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,^{[1]} orbifold notation,^{[2]} 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.^{[3]}
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.^{[4]}
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Transcription
Involutional symmetry
There are four involutional groups: no symmetry (C_{1}), reflection symmetry (C_{s}), 2fold rotational symmetry (C_{2}), and central point symmetry (C_{i}).
Intl  Geo 
Orbifold  Schönflies  Conway  Coxeter  Order  Abstract  Fund. domain  

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  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  Fund. domain  

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  Fund. domain 

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  Fund. domain  

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 trianglefaced regular polyhedra with these symmetries.
Intl  Geo 
Orbifold  Schönflies  Conway  Coxeter  Order  Abstract  Fund. domain 

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} 
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} 
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]^{+} = 
See also
 Crystallographic point group
 Triangle group
 List of planar symmetry groups
 Point groups in two dimensions
References
 ^ Johnson, 2015
 ^ Conway, John H. (2008). The symmetries of things. Wellesley, Mass: A.K. Peters. ISBN 9781568812205. OCLC 181862605.
 ^ Conway, John; Smith, Derek A. (2003). On quaternions and octonions: their geometry, arithmetic, and symmetry. Natick, Mass: A.K. Peters. ISBN 9781568811345. OCLC 560284450.
 ^ Sands, "Introduction to Crystallography", 1993
Further reading
 Peter R. Cromwell, Polyhedra (1997), Appendix I
 Sands, Donald E. (1993). "Crystal Systems and Geometry". Introduction to Crystallography. Mineola, New York: Dover Publications, Inc. p. 165. ISBN 0486678393.
 On Quaternions and Octonions, 2003, John Horton Conway and Derek A. Smith ISBN 9781568811345
 The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim GoodmanStrauss, ISBN 9781568812205
 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, WileyInterscience Publication, 1995, ISBN 9780471010036 [1]
 (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10]
 (Paper 23) H.S.M. Coxeter, Regular and SemiRegular Polytopes II, [Math. Zeit. 188 (1985) 559–591]
 (Paper 24) H.S.M. Coxeter, Regular and SemiRegular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
 N.W. Johnson: Geometries and Transformations, (2018) ISBN 9781107103405 Chapter 11: Finite symmetry groups, Table 11.4 Finite Groups of Isometries in 3space
External links
 Finite spherical symmetry groups
 Weisstein, Eric W. "Schoenflies symbol". MathWorld.
 Weisstein, Eric W. "Crystallographic point groups". MathWorld.
 Simplest Canonical Polyhedra of Each Symmetry Type, by David I. McCooey