# List of spherical symmetry groups

Selected point groups in three dimensions :
| 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 symmetry O$_h$, (*432) [4,3] = | Icosahedral symmetry I$_h$, (*532) [5,3] = |
Finite spherical symmetry groups are also called point groups in three dimensions. There are five fundamental symmetry classes which have triangular fundamental domains: dihedral, cyclic, tetrahedral, octahedral, and icosahedral symmetry.
This article lists the groups by Schoenflies notation, Coxeter notation, orbifold notation, and order. John Conway uses a variation of the Schoenflies notation, based on the groups' quaternion algebraic structure, labeled by one or two upper case letters, and whole number subscripts. The group order is defined as the subscript, unless the order is doubled for symbols with a plus or minus, "±", prefix, which implies a central inversion.
Hermann–Mauguin notation (International notation) is also given. The crystallography groups, 32 in total, are a subset with element orders 2, 3, 4 and 6.

## Involutional symmetry

There are four involutional groups: no symmetry (C$_1$), reflection symmetry (C$_s$), 2-fold rotational symmetry (C$_2$), and central point symmetry (C$_i$).
| Intl | Geo | Orbifold | Schönflies | Conway | Coxeter | Order | Abstract |
| 1 | 1 | 11 | C$_1$ | C$_1$ | ][ [ ]$^+$ | 1 | Z$_1$ |
| 2 | 2 | 22 | D$_1$ = C$_2$ | D$_2$ = C$_2$ | [2]$^+$ | 2 | Z$_2$ |
| 1 | 22 | × | C$_i$ = S$_2$ | CC$_2$ | [2$^+$,2$^+$] | 2 | Z$_2$ |
| 2 = m | 1 | * | C$_s$ = C$_{1v}$ = C$_{1h}$ | ±C$_1$ = CD$_2$ | [ ] | 2 | Z$_2$ |

## Cyclic symmetry

There are four infinite cyclic symmetry families, with n = 2 or higher. (n may be 1 as a special case as no symmetry)
| Intl | Geo | Orbifold | Schönflies | Conway | Coxeter | Coxeter | Order | Abstract | Fund. domain |
| 4 | 42 | 2× | S$_4$ | CC$_4$ | [2$^+$,4$^+$] | | 4 | Z$_4$ | |
| 2/m | 22 | 2* | C$_{2h}$ = D$_{1d}$ | ±C$_2$ = ±D$_2$ | [2,2$^+$] [2$^+$,2] | | 4 | Z$_4$ | |
| Intl | Geo | Orbifold | Schönflies | Conway | Coxeter | Order | Abstract |
| 2 3 4 5 6 n | 2 3 4 5 6 n | 22 33 44 55 66 nn | C$_2$ C$_3$ C$_4$ C$_5$ C$_6$ C$_n$ | C$_2$ C$_3$ C$_4$ C$_5$ C$_6$ C$_n$ | [2]$^+$ [3]$^+$ [4]$^+$ [5]$^+$ [6]$^+$ [n]$^+$ | 2 3 4 5 6 n | Z$_2$ Z$_3$ Z$_4$ Z$_5$ Z$_6$ Z$_n$ |
| 2mm 3m 4mm 5m 6mm nm (n is odd) nmm (n is even) | 2 3 4 5 6 n | *22 *33 *44 *55 *66 *nn | C$_{2v}$ C$_{3v}$ C$_{4v}$ C$_{5v}$ C$_{6v}$ C$_{nv}$ | CD$_4$ CD$_6$ CD$_8$ CD$_{10}$ CD$_{12}$ CD$_{2n}$ | [2] [3] [4] [5] [6] [n] | 4 6 8 10 12 2n | D$_4$ D$_6$ D$_8$ D$_{10}$ D$_{12}$ D$_{2n}$ |
| 3 8 5 12 - | 62 82 10.2 12.2 2n.2 | 3× 4× 5× 6× n× | S$_6$ S$_8$ S$_{10}$ S$_{12}$ S$_{2n}$ | ±C$_3$ CC$_8$ ±C$_5$ CC$_{12}$ CC$_{2n}$ / ±C$_n$ | [2$^+$,6$^+$] [2$^+$,8$^+$] [2$^+$,10$^+$] [2$^+$,12$^+$] [2$^+$,2n$^+$] | 6 8 10 12 2n | Z$_6$ Z$_8$ Z$_{10}$ Z$_{12}$ Z$_{2n}$ |
| 3/m=6 4/m 5/m=10 6/m n/m | 32 42 52 62 n2 | 3* 4* 5* 6* n* | C$_{3h}$ C$_{4h}$ C$_{5h}$ C$_{6h}$ C$_{nh}$ | CC$_6$ ±C$_4$ CC$_{10}$ ±C$_6$ ±C$_n$ / CC$_{2n}$ | [2,3$^+$] [2,4$^+$] [2,5$^+$] [2,6$^+$] [2,n$^+$] | 6 8 10 12 2n | Z$_6$ Z$_2$×Z$_4$ Z$_{10}$ Z$_2$×Z$_6$ Z$_2$×Z$_n$ ≅Z$_{2n}$ (odd n) |

## Dihedral symmetry

There are three infinite dihedral symmetry families, with n = 2 or higher (n may be 1 as a special case).
| Intl | Geo | Orbifold | Schönflies | Conway | Coxeter | Order | Abstract |
| 222 | 2.2 | 222 | D$_2$ | D$_4$ | [2,2]$^+$ | 4 | D$_4$ |
| 42m | 42 | 2*2 | D$_{2d}$ | DD$_8$ | [2$^+$,4] | 8 | D$_4$ |
| mmm | 22 | *222 | D$_{2h}$ | ±D$_4$ | [2,2] | 8 | Z$_2$×D$_4$ |
| Intl | Geo | Orbifold | Schönflies | Conway | Coxeter | Order | Abstract |
| 32 422 52 622 | 3.2 4.2 5.2 6.2 n.2 | 223 224 225 226 22n | D$_3$ D$_4$ D$_5$ D$_6$ D$_n$ | D$_6$ D$_8$ D$_{10}$ D$_{12}$ D$_{2n}$ | [2,3]$^+$ [2,4]$^+$ [2,5]$^+$ [2,6]$^+$ [2,n]$^+$ | 6 8 10 12 2n | D$_6$ D$_8$ D$_{10}$ D$_{12}$ D$_{2n}$ |
| 3m 82m 5m 12.2m | 62 82 10.2 12.2 n2 | 2*3 2*4 2*5 2*6 2*n | D$_{3d}$ D$_{4d}$ D$_{5d}$ D$_{6d}$ D$_{nd}$ | ±D$_6$ DD$_{16}$ ±D$_{10}$ DD$_{24}$ DD$_{4n}$ / ±D$_{2n}$ | [2$^+$,6] [2$^+$,8] [2$^+$,10] [2$^+$,12] [2$^+$,2n] | 12 16 20 24 4n | D$_{12}$ D$_{16}$ D$_{20}$ D$_{24}$ D$_{4n}$ |
| 6m2 4/mmm 10m2 6/mmm | 32 42 52 62 n2 | *223 *224 *225 *226 *22n | D$_{3h}$ D$_{4h}$ D$_{5h}$ D$_{6h}$ D$_{nh}$ | DD$_{12}$ ±D$_8$ DD$_{20}$ ±D$_{12}$ ±D$_{2n}$ / DD$_{4n}$ | [2,3] [2,4] [2,5] [2,6] [2,n] | 12 16 20 24 4n | D$_{12}$ Z$_2$×D$_8$ D$_{20}$ Z$_2$×D$_{12}$ Z$_2$×D$_{2n}$ ≅D$_{4n}$ (odd n) |

## Polyhedral symmetry

There are three types of polyhedral symmetry: tetrahedral symmetry, octahedral symmetry, and icosahedral symmetry, named after the triangle-faced regular polyhedra with these symmetries.
Tetrahedral symmetry :
| Intl | Geo | Orbifold | Schönflies | Conway | Coxeter | Order | Abstract |
| 23 | 3.3 | 332 | T | T | [3,3]$^+$ | 12 | A$_4$ |
| m3 | 43 | 3*2 | T$_h$ | ±T | [4,3$^+$] | 24 | 2×A$_4$ |
| 43m | 33 | *332 | T$_d$ | TO | [3,3] | 24 | S$_4$ |
Octahedral symmetry :
| Intl | Geo | Orbifold | Schönflies | Conway | Coxeter | Order | Abstract | Fund. domain |
| 432 | 4.3 | 432 | O | O | [4,3]$^+$ | 24 | S$_4$ | |
| m3m | 43 | *432 | O$_h$ | ±O | [4,3] | 48 | 2×S$_4$ | |
Icosahedral symmetry :
| Intl | Geo | Orbifold | Schönflies | Conway | Coxeter | Order | Abstract | Fund. domain |
| 532 | 5.3 | 532 | I | I | [5,3]$^+$ | 60 | A$_5$ | |
| 532/m | 53 | *532 | I$_h$ | ±I | [5,3] | 120 | 2×A$_5$ | |

## Continuous symmetries

All of the discrete point symmetries are subgroups of certain continuous symmetries. They can be classified as products of orthogonal groups O(n) or special orthogonal groups SO(n). O(1) is a single orthogonal reflection, dihedral symmetry order 2, Dih$_1$. SO(1) is just the identity. Half turns, C$_2$, are needed to complete.
| Rank 3 groups | Other names | Example geometry | Example finite subgroups |
| O(3) | | Full symmetry of the sphere | [3,3] = , [4,3] = , [5,3] = [4,3$^+$] = |
| SO(3) | Sphere group | Rotational symmetry | [3,3]$^+$ = , [4,3]$^+$ = , [5,3]$^+$ = |
| O(2)×O(1) O(2)⋊C$_2$ | Dih$_∞$×Dih$_1$ Dih$_∞$⋊C$_2$ | Full symmetry of a spheroid, torus, cylinder, bicone or hyperboloid Full circular symmetry with half turn | [p,2] = [p]×[ ] = [2p,2$^+$] = , [2p$^+$,2$^+$] = |
| SO(2)×O(1) | C$_∞$×Dih$_1$ | Rotational symmetry with reflection | [p$^+$,2] = [p]$^+$×[ ] = |
| SO(2)⋊C$_2$ | C$_∞$⋊C$_2$ | Rotational symmetry with half turn | [p,2]$^+$ = |
| O(2)×SO(1) | Dih$_∞$ Circular symmetry | Full symmetry of a hemisphere, cone, paraboloid or any surface of revolution | [p,1] = [p] = |
| SO(2)×SO(1) | C$_∞$ Circle group | Rotational symmetry | [p,1]$^+$ = [p]$^+$ = |