List of spherical symmetry groups
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] = 
Spherical symmetry groups are also called point groups in three dimensions; however, this article is limited to the finite symmetries. 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]}
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}).


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)


Dihedral symmetry
There are three infinite dihedral symmetry families, with n as 2 or higher. (n may be 1 as a special case)


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.


See also
 Crystallographic point group
 Triangle group
 List of planar symmetry groups
 Point groups in two dimensions
Notes
References
 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 GoodmanStrass, 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
 (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, (2015) Chapter 11: Finite symmetry groups
External links
 Simplest Canonical Polyhedra of Each Symmetry Type, by David I. McCooey