This article includes a list ofgeneral references, butit lacks sufficient correspondinginline citations. Please help toimprove this article byintroducing more precise citations.(May 2013) (Learn how and when to remove this message)

Selectedpoint groups in three dimensions

Involutional symmetry Cs, (*) [ ] =	Cyclic symmetry Cnv, (*nn) [n] =	Dihedral symmetry Dnh, (*n22) [n,2] =
Polyhedral group, [n,3], (*n32)
Tetrahedral symmetry Td, (*332) [3,3] =	Octahedral symmetry Oh, (*432) [4,3] =	Icosahedral symmetry Ih, (*532) [5,3] =

A regularoctahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. Acube has the same set of symmetries, since it is the polyhedron that isdual to an octahedron.

The group of orientation-preserving symmetries is S₄, thesymmetric 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.

Chiral andfull (orachiral)octahedral symmetry are thediscrete point symmetries (or equivalently,symmetries on the sphere) with the largestsymmetry groups compatible withtranslational symmetry. They are among thecrystallographic point groups of thecubic crystal system.

Examples
${\displaystyle (0,0)=0}$	${\displaystyle (3,0)=7}$	${\displaystyle (0,3)=3}$	${\displaystyle (7,1)=1^{\prime }}$	${\displaystyle (4,5)=9^{\prime }}$
${\displaystyle (7,0)=0^{\prime }}$	${\displaystyle (4,0)=7^{\prime }}$	${\displaystyle (7,3)=3^{\prime }}$	${\displaystyle (0,1)=1}$	${\displaystyle (3,5)=9}$
A complete list can be found inthe Wikiversity article.

As thehyperoctahedral group of dimension 3 the full octahedral group is thewreath product${\displaystyle {\mathrm{S}}_2\wr {\mathrm{S}}_3\simeq {\mathrm{S}}_2^3⋊{\mathrm{S}}_3}$, and a natural way to identify its elements is as pairs(m,n) with${\displaystyle m\in [0,2^3)}$ and${\displaystyle n\in [0,3!)}$.
But as it is also thedirect productS₄ × S₂, one can simply identify the elements of tetrahedral subgroupT_d as${\displaystyle a\in [0,4!)}$ and their inversions as${\displaystyle a^{\prime }}$.

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

Arotoreflection is a combination of rotation and reflection.

Illustration of rotoreflections
The reflection 7′ applied on the 120° rotation 4 gives the 60° rotoreflection 8′. ${\displaystyle 7^{\prime }\circ 4=8^{\prime }}$
The reflection 7′ applied on the 90° rotation 22′ gives the 90° rotoreflection 17. ${\displaystyle 7^{\prime }\circ {22}^{\prime }=17}$

Gyration axes
C4	C3 >	C2
3	4	6

O,432, or [4,3]⁺ of order 24, ischiral octahedral symmetry orrotational octahedral symmetry . This group is like chiraltetrahedral symmetry T, but the C₂ axes are now C₄ axes, and additionally there are 6 C₂ axes, through the midpoints of the edges of the cube. T_d and O are isomorphic as abstract groups: they both correspond to S₄, thesymmetric group on 4 objects. T_d is the union of T and the set obtained by combining each element of O  \  T with inversion. O is the rotation group of thecube and the regularoctahedron.

Chiral octahedral symmetry

Orthogonal projection	Stereographic projection
2-fold	4-fold	3-fold	2-fold

O_h,*432, [4,3], orm3m of order 48 –achiral octahedral symmetry orfull octahedral symmetry. This group has the same rotation axes as O, but with mirror planes, comprising both the mirror planes of T_d and T_h. This group is isomorphic to S₄.C₂, and is the full symmetry group of thecube andoctahedron. It is thehyperoctahedral group forn = 3. See alsothe isometries of the cube.

Compound of cube and octahedron

Each face of thedisdyakis dodecahedron is a fundamental domain.

The octahedral group O_h with fundamental domain

With the 4-fold axes as coordinate axes, a fundamental domain ofO_h 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 thecube is given byz = 1, and theoctahedron byx +y +z = 1 (or the corresponding inequalities, to get the solid instead of the surface).ax +by +cz = 1 gives a polyhedron with 48 faces, e.g. the disdyakis dodecahedron.

Faces are 8-by-8 combined to larger faces fora =b = 0 (cube) and 6-by-6 fora =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:D_2h, andT_d. D_2h symmetry can be doubled to D_4h by restoring 2 mirrors from one of three orientations.

Octahedral symmetry and reflective subgroups
Orthographic projection Stereographic projection 4-fold 3-fold 2-fold Oh, [4,3],, full octahedral symmetry (3+6 mirrors) Td, [3,3]=[1+,4,3], =, tetrahedral subgroup (6 mirrors) C3v, [3],, dihedral subgroup (3 mirrors)	Orthographic projection Stereographic projection 4-fold 3-fold 2-fold D4h, [4,2], dihedral subgroup (1+2+2 mirrors) D2h, [2,2]=[4,3*], = dihedral subgroup (1+1+1 mirrors) C4v, [4],, dihedral subgroup (2+2 mirrors)

Take the set of all 3×3permutation matrices and assign a + or − sign to each of the three 1s. There are${\displaystyle 3!=6}$ permutations and${\displaystyle 2^3=8}$ sign combinations for a total of 48 matrices, giving the full octahedral group. 24 of these matrices have adeterminant 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 aCoxeter–Dynkin diagram. The product of the reflections produce 3 rotational generators.

[4,3],

	Reflections			Rotations			Rotoreflection
Generators	R0	R1	R2	R0R1	R1R2	R0R2	R0R1R2
Group
Order	2	2	2	4	3	2	6
Matrix

O Td Th Cycle graphs of subgroups of order 24

Rotational subgroups Reflective subgroups Subgroups containing inversion

Schoenflies notation	Coxeter		Orb.	H-M	Structure	Cyc.	Order	Index
Oh	[4,3]		*432	m3m	S4×S2		48	1
Td	[3,3]		*332	43m	S4		24	2
D4h	[2,4]		*224	4/mmm	D2×D8		16	3
D2h	[2,2]		*222	mmm	D3 2 = D2×D4		8	6
C4v	[4]		*44	4mm	D8		8	6
C3v	[3]		*33	3m	D6 = S3		6	8
C2v	[2]		*22	mm2	D2 2 = D4		4	12
Cs = C1v	[ ]		*	2 or m	D2		2	24
Th	[3+,4]		3*2	m3	A4×S2		24	2
C4h	[4+,2]		4*	4/m	Z4×D2		8	6
D3d	[2+,6]		2*3	3m	D12 = Z2×D6		12	4
D2d	[2+,4]		2*2	42m	D8		8	6
C2h = D1d	[2+,2]		2*	2/m	Z2×D2		4	12
S6	[2+,6+]		3×	3	Z6 = Z2×Z3		6	8
S4	[2+,4+]		2×	4	Z4		4	12
S2	[2+,2+]		×	1	S2		2	24
O	[4,3]+		432	432	S4		24	2
T	[3,3]+		332	23	A4		12	4
D4	[2,4]+		224	422	D8		8	6
D3	[2,3]+		223	322	D6 = S3		6	8
D2	[2,2]+		222	222	D4 = Z2 2		4	12
C4	[4]+		44	4	Z4		4	12
C3	[3]+		33	3	Z3 = A3		3	16
C2	[2]+		22	2	Z2		2	24
C1	[ ]+		11	1	Z1		1	48

Octahedral subgroups inCoxeter notation[1]

The cube has 48 isometries (symmetry elements), forming thesymmetry group O_h, isomorphic toS₄ × Z₂. They can be categorized as follows:

• O (the identity and 23 proper rotations) with the followingconjugacy classes (in parentheses are given the permutations of the body diagonals and theunit 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 ± i )/√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 ± j )/√2, etc.)
  • rotation about a body diagonal by an angle of 120°: 4 axes, 2 per axis, together 8 ((1 2 3), etc.; (1 ± i ± j ± k)/2)
• The same withinversion (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 regularhexagon).

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 (likedice have), the symmetry group is a subgroup of O_h.

Examples:

• C_4v, [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_2h, [2,2], (*222): if opposite faces have the same colors, different for each set of two, the cube has 8 isometries, like acuboid.
• D_4h, [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 squareprism (square box).
• C_2v, [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_s, [ ], (*):
  • 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_3v, [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_2d, [2⁺,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.
• T_h, [3⁺,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 donot 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 −x).
• T_d, [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 theother proper rotations.
• T, [3,3]⁺, (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_h, [4,3], (*432), is preservedif and only if all faces have the same pattern such that the full symmetry of thesquare is preserved, with for the square a symmetry group,Dih₄, [4], of order 8.

The full symmetry of the cube under proper rotations, O, [4,3]⁺, (432), is preserved if and only if all faces have the same pattern with4-fold rotational symmetry, Z₄, [4]⁺.

InRiemann surface theory, theBolza 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 thesystole among all genus 2 hyperbolic surfaces.

Class	Name	Picture	Faces	Edges	Vertices	Dual name	Picture
Archimedean solid (Catalan solid)	snub cube		38	60	24	pentagonal icositetrahedron

Class	Name	Picture	Faces	Edges	Vertices	Dual name	Picture
Platonic solid	Cube		6	12	8	Octahedron
Archimedean solid (dualCatalan solid)	Cuboctahedron		14	24	12	Rhombic dodecahedron
	Truncated cube		14	36	24	Triakis octahedron
	Truncated octahedron		14	36	24	Tetrakis hexahedron
	Rhombicuboctahedron		26	48	24	Deltoidal icositetrahedron
	Truncated cuboctahedron		26	72	48	Disdyakis dodecahedron
Regular compound polyhedron	Stellated octahedron		8	12	8	Self-dual
	Cube and octahedron		14	24	14	Self-dual

• Tetrahedral symmetry
• Icosahedral symmetry
• Binary octahedral group
• Hyperoctahedral group
• Learning materials related toFull octahedral group at Wikiversity

• Peter R. Cromwell,Polyhedra (1997), p. 295
• The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss,ISBN 978-1-56881-220-5
• Kaleidoscopes: Selected Writings ofH.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[1]
• N.W. Johnson:Geometries and Transformations, (2018)ISBN 978-1-107-10340-5 Chapter 11:Finite symmetry groups, 11.5 Spherical Coxeter groups

• Weisstein, Eric W."Octahedral group".MathWorld.
• Groupprops:Direct product of S4 and Z2

• Finite groups
• Rotational symmetry

• Articles with short description
• Short description matches Wikidata
• Articles lacking in-text citations from May 2013
• All articles lacking in-text citations
• Pages using multiple image with manual scaled images