View source for Octahedral symmetry

{{Short description|3D symmetry group}}
{{more footnotes|date=May 2013}}
{{3d point group navigator}}
{{redirect|Cube group|the mathematical group pertaining specifically to the 3x3x3 twisty puzzle cube|Rubik's Cube group}}
[[File:Full octahedral group; cycle graph.svg|thumb|330px|[[Cycle graph (algebra)|Cycle graph]]{{br}}The four hexagonal cycles have the inversion (the black knot on top) in common. The hexagons are symmetric, so e.g. 3 and 4 are in the same cycle.]]
A regular [[octahedron]] has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A [[cube]] has the same set of symmetries, since it is the polyhedron that is [[dual polyhedron|dual]] to an octahedron.

The group of orientation-preserving symmetries is S<sub>4</sub>, the [[symmetric group]] or the group of permutations of four objects, since there is exactly one such symmetry for each permutation of the four diagonals of the cube.

== Details ==

'''Chiral''' and '''full''' (or '''achiral''') '''octahedral symmetry''' are the [[Point groups in three dimensions|discrete point symmetries]] (or equivalently, [[List of spherical symmetry groups|symmetries on the sphere]]) with the largest [[symmetry group]]s compatible with [[translational symmetry]]. They are among the [[Crystal system#Overview of point groups by crystal system|crystallographic point groups]] of the [[cubic crystal system]].

{| class=wikitable
|+ [[Conjugacy class]]es
!colspan="2"|Elements of O
!colspan="2"|Inversions of elements of O
|-
|identity
|style="background:#fff;"|'''0'''
|inversion
|style="background:#ffff7f;"|'''0′'''
|-
|3 &times; rotation by 180° about a 4-fold axis
|style="background:#fff;"|'''7''', '''16''', '''23'''
|3 &times; reflection in a plane perpendicular to a 4-fold axis
|style="background:#ffff7f;"|'''7′''', '''16′''', '''23′'''
|-
|8 &times; rotation by 120° about a 3-fold axis
|style="background:#fff;"|3, 4, 8, 11, 12, 15,  19, 20
|8 &times; rotoreflection by 60°
|style="background:#ffff7f;"|3′, 4′, 8′, 11′, 12′, 15′,  19′, 20′
|-
|6 &times; rotation by 180° about a 2-fold axis
|style="background:#3375ff;"|1′, 2′, 5′, 6′, 14′, 21′
|6 &times; reflection in a plane perpendicular to a 2-fold axis
|style="background:#33d42a;"|1, 2, 5, 6, 14, 21
|-
|6 &times; rotation by 90° about a 4-fold axis
|style="background:#ef2500;"|9′, 10′, 13′, 17′, 18′, 22′
|6 &times; rotoreflection by 90°
|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>
|-
| [[File:Cube permutation 7 0.svg|180px]]{{br}}<math>(7,0) = 0'</math>
| [[File:Cube permutation 4 0.svg|180px]]{{br}}<math>(4,0) = 7'</math>
| [[File:Cube permutation 7 3.svg|180px]]{{br}}<math>(7,3) = 3'</math>
| [[File:Cube permutation 0 1.svg|180px]]{{br}}<math>(0,1) = 1</math>
| [[File:Cube permutation 3 5.svg|180px]]{{br}}<math>(3,5) = 9</math>
|-
|colspan="5"| A complete list can be found in [[v:Full octahedral group|the Wikiversity article]].
|}

As the [[hyperoctahedral group]] of dimension 3 the full octahedral group is the [[wreath product]] <math>\mathrm{S}_2 \wr \mathrm{S}_3 \simeq \mathrm{S}_2^3 \rtimes \mathrm{S}_3</math>, and a natural way to identify its elements is as pairs {{nowrap|(''m'', ''n'')}} with <math>m \in [0, 2^3)</math> and <math>n \in [0, 3!)</math>.{{br}}But as it is also the [[Direct product of groups|direct product]] {{nowrap|S<sub>4</sub> × S<sub>2</sub>}}, one can simply identify the elements of tetrahedral subgroup ''T''<sub>''d''</sub> as <math>a \in [0, 4!)</math> and their inversions as <math>a'</math>.

So e.g. the identity {{nowrap|(0, 0)}} is represented as 0 and the inversion {{nowrap|(7, 0)}} as 0′.{{br}}
{{nowrap|(3, 1)}} is represented as 6 and {{nowrap|(4, 1)}} as 6′.

A [[Improper rotation|rotoreflection]] is a combination of rotation and reflection.

{| 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 7′
 | image2 = Cube permutation 0 4.svg
 | width2 =  230
 | caption2 = applied on the 120° rotation 4
 | image3 = Cube permutation 4 4.svg
 | width3 =  230
 | caption3 = gives the 60° rotoreflection 8′.
}}
{{br}}<math>7' \circ 4 = 8'</math>
|-
|
{{multiple image
 | align = left
 | image1 = Cube permutation 4 0.svg
 | width1 =  230
 | caption1 = The reflection 7′
 | image2 = Cube permutation 1 1.svg
 | width2 =  230
 | caption2 = applied on the 90° rotation 22′
 | image3 = Cube permutation 5 1.svg
 | width3 =  230
 | caption3 = gives the 90° rotoreflection 17.
}}
{{br}}<math>7' \circ 22' = 17</math>
|}

=== Chiral octahedral symmetry ===

{| class=wikitable align=right
|- valign=top
!colspan=3|Gyration axes
|- align=center
!C<sub>4</sub>{{br}}[[File:Monomino.png|12px]]||C<sub>3</sub>{{br}}>[[File:3-fold rotation axis.svg|12px]]||C<sub>2</sub>{{br}}[[File:Rhomb.svg|12px]]
|- align=center
|3||4||6
|}

'''O''', '''432''', or [4,3]<sup>+</sup> of order 24, is '''chiral octahedral symmetry''' or '''rotational octahedral symmetry''' .  This group is like chiral [[tetrahedral symmetry]] T, but the C<sub>2</sub> axes are now C<sub>4</sub> axes, and additionally there are 6 C<sub>2</sub> axes, through the midpoints of the edges of the cube.  T<sub>d</sub> and O are isomorphic as abstract groups: they both correspond to S<sub>4</sub>, the [[symmetric group]] on 4 objects. T<sub>d</sub> is the union of T and the set obtained by combining each element of O{{hsp}}&nbsp;\&nbsp;{{hsp}}T with inversion. O is the rotation group of the [[Cube (geometry)|cube]] and the regular [[octahedron]].

{| class=wikitable
|+ Chiral octahedral symmetry
!Orthogonal projection
!colspan=3|Stereographic projection
|-
!2-fold
!4-fold
!3-fold
!2-fold
|-
|[[File:Sphere_symmetry_group_o.png|150px]]
|[[File:Disdyakis_dodecahedron_stereographic_D4_gyrations.png|150px]]
|[[File:Disdyakis_dodecahedron_stereographic_D3_gyrations.png|150px]]
|[[File:Disdyakis_dodecahedron_stereographic_D2_gyrations.png|150px]]
|}

=== <span id="Achiral octahedral symmetry">Full octahedral symmetry</span> ===
'''O<sub>h</sub>''', '''*432''', [4,3], or '''m3m''' of order 48 – '''achiral octahedral symmetry''' or '''full octahedral symmetry'''. This group has the  same rotation axes as O, but with mirror planes, comprising both the mirror planes of T<sub>d</sub> and T<sub>h</sub>.  This group is isomorphic to S<sub>4</sub>.C<sub>2</sub>, and is the full symmetry group of the [[Cube (geometry)|cube]] and [[octahedron]]. It is the [[hyperoctahedral group]] for {{nowrap|1=''n'' = 3}}. See also [[Cube#The isometries of the cube|the isometries of the cube]].

{{multiple image
 | align = left
 | total_width = 450
 | image1 = Polyhedron pair 6-8.png |width1=4000|height1=4000
 | caption1 = [[Compound of cube and octahedron]]
 | image2 = Polyhedron great rhombi 6-8 dual max.png |width2=4000|height2=4000
 | caption2 = Each face of the [[disdyakis dodecahedron]]  is a fundamental domain.
 | image3 = Sphere symmetry group oh.png |width3=649|height3=625
 | caption3 = The octahedral group O<sub>h</sub> with fundamental domain
}}

With the 4-fold axes as coordinate axes, a fundamental domain of '''O<sub>h</sub>''' is given by 0 ≤ ''x'' ≤ ''y'' ≤ ''z''. An object with this symmetry is characterized by the part of the object in the fundamental domain, for example the [[cube]] is given by {{nowrap|1=''z'' = 1}}, and the [[octahedron]] by {{nowrap|1=''x'' + ''y'' + ''z'' = 1}} (or the corresponding inequalities, to get the solid instead of the surface).
{{nowrap|1=''ax'' + ''by'' + ''cz'' = 1}} gives a polyhedron with 48 faces, e.g. the disdyakis dodecahedron.

Faces are 8-by-8 combined to larger faces for {{nowrap|1=''a'' = ''b'' = 0}} (cube) and 6-by-6 for {{nowrap|1=''a'' = ''b'' = ''c''}} (octahedron).

The 9 mirror lines of full octahedral symmetry can be divided into two subgroups of 3 and 6 (drawn in purple and red), representing in two orthogonal subsymmetries: [[Dihedral symmetry in three dimensions|D<sub>2h</sub>]], and [[Tetrahedral symmetry|T<sub>d</sub>]]. D<sub>2h</sub> symmetry can be doubled to D<sub>4h</sub> by restoring 2 mirrors from one of three orientations.
{{Clear}}

{| class="wikitable collapsible collapsed"
!colspan=2| Octahedral symmetry and reflective subgroups
|- valign=top
|
{| class=wikitable
|-
!rowspan=2|[[Orthographic projection|Orthographic{{br}}projection]]
!colspan=3|[[Stereographic projection]]
|-
!4-fold
!3-fold
!2-fold
|-
!colspan=4|O<sub>h</sub>, [4,3], {{CDD|node_c2|4|node_c1|3|node_c1}}, full octahedral symmetry (3+6 mirrors)
|-
|[[File:Sphere symmetry group oh.png|100px]]
|[[File:Disdyakis dodecahedron stereographic D4.png|100px]]
|[[File:Disdyakis dodecahedron stereographic D3.png|100px]]
|[[File:Disdyakis dodecahedron stereographic D2.png|100px]]
|-
!colspan=4|T<sub>d</sub>, [3,3]=[1<sup>+</sup>,4,3], {{CDD|node_h0|4|node_c1|3|node_c1}} = {{CDD|nodeab_c1|split2|node_c1}}, tetrahedral subgroup (6 mirrors)
|-
|[[File:Sphere symmetry group td.png|100px]]
|[[File:Tetrakis hexahedron stereographic D4.png|100px]]
|[[File:Tetrakis hexahedron stereographic D3.png|100px]]
|[[File:Tetrakis hexahedron stereographic D2.png|100px]]
|-
!colspan=4|C<sub>3v</sub>, [3], {{CDD|node_c1|3|node_c1}}, dihedral subgroup (3 mirrors)
|-
![[File:Trigonal_hosohedron_ortho.png|100px]]
|[[File:trigonal hosohedron stereographic D4.png|100px]]
|[[File:trigonal hosohedron stereographic D3.png|100px]]
|[[File:trigonal hosohedron stereographic D2.png|100px]]
|}
|
{| class=wikitable
|-
!rowspan=2|Orthographic{{br}}projection
!colspan=3|Stereographic projection
|-
!4-fold
!3-fold
!2-fold
|-
!colspan=4|D<sub>4h</sub>, [4,2], {{CDD|node_c2|4|node_c1|2|node_c3}} dihedral subgroup (1+2+2 mirrors)
|-
![[File:Octagonal bipyramidal orthogonal.png|100px]]
|[[File:octagonal bipyramidal stereographic D4.png|100px]]
|[[File:octagonal bipyramidal stereographic D3.png|100px]]
|[[File:octagonal bipyramidal stereographic D2.png|100px]]
|-
!colspan=4|D<sub>2h</sub>, [2,2]=[4,3*], {{CDD|node_c2|2|node_c1|2|node_c3}} = dihedral subgroup (1+1+1 mirrors)
|-
![[File:Octahedron orthographic.png|100px]]
|[[File:octahedron stereographic D4.png|100px]]
|[[File:octahedron stereographic D3.png|100px]]
|[[File:octahedron stereographic D2.png|100px]]
|-
!colspan=4|C<sub>4v</sub>, [4], {{CDD|node_c2|4|node_c1}}, dihedral subgroup (2+2 mirrors)
|-
![[File:Tetragonal_hosohedron_ortho.png|100px]]
|[[File:Tetragonal hosohedron stereographic D4.png|100px]]
|[[File:Tetragonal hosohedron stereographic D3.png|100px]]
|[[File:Tetragonal hosohedron stereographic D2.png|100px]]
|}
|}

=== Rotation matrices ===
Take the set of all 3×3 [[permutation matrix|permutation matrices]] and assign a + or − sign to each of the three 1s. There are <math>3!=6</math> permutations and <math>2^3=8</math> sign combinations for a total of 48 matrices, giving the full octahedral group. 24 of these matrices have a [[determinant]] of +1; these are the rotation matrices of the chiral octahedral group. The other 24 matrices have a determinant of −1 and correspond to a reflection or inversion.

Three reflectional generator matrices are needed for octahedral symmetry, which represent the three mirrors of a [[Coxeter–Dynkin diagram]]. The product of the reflections produce 3 rotational generators.

{| class=wikitable
|+ [4,3], {{CDD|node_n0|4|node_n1|3|node_n2}}
!
!colspan=3|Reflections
!colspan=3|Rotations
![[Rotoreflection]]
|-
!Generators
! R<sub>0</sub>
! R<sub>1</sub>
! R<sub>2</sub>
! R<sub>0</sub>R<sub>1</sub>
! R<sub>1</sub>R<sub>2</sub>
! R<sub>0</sub>R<sub>2</sub>
! R<sub>0</sub>R<sub>1</sub>R<sub>2</sub>
|- align=center
!Group
|{{CDD|node_n0}}
|{{CDD|node_n1}}
|{{CDD|node_n2}}
|{{CDD|node_h2|4|node_h2}}
|{{CDD|node_h2|3|node_h2}}
|{{CDD|node_h2|2x|node_h2}}
| {{CDD|node_h2|6|node_h4|2x|node_h2}}
|- align=center
!Order
|2||2||2||4||3||2||6
|- align=center
!Matrix
|
<math>\left [\begin{smallmatrix}
1 & \phantom{+}0 & \phantom{+}0 \\
0 & \phantom{+}1 & \phantom{+}0 \\
0 & \phantom{+}0 & -1 \\
\end{smallmatrix}\right ]
</math>
|
<math>\left [\begin{smallmatrix}
1 & 0 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0 \\
\end{smallmatrix}\right ]
</math>
|
<math>\left [\begin{smallmatrix}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 1 \\
\end{smallmatrix}\right ]
</math>
|
<math>\left [\begin{smallmatrix}
1 & \phantom{+}0 & \phantom{+}0 \\
0 & \phantom{+}0 & \phantom{+}1 \\
0 & -1 & \phantom{+}0 \\
\end{smallmatrix}\right ]
</math>
|
<math>\left [\begin{smallmatrix}
0 & 1 & 0 \\
0 & 0 & 1 \\
1 & 0 & 0 \\
\end{smallmatrix}\right ]
</math>
|
<math>\left [\begin{smallmatrix}
0 & \phantom{+}1 & \phantom{+}0 \\
1 & \phantom{+}0 & \phantom{+}0 \\
0 & \phantom{+}0 & -1 \\
\end{smallmatrix}\right ]
</math>
|
<math>\left [\begin{smallmatrix}
\phantom{+}0 & 1 & 0 \\
\phantom{+}0 & 0 & 1 \\
-1 & 0 & 0 \\
\end{smallmatrix}\right ]
</math>
|}

=== Subgroups of full octahedral symmetry ===

{| align=left
|
{{multiple image
 | align = left
 | total_width = 610
 | image1 = Subgroup of Oh; S4 blue red; cycle graph.svg
 | width1 =  1155 | height1 = 554
 | caption1 = O
 | image2 = Subgroup of Oh; S4 green orange; cycle graph.svg
 | width2 = 1155 | height2 = 554
 | caption2 = T<sub>d</sub>
 | image3 = Subgroup of Oh; A4xC2; cycle graph.svg
 | width3 = 950 | height3 = 422
 | caption3 = T<sub>h</sub>
 | footer = Cycle graphs of subgroups of order 24
}}
|-
|
[[File:Full octahedral group; subgroups Hasse diagram.svg|left|thumb|600px|Subgroups ordered in a [[Hasse diagram]]]]
|- 
|
{{multiple image
 | align = left
 | total_width = 610
 | image1 = Full octahedral group; subgroups Hasse diagram; rotational.svg
 | width1 =  782 | height1 = 990
 | caption1 = Rotational subgroups
 | image2 = Full octahedral group; subgroups Hasse diagram; reflective.svg
 | width2 = 861 | height2 = 1200
 | caption2 = Reflective subgroups
 | image3 = Full octahedral group; subgroups Hasse diagram; inversion.svg
 | width3 =  1100 | height3 = 1300
 | caption3 = Subgroups containing inversion

}}
|}

{| class="wikitable sortable"
![[Schoenflies notation]]
!colspan=2|[[Coxeter notation|Coxeter]]
![[Orbifold notation|Orb.]]
![[Hermann–Mauguin notation|H-M]]
![[List of small groups|Structure]]
![[Cycle diagram|Cyc.]]
![[Symmetry number|Order]]
![[Index of a subgroup|Index]]
|-align=center BGCOLOR="#e0f0f0"
|O<sub>h</sub> ||[4,3]||{{CDD|node|4|node|3|node}} ||*432||m{{overline|3}}m ||[[Symmetric group|S<sub>4</sub>]]×S<sub>2</sub>||||48||1
|-align=center BGCOLOR="#e0f0f0"
|T<sub>d</sub> ||[3,3]||{{CDD|node|3|node|3|node}} ||*332 ||{{overline|4}}3m ||S<sub>4</sub>||[[File:Subgroup of Oh; S4 green orange; cycle graph.svg|40px]]||24 ||2
|-align=center BGCOLOR="#e0f0f0"
|D<sub>4h</sub> ||[2,4]||{{CDD|node|2|node|4|node}} ||*224 ||4/mmm ||D<sub>2</sub>×D<sub>8</sub>||[[File:GroupDiagramMiniC2D8.svg|25px]]||16 ||3
|-align=center BGCOLOR="#e0f0f0"
|D<sub>2h</sub>  ||[2,2]||{{CDD|node|2|node|2|node}} ||*222 ||mmm |||D{{supsub|3|2}} = D<sub>2</sub>×D<sub>4</sub>||[[File:GroupDiagramMiniC2x3.svg|25px]]||8 ||6
|-align=center BGCOLOR="#e0f0f0"
|C<sub>4v</sub> ||[4]||{{CDD|node|4|node}} ||*44 ||4mm ||[[Dihedral group|D<sub>8</sub>]]||[[File:GroupDiagramMiniD8.svg|25px]]||8 ||6
|-align=center BGCOLOR="#e0f0f0"
|C<sub>3v</sub> ||[3]||{{CDD|node|3|node}} ||*33 ||3m ||D<sub>6</sub> = S<sub>3</sub>||[[File:GroupDiagramMiniD6.svg|25px]]||6 ||8
|-align=center BGCOLOR="#e0f0f0"
|C<sub>2v</sub> ||[2]||{{CDD|node|2|node}} ||*22 ||mm2 ||D{{supsub|2|2}} = D<sub>4</sub>||[[File:GroupDiagramMiniD4.svg|25px]]||4 ||12
|-align=center BGCOLOR="#e0f0f0"
|C<sub>s</sub> = C<sub>1v</sub> ||[&nbsp;]||{{CDD|node}} ||* ||{{overline|2}} or m ||D<sub>2</sub>||[[File:GroupDiagramMiniC2.svg|25px]]||2 ||24
|-align=center BGCOLOR="#f0f0e0"
|T<sub>h</sub> ||[3<sup>+</sup>,4]||{{CDD|node_h2|3|node_h2|4|node}} ||3*2 ||m{{overline|3}} ||[[Alternating group|A<sub>4</sub>]]×S<sub>2</sub>||[[File:Subgroup of Oh; A4xC2; cycle graph.svg|40px]]||24 ||2
|-align=center BGCOLOR="#f0f0e0"
|C<sub>4h</sub> ||[4<sup>+</sup>,2]||{{CDD|node_h2|4|node_h2|2|node}} ||4* ||4/m ||[[Cyclic group|Z<sub>4</sub>]]×D<sub>2</sub>||[[File:GroupDiagramMiniC2C4.svg|25px]]||8 ||6
|-align=center BGCOLOR="#f0f0e0"
|D<sub>3d</sub> ||[2<sup>+</sup>,6]||{{CDD|node_h2|2x|node_h2|6|node}} ||2*3 ||{{overline|3}}m ||D<sub>12</sub> = Z<sub>2</sub>×D<sub>6</sub>||[[File:GroupDiagramMiniD12.svg|25px]]||12 ||4
|-align=center BGCOLOR="#f0f0e0"
|D<sub>2d</sub> ||[2<sup>+</sup>,4]||{{CDD|node_h2|2x|node_h2|4|node}} ||2*2 ||{{overline|4}}2m ||D<sub>8</sub>||[[File:GroupDiagramMiniD8.svg|25px]]||8 ||6
|-align=center BGCOLOR="#f0f0e0"
| C<sub>2h</sub> = D<sub>1d</sub> ||[2<sup>+</sup>,2]||{{CDD|node_h2|2x|node_h2|2|node}} ||2* ||2/m ||Z<sub>2</sub>×D<sub>2</sub>||[[File:GroupDiagramMiniD4.svg|25px]]||4 ||12
|-align=center BGCOLOR="#e0e0e0"
|S<sub>6</sub> ||[2<sup>+</sup>,6<sup>+</sup>]||{{CDD|node_h2|2x|node_h4|6|node_h2}} ||3× ||{{overline|3}} ||Z<sub>6</sub> = Z<sub>2</sub>×Z<sub>3</sub>||[[File:GroupDiagramMiniC6.svg|25px]]||6 ||8
|-align=center BGCOLOR="#e0e0e0"
|S<sub>4</sub> ||[2<sup>+</sup>,4<sup>+</sup>]||{{CDD|node_h2|2x|node_h4|4|node_h2}} ||2× ||{{overline|4}} ||Z<sub>4</sub>||[[File:GroupDiagramMiniC4.svg|25px]]||4 ||12
|-align=center BGCOLOR="#e0e0e0"
|S<sub>2</sub> ||[2<sup>+</sup>,2<sup>+</sup>]||{{CDD|node_h2|2x|node_h4|2x|node_h2}} ||× ||{{overline|1}} ||S<sub>2</sub>||[[File:GroupDiagramMiniC2.svg|25px]]||2 ||24
|-align=center BGCOLOR="#f0e0f0"
| O ||[4,3]<sup>+</sup>||{{CDD|node_h2|4|node_h2|3|node_h2}} ||432 ||432 ||S<sub>4</sub>||[[File:Subgroup of Oh; S4 blue red; cycle graph.svg|40px]]||24 ||2
|-align=center BGCOLOR="#f0e0f0"
|T ||[3,3]<sup>+</sup>||{{CDD|node_h2|3|node_h2|3|node_h2}} ||332 ||23 ||A<sub>4</sub>||[[File:GroupDiagramMiniA4.svg|25px]]||12 ||4
|-align=center BGCOLOR="#f0e0f0"
|D<sub>4</sub> ||[2,4]<sup>+</sup>||{{CDD|node_h2|2x|node_h2|4|node_h2}} ||224 ||422 ||D<sub>8</sub>||[[File:GroupDiagramMiniD8.svg|25px]]||8 ||6
|-align=center BGCOLOR="#f0e0f0"
|D<sub>3</sub> ||[2,3]<sup>+</sup>||{{CDD|node_h2|2x|node_h2|3|node_h2}} ||223 ||322 ||D<sub>6</sub> = S<sub>3</sub>||[[File:GroupDiagramMiniD6.svg|25px]]||6 ||8
|-align=center BGCOLOR="#f0e0f0"
|D<sub>2</sub> ||[2,2]<sup>+</sup>||{{CDD|node_h2|2x|node_h2|2x|node_h2}} ||222 ||222 ||D<sub>4</sub> = Z{{supsub|2|2}}||[[File:GroupDiagramMiniD4.svg|25px]]||4 ||12
|-align=center BGCOLOR="#f0e0f0"
|C<sub>4</sub> ||[4]<sup>+</sup>||{{CDD|node_h2|4|node_h2}} ||44 ||4 ||Z<sub>4</sub>||[[File:GroupDiagramMiniC4.svg|25px]]||4 ||12
|-align=center BGCOLOR="#f0e0f0"
|C<sub>3</sub> ||[3]<sup>+</sup>||{{CDD|node_h2|3|node_h2}} ||33 ||3 ||Z<sub>3</sub> = A<sub>3</sub>||[[File:GroupDiagramMiniC3.svg|25px]]||3 ||16
|-align=center BGCOLOR="#f0e0f0"
|C<sub>2</sub> ||[2]<sup>+</sup>||{{CDD|node_h2|2x|node_h2}} ||22 ||2 ||Z<sub>2</sub>||[[File:GroupDiagramMiniC2.svg|25px]]||2 ||24
|-align=center BGCOLOR="#f0e0f0"
| C<sub>1</sub> ||[&nbsp;]<sup>+</sup>||{{CDD|node_h2}} ||11 ||1 ||Z<sub>1</sub>||[[File:GroupDiagramMiniC1.svg|25px]]||1 ||48
|}

{| width=480 
|[[File:Octahedral_symmetry_tree_conway.png|480px]]
|-
|Octahedral subgroups in [[Coxeter notation]]<ref>John Conway, ''The Symmetries of Things'', Fig 20.8, p. 280</ref>
|}

== The isometries of the cube ==
<!-- This section is linked from [[Conjugacy class]] -->
[[File:48 элементов симметрии куба.gif|thumb|right|48 symmetry elements of a cube]]
The cube has 48 isometries (symmetry elements), forming the [[symmetry group]] O<sub>h</sub>, isomorphic to [[Symmetric group|S<sub>4</sub>]]&thinsp;×&thinsp;Z<sub>2</sub>. They can be categorized as follows:
* O (the identity and 23 proper rotations) with the following [[conjugacy class]]es (in parentheses are given the permutations of the body diagonals and the [[Quaternions and spatial rotation|unit quaternion representation]]):
** identity (identity; 1)
** rotation about an axis from the center of a face to the center of the opposite face  by an angle of 90°: 3 axes, 2 per axis, together 6 ((1 2 3 4), etc.; ((1&thinsp;±&thinsp;''i''{{space|hair}})/{{radic|2}}, etc.)
** ditto by an angle of 180°: 3 axes, 1 per axis, together 3 ((1 2) (3 4), etc.; ''i'', ''j'', ''k'')
** rotation about an axis from the center of an edge to the center of the opposite edge by an angle of 180°: 6 axes, 1 per axis, together 6 ((1 2), etc.; ((''i''&thinsp;±&thinsp;''j''{{space|hair}})/{{radic|2}}, etc.)
** rotation about a body diagonal by an angle of 120°: 4 axes, 2 per axis, together 8 ((1 2 3), etc.; (1&thinsp;±&thinsp;''i''&thinsp;±&thinsp;''j''&thinsp;±&thinsp;''k'')/2)
* The same with [[inversion in a point|inversion]] ('''x''' is mapped to −'''x''') (also 24 isometries). Note that rotation by an angle of 180° about an axis combined with inversion is just reflection in the perpendicular plane. The combination of inversion and rotation about a body diagonal by an angle of 120° is rotation about the body diagonal by an angle of 60°, combined with reflection in the perpendicular plane (the rotation itself does not map the cube to itself; the intersection of the reflection plane with the cube is a regular [[hexagon]]).

An isometry of the cube can be identified in various ways:
* by the faces three given adjacent faces (say 1, 2, and 3 on a die) are mapped to
* by the image of a cube with on one face a non-symmetric marking: the face with the marking, whether it is normal or a mirror image, and the orientation
* by a permutation of the four body diagonals (each of the 24 permutations is possible), combined with a toggle for inversion of the cube, or not

For cubes with colors or markings (like [[dice]] have), the symmetry group is a subgroup of O<sub>h</sub>.

Examples:
* ''C''<sub>4v</sub>, [4], (*422): if one face has a different color (or two opposite faces have colors different from each other and from the other four), the cube has 8 isometries, like a square has in 2D.
* ''D''<sub>2h</sub>, [2,2], (*222): if opposite faces have the same colors, different for each set of two, the cube has 8 isometries, like a [[cuboid]].
* ''D''<sub>4h</sub>, [4,2], (*422): if two opposite faces have the same color, and all other faces have one different color, the cube has 16 isometries, like a square [[Prism (geometry)|prism]] (square box).
* ''C''<sub>2v</sub>, [2], (*22):
** if two adjacent faces have the same color, and all other faces have one different color, the cube has 4 isometries.
** if three faces, of which two opposite to each other, have one color and the other three one other color, the cube has 4 isometries.
** if two opposite faces have the same color, and two other opposite faces also, and the last two have different colors, the cube has 4 isometries, like a piece of blank paper with a shape with a mirror symmetry.
* ''C''<sub>s</sub>, [&nbsp;], (*):
** if two adjacent faces have colors different from each other, and the other four have a third color, the cube has 2 isometries.
** if two opposite faces have the same color, and all other faces have  different colors, the cube has 2 isometries, like an asymmetric piece of blank paper.
* ''C''<sub>3v</sub>, [3], (*33): if three faces, of which none opposite to each other, have one color and the other three one other color, the cube has 6 isometries.

For some larger subgroups a cube with that group as symmetry group is not possible with just coloring whole faces. One has to draw some pattern on the faces.

Examples:
* ''D''<sub>2d</sub>, [2<sup>+</sup>,4], (2*2): if one face has a line segment dividing the face into two equal rectangles, and the opposite has the same in perpendicular direction, the cube has 8 isometries; there is a symmetry plane and 2-fold rotational symmetry with an axis at an angle of 45° to that plane, and, as a result, there is also another symmetry plane perpendicular to the first, and another axis of 2-fold rotational symmetry perpendicular to the first.
* [[Pyritohedral symmetry|T<sub>h</sub>]], [3<sup>+</sup>,4], (3*2): if each face has a line segment dividing the face into two equal rectangles, such that the line segments of adjacent faces do ''not'' meet at the edge, the cube has 24 isometries: the even permutations of the body diagonals and the same combined with inversion ('''x''' is mapped to &minus;'''x''').
* T<sub>d</sub>, [3,3], (*332): if the cube consists of eight smaller cubes, four white and four black, put together alternatingly in all three standard directions, the cube has again 24 isometries: this time the even permutations of the body diagonals and the inverses of the ''other'' proper rotations.
* T, [3,3]<sup>+</sup>, (332): if each face has the same pattern with 2-fold rotational symmetry, say the letter S, such that at all edges a top of one S meets a side of the other S, the cube has 12 isometries: the even permutations of the body diagonals.

The full symmetry of the cube, O<sub>h</sub>, [4,3], (*432), is preserved [[if and only if]] all faces have the same pattern such that the full symmetry of the [[square]] is preserved, with for the square a symmetry group, [[Dihedral symmetry|Dih<sub>4</sub>]], [4], of order 8.

The full symmetry of the cube under proper rotations, O, [4,3]<sup>+</sup>, (432), is preserved if and only if all faces have the same pattern with [[Cyclic symmetry|4-fold rotational symmetry]], Z<sub>4</sub>, [4]<sup>+</sup>.

== Octahedral symmetry of the Bolza surface ==
In [[Riemann surface]] theory, the [[Bolza surface]], sometimes called the Bolza curve, is obtained as the ramified double cover of the Riemann sphere, with ramification locus at the set of vertices of the regular inscribed octahedron.  Its automorphism group includes the hyperelliptic involution which flips the two sheets of the cover.  The quotient by the order 2 subgroup generated by the hyperelliptic involution yields precisely the group of symmetries of the octahedron.  Among the many remarkable properties of the Bolza surface is the fact that it maximizes the [[systolic geometry|systole]] among all genus 2 hyperbolic surfaces.

== Solids with octahedral chiral symmetry ==

{| class=wikitable
|-
!Class
! Name
! Picture
! Faces
! Edges
! Vertices
! Dual name
! Picture
|- align=center
![[Archimedean solid]]{{br}}([[Catalan solid]])
| [[snub cube]]|| [[File:Polyhedron snub 6-8 right max.png|60px]]|| 38 || 60 || 24
|[[pentagonal icositetrahedron]]||[[File:Polyhedron snub 6-8 right dual max.png|60px]]
|}

== Solids with full octahedral symmetry ==

{| class=wikitable
|-
!Class
!Name 
!Picture
!Faces
!Edges
!Vertices
!Dual name 
!Picture
|- align=center
![[Platonic solid]]
|[[Cube]]||[[File:Polyhedron 6 max.png|60px|Hexahedron (cube)]]||6||12||8
|[[Octahedron]]||[[File:Polyhedron 8 max.png|60px|Octahedron]]
|- align=center
!rowspan=5|[[Archimedean solid]]{{br}}(dual [[Catalan solid]])
| [[Cuboctahedron]]|| [[File:Polyhedron 6-8 max.png|60px]]|| 14 || 24 || 12
| [[Rhombic dodecahedron]]|| [[File:Polyhedron 6-8 dual max.png|60px]]
|- align=center
| [[Truncated cube]]|| [[File:Polyhedron truncated 6 max.png|60px]]|| 14 || 36|| 24
| [[Triakis octahedron]]|| [[File:Polyhedron truncated 6 dual max.png|60px]]
|- align=center
| [[Truncated octahedron]]|| [[File:Polyhedron truncated 8 max.png|60px]]|| 14 || 36 || 24
| [[Tetrakis hexahedron]]|| [[File:Polyhedron truncated 8 dual max.png|60px]]
|- align=center
| [[Rhombicuboctahedron]]|| [[File:Polyhedron small rhombi 6-8 max.png|60px]]|| 26 || 48 || 24
| [[Deltoidal icositetrahedron]]|| [[File:Polyhedron small rhombi 6-8 dual max.png|60px]]
|- align=center
| [[Truncated cuboctahedron]]|| [[File:Polyhedron great rhombi 6-8 max.png|60px]]|| 26 || 72 || 48
| [[Disdyakis dodecahedron]]|| [[File:Polyhedron great rhombi 6-8 dual max.png|60px]]
|- align=center
!rowspan=2|[[Regular compound polyhedron|Regular{{br}}compound{{br}}polyhedron]]
|[[Stellated octahedron ]]||[[File:Polyhedron stellated 8 max.png|60px]]||8 ||12 ||8||colspan=2|Self-dual
|- align=center
|[[Compound of cube and octahedron|Cube and octahedron]]
|[[File:Polyhedron pair 6-8 max.png|60px]]
|14||24||14||colspan=2|Self-dual
|}

== See also ==
* [[Tetrahedral symmetry]]
* [[Icosahedral symmetry]]
* [[Binary octahedral group]]
* [[Hyperoctahedral group]]
* {{Wikiversity-inline|Full octahedral group}}

== References ==
{{Reflist}}

== Further reading ==
* Peter R. Cromwell, ''Polyhedra'' (1997), p.&nbsp;295
* ''The Symmetries of Things'' 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, {{isbn|978-1-56881-220-5}}
* ''Kaleidoscopes: Selected Writings of [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]'', edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, {{isbn|978-0-471-01003-6}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
* [[Norman Johnson (mathematician)|N.W. Johnson]]: ''Geometries and Transformations'', (2018) {{ISBN|978-1-107-10340-5}} Chapter 11: ''Finite symmetry groups'', 11.5 Spherical Coxeter groups

== External links ==
* {{mathworld | urlname = OctahedralGroup | title = Octahedral group }}
* Groupprops: [http://groupprops.subwiki.org/wiki/Direct_product_of_S4_and_Z2 Direct product of S4 and Z2]

{{DEFAULTSORT:Octahedral Symmetry}}
[[Category:Finite groups]]
[[Category:Rotational symmetry]]