Involutional symmetry C_{s}, (*) [ ] = 
Cyclic symmetry C_{nv}, (*nn) [n] = 
Dihedral symmetry D_{nh}, (*n22) [n,2] =  
Polyhedral group, [n,3], (*n32)  

Tetrahedral symmetry T_{d}, (*332) [3,3] = 
Octahedral symmetryOh, (*432)[4,3] = Icosahedral symmetry I_{h}, (*532) [5,3] = 
A regular octahedron has 24 rotational (or orientationpreserving) 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 to an octahedron.
The group of orientationpreserving symmetries is S_{4}, 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.
Chiral and full (or achiral) octahedral symmetry are the discrete point symmetries (or equivalently, symmetries on the sphere) with the largest symmetry groups compatible with translational symmetry. They are among the crystallographic point groups of the cubic crystal system.
Elements of O  Inversions of elements of O  

identity  0  inversion  0′ 
3 × rotation by 180° about a 4fold axis  7, 16, 23  3 × reflection in a plane perpendicular to a 4fold axis  7′, 16′, 23′ 
8 × rotation by 120° about a 3fold axis  3, 4, 8, 11, 12, 15, 19, 20  8 × rotoreflection by 60°  3′, 4′, 8′, 11′, 12′, 15′, 19′, 20′ 
6 × rotation by 180° about a 2fold axis  1′, 2′, 5′, 6′, 14′, 21′  6 × reflection in a plane perpendicular to a 2fold axis  1, 2, 5, 6, 14, 21 
6 × rotation by 90° about a 4fold axis  9′, 10′, 13′, 17′, 18′, 22′  6 × rotoreflection by 90°  9, 10, 13, 17, 18, 22 
Examples  











A complete list can be found in the Wikiversity article. 
As the hyperoctahedral group of dimension 3 the full octahedral group is the wreath product ,
and a natural way to identify its elements is as pairs (m, n) with and .
But as it is also the direct product S_{4} × S_{2}, one can simply identify the elements of tetrahedral subgroup T_{d} as and their inversions as a′.
So e.g. the identity (0, 0) is represented as 0 and the inversion (7, 0) as 0′.
(3, 1) is represented as 6 and (4, 1) as 6′.
A rotoreflection is a combination of rotation and reflection.
Illustration of rotoreflections  

 

Gyration axes  

C_{4} 
C_{3} > 
C_{2} 
3  4  6 
O, 432, or [4,3]^{+} of order 24, is chiral octahedral symmetry or rotational octahedral symmetry . This group is like chiral tetrahedral symmetry T, but the C_{2} axes are now C_{4} axes, and additionally there are 6 C_{2} axes, through the midpoints of the edges of the cube. T_{d} and O are isomorphic as abstract groups: they both correspond to S_{4}, the symmetric 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 the cube and the regular octahedron.
Orthogonal projection  Stereographic projection  

2fold  4fold  3fold  2fold 
O_{h}, *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_{d} and T_{h}. This group is isomorphic to S_{4}.C_{2}, and is the full symmetry group of the cube and octahedron. It is the hyperoctahedral group for n = 3. See also the isometries of the cube.
With the 4fold axes as coordinate axes, a fundamental domain of O_{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 the cube is given by z = 1, and the octahedron by x + 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 8by8 combined to larger faces for a = b = 0 (cube) and 6by6 for 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: D_{2h}, and T_{d}. D_{2h} symmetry can be doubled to D_{4h} by restoring 2 mirrors from one of three orientations.
Octahedral symmetry and reflective subgroups  



Take the set of all 3×3 permutation matrices and assign a + or − sign to each of the three 1s. There are permutations and 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 CoxeterDynkin diagram. The product of the reflections produce 3 rotational generators.
Reflections  Rotations  Rotoreflection  

Generators  R_{0}  R_{1}  R_{2}  R_{0}R_{1}  R_{1}R_{2}  R_{0}R_{2}  R_{0}R_{1}R_{2} 
Group  
Order  2  2  2  4  3  2  6 
Matrix 









Schoe.  Coxeter  Orb.  HM  Structure  Cyc.  Order  Index  

O_{h}  [4,3]  *432  m3m  S_{4}×S_{2}  48  1  
T_{d}  [3,3]  *332  43m  S_{4}  24  2  
D_{4h}  [2,4]  *224  4/mmm  D_{2}×D_{8}  16  3  
D_{2h}  [2,2]  *222  mmm  D_{2}^{3}=D_{2}×D_{4}  8  6  
C_{4v}  [4]  *44  4mm  D_{8}  8  6  
C_{3v}  [3]  *33  3m  D_{6}=S_{3}  6  8  
C_{2v}  [2]  *22  mm2  D_{2}^{2}=D_{4}  4  12  
C_{s}=C_{1v}  [ ]  *  2 or m  D_{2}  2  24  
T_{h}  [3^{+},4]  3*2  m3  A_{4}×S_{2}  24  2  
C_{4h}  [4^{+},2]  4*  4/m  Z_{4}×D_{2}  8  6  
D_{3d}  [2^{+},6]  2*3  3m  D_{12}=Z_{2}×D_{6}  12  4  
D_{2d}  [2^{+},4]  2*2  42m  D_{8}  8  6  
C_{2h} = D_{1d}  [2^{+},2]  2*  2/m  Z_{2}×D_{2}  4  12  
S_{6}  [2^{+},6^{+}]  3×  3  Z_{6}=Z_{2}×Z_{3}  6  8  
S_{4}  [2^{+},4^{+}]  2×  4  Z_{4}  4  12  
S_{2}  [2^{+},2^{+}]  ×  1  S_{2}  2  24  
O  [4,3]^{+}  432  432  S_{4}  24  2  
T  [3,3]^{+}  332  23  A_{4}  12  4  
D_{4}  [2,4]^{+}  224  422  D_{8}  8  6  
D_{3}  [2,3]^{+}  223  322  D_{6}=S_{3}  6  8  
D_{2}  [2,2]^{+}  222  222  D_{4}=Z_{2}^{2}  4  12  
C_{4}  [4]^{+}  44  4  Z_{4}  4  12  
C_{3}  [3]^{+}  33  3  Z_{3}=A_{3}  3  16  
C_{2}  [2]^{+}  22  2  Z_{2}  2  24  
C_{1}  [ ]^{+}  11  1  Z_{1}  1  48 
Octahedral subgroups in Coxeter notation^{[1]} 
The cube has 48 isometries (symmetry elements), forming the symmetry group O_{h}, isomorphic to S_{4} × Z_{2}. They can be categorized as follows:
An isometry of the cube can be identified in various ways:
For cubes with colors or markings (like dice have), the symmetry group is a subgroup of O_{h}.
Examples:
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:
The full symmetry of the cube, O_{h}, [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, Dih_{4}, [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 with 4fold rotational symmetry, Z_{4}, [4]^{+}.
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 systole 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 (dual Catalan 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  Selfdual  
Cube and octahedron  14  24  14  Selfdual 