color:gray;">(4, 1):</span> [[0, 1, 0], [ 1, 0, 0], [0, 0,-1]], <span style="color:gray;">(4, 2):</span> [[ 1, 0, 0], [0, 0, 1], [0,-1, 0]], <span style="color:gray;">(4, 3):</span> [[0, 1, 0], [0, 0, 1], [-1, 0, 0]], <span style="color:gray;">(4, 4):</span> [[0, 0, 1], [ 1, 0, 0], [0,-1, 0]], <span style="color:gray;">(4, 5):</span> [[0, 0, 1], [0, 1, 0], [-1, 0, 0]],</span>
 <span style="white-space: nowrap;"><span style="color:gray;">(5, 0):</span> [[-1, 0, 0], [0, 1, 0], [0, 0,-1]], <span style="color:gray;">(5, 1):</span> [[0,-1, 0], [ 1, 0, 0], [0, 0,-1]], <span style="color:gray;">(5, 2):</span> [[-1, 0, 0], [0, 0, 1], [0,-1, 0]], <span style="color:gray;">(5, 3):</span> [[0,-1, 0], [0, 0, 1], [-1, 0, 0]], <span style="color:gray;">(5, 4):</span> [[0, 0,-1], [ 1, 0, 0], [0,-1, 0]], <span style="color:gray;">(5, 5):</span> [[0, 0,-1], [0, 1, 0], [-1, 0, 0]],</span>
 <span style="white-space: nowrap;"><span style="color:gray;">(6, 0):</span> [[ 1, 0, 0], [0,-1, 0], [0, 0,-1]], <span style="color:gray;">(6, 1):</span> [[0, 1, 0], [-1, 0, 0], [0, 0,-1]], <span style="color:gray;">(6, 2):</span> [[ 1, 0, 0], [0, 0,-1], [0,-1, 0]], <span style="color:gray;">(6, 3):</span> [[0, 1, 0], [0, 0,-1], [-1, 0, 0]], <span style="color:gray;">(6, 4):</span> [[0, 0, 1], [-1, 0, 0], [0,-1, 0]], <span style="color:gray;">(6, 5):</span> [[0, 0, 1], [0,-1, 0], [-1, 0, 0]],</span>
 <span style="white-space: nowrap;"><span style="color:gray;">(7, 0):</span> [[-1, 0, 0], [0,-1, 0], [0, 0,-1]], <span style="color:gray;">(7, 1):</span> [[0,-1, 0], [-1, 0, 0], [0, 0,-1]], <span style="color:gray;">(7, 2):</span> [[-1, 0, 0], [0, 0,-1], [0,-1, 0]], <span style="color:gray;">(7, 3):</span> [[0,-1, 0], [0, 0,-1], [-1, 0, 0]], <span style="color:gray;">(7, 4):</span> [[0, 0,-1], [-1, 0, 0], [0,-1, 0]], <span style="color:gray;">(7, 5):</span> [[0, 0,-1], [0,-1, 0], [-1, 0, 0]]</span>
|}

{| class="collapsible collapsed" style="width: 100%; border: 1px solid #ddd; background-color: white;"
!style="background-color: #eee; font-weight: normal;"| '''Truncated cuboctahedron''' coordinates
|-
|
The points of the {{w|truncated cuboctahedron}} are the <math>2^3 \cdot 6! = 48</math> permutations of <math>\bigl(\pm 1, \pm (\sqrt{2} + 1), \pm (2 \sqrt{2} + 1)\bigr)</math>.<br>
E.g. <math>(5, 4) = (1, -b, a) = (1, -2 \sqrt{2} -1, \sqrt{2} +1)</math>.
 a = sqrt(2) + 1
 b = 2*sqrt(2) + 1

 <span style="white-space: nowrap;"><span style="color:gray;">(0, 0):</span> (-b, -a, -1), <span style="color:gray;">(0, 1):</span> (-a, -b, -1), <span style="color:gray;">(0, 2):</span> (-b, -1, -a), <span style="color:gray;">(0, 3):</span> (-a, -1, -b), <span style="color:gray;">(0, 4):</span> (-1, -b, -a), <span style="color:gray;">(0, 5):</span> (-1, -a, -b),</span>
 <span style="white-space: nowrap;"><span style="color:gray;">(1, 0):</span> ( b, -a, -1), <span style="color:gray;">(1, 1):</span> ( a, -b, -1), <span style="color:gray;">(1, 2):</span> ( b, -1, -a), <span style="color:gray;">(1, 3):</span> ( a, -1, -b), <span style="color:gray;">(1, 4):</span> ( 1, -b, -a), <span style="color:gray;">(1, 5):</span> ( 1, -a, -b),</span>
 <span style="white-space: nowrap;"><span style="color:gray;">(2, 0):</span> (-b, &nbsp;a, -1), <span style="color:gray;">(2, 1):</span> (-a, &nbsp;b, -1), <span style="color:gray;">(2, 2):</span> (-b, &nbsp;1, -a), <span style="color:gray;">(2, 3):</span> (-a, &nbsp;1, -b), <span style="color:gray;">(2, 4):</span> (-1, &nbsp;b, -a), <span style="color:gray;">(2, 5):</span> (-1, &nbsp;a, -b),</span>
 <span style="white-space: nowrap;"><span style="color:gray;">(3, 0):</span> ( b, &nbsp;a, -1), <span style="color:gray;">(3, 1):</span> ( a, &nbsp;b, -1), <span style="color:gray;">(3, 2):</span> ( b, &nbsp;1, -a), <span style="color:gray;">(3, 3):</span> ( a, &nbsp;1, -b), <span style="color:gray;">(3, 4):</span> ( 1, &nbsp;b, -a), <span style="color:gray;">(3, 5):</span> ( 1, &nbsp;a, -b),</span>
 <span style="white-space: nowrap;"><span style="color:gray;">(4, 0):</span> (-b, -a, &nbsp;1), <span style="color:gray;">(4, 1):</span> (-a, -b, &nbsp;1), <span style="color:gray;">(4, 2):</span> (-b, -1, &nbsp;a), <span style="color:gray;">(4, 3):</span> (-a, -1, &nbsp;b), <span style="color:gray;">(4, 4):</span> (-1, -b, &nbsp;a), <span style="color:gray;">(4, 5):</span> (-1, -a, &nbsp;b),</span>
 <span style="white-space: nowrap;"><span style="color:gray;">(5, 0):</span> ( b, -a, &nbsp;1), <span style="color:gray;">(5, 1):</span> ( a, -b, &nbsp;1), <span style="color:gray;">(5, 2):</span> ( b, -1, &nbsp;a), <span style="color:gray;">(5, 3):</span> ( a, -1, &nbsp;b), <span style="color:gray;">(5, 4):</span> ( 1, -b, &nbsp;a), <span style="color:gray;">(5, 5):</span> ( 1, -a, &nbsp;b),</span>
 <span style="white-space: nowrap;"><span style="color:gray;">(6, 0):</span> (-b, &nbsp;a, &nbsp;1), <span style="color:gray;">(6, 1):</span> (-a, &nbsp;b, &nbsp;1), <span style="color:gray;">(6, 2):</span> (-b, &nbsp;1, &nbsp;a), <span style="color:gray;">(6, 3):</span> (-a, &nbsp;1, &nbsp;b), <span style="color:gray;">(6, 4):</span> (-1, &nbsp;b, &nbsp;a), <span style="color:gray;">(6, 5):</span> (-1, &nbsp;a, &nbsp;b),</span>
 <span style="white-space: nowrap;"><span style="color:gray;">(7, 0):</span> ( b, &nbsp;a, &nbsp;1), <span style="color:gray;">(7, 1):</span> ( a, &nbsp;b, &nbsp;1), <span style="color:gray;">(7, 2):</span> ( b, &nbsp;1, &nbsp;a), <span style="color:gray;">(7, 3):</span> ( a, &nbsp;1, &nbsp;b), <span style="color:gray;">(7, 4):</span> ( 1, &nbsp;b, &nbsp;a), <span style="color:gray;">(7, 5):</span> ( 1, &nbsp;a, &nbsp;b)</span>
|}

<small>These are Python dictionaries without the surrounding braces. (Dicts work only in one direction, but [https://pypi.python.org/pypi/bidict bidict] can be used to get back to the pairs.)</small>

A dictionary of these permutations and their properties (including conjugacy class and inverse) can be found [https://pastebin.com/raw/JBx0TSsC here].

A dictionary of all the subgroups can be found [https://github.com/watchduck/full_octahedral_group/blob/master/projects/p03_subgroups/store_dicts.py here] (as a bijection from naive names to tuples of S<sub>4</sub> based numbers).





[[Category:Full octahedral group]]