List of spherical symmetry groups

Selected point groups in three dimensions
Polyhedral group, [n,3], (*n32)
Involutional symmetry C_s, (*) [ ] =	Cyclic symmetry C_nv, (*nn) [n] =	Dihedral symmetry D_nh, (*n22) [n,2] =
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]

Involutional symmetry

There are four involutional groups: no symmetry (C₁), reflection symmetry (C_s), 2-fold rotational symmetry (C₂), and central point symmetry (C_i).

Intl	Geo	Orbifold	Schönflies	Conway	Coxeter	Order	Abstract
1	1	11	C₁	C₁	][ [ ]⁺	1	Z₁
2	2	22	D₁ = C₂	D₂ = C₂	[2]⁺	2	Z₂
1	22	×	C_i = S₂	CC₂	[2⁺,2⁺]	2	Z₂
2 = m	1	*	C_s = C_1v = C_1h	±C₁ = CD₂	[ ]	2	Z₂

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×	S₄	CC₄	[2⁺,4⁺]		4	Z₄
2/m	22	2*	C_2h = D_1d	±C₂ = ±D₂	[2,2⁺] [2⁺,2]		4	Z₄

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	C₂ C₃ C₄ C₅ C₆ C_n	C₂ C₃ C₄ C₅ C₆ C_n	[2]⁺ [3]⁺ [4]⁺ [5]⁺ [6]⁺ [n]⁺	2 3 4 5 6 n	Z₂ Z₃ Z₄ Z₅ Z₆ 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	CD₄ CD₆ CD₈ CD₁₀ CD₁₂ CD_2n	[2] [3] [4] [5] [6] [n]	4 6 8 10 12 2n	D₄ D₆ D₈ D₁₀ D₁₂ D_2n
3 8 5 12 -	62 82 10.2 12.2 2n.2	3× 4× 5× 6× n×	S₆ S₈ S₁₀ S₁₂ S_2n	±C₃ CC₈ ±C₅ CC₁₂ CC_2n / ±C_n	[2⁺,6⁺] [2⁺,8⁺] [2⁺,10⁺] [2⁺,12⁺] [2⁺,2n⁺]	6 8 10 12 2n	Z₆ Z₈ Z₁₀ Z₁₂ 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	CC₆ ±C₄ CC₁₀ ±C₆ ±C_n / CC_2n	[2,3⁺] [2,4⁺] [2,5⁺] [2,6⁺] [2,n⁺]	6 8 10 12 2n	Z₆ Z₂×Z₄ Z₁₀ Z₂×Z₆ Z₂×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	D₂	D₄	[2,2]⁺	4	D₄
42m	42	2*2	D_2d	DD₈	[2⁺,4]	8	D₄
mmm	22	*222	D_2h	±D₄	[2,2]	8	Z₂×D₄

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	D₃ D₄ D₅ D₆ D_n	D₆ D₈ D₁₀ D₁₂ D_2n	[2,3]⁺ [2,4]⁺ [2,5]⁺ [2,6]⁺ [2,n]⁺	6 8 10 12 2n	D₆ D₈ D₁₀ D₁₂ D_2n
3m 82m 5m 12.2m	62 82 10.2 12.2 n2	23 24 25 26 2*n	D_3d D_4d D_5d D_6d D_nd	±D₆ DD₁₆ ±D₁₀ DD₂₄ DD_4n / ±D_2n	[2⁺,6] [2⁺,8] [2⁺,10] [2⁺,12] [2⁺,2n]	12 16 20 24 4n	D₁₂ D₁₆ D₂₀ D₂₄ 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	DD₁₂ ±D₈ DD₂₀ ±D₁₂ ±D_2n / DD_4n	[2,3] [2,4] [2,5] [2,6] [2,n]	12 16 20 24 4n	D₁₂ Z₂×D₈ D₂₀ Z₂×D₁₂ Z₂×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	A₄
m3	43	3*2	T_h	±T	[4,3⁺]	24	2×A₄
43m	33	*332	T_d	TO	[3,3]	24	S₄

Octahedral symmetry
Intl	Geo	Orbifold	Schönflies	Conway	Coxeter	Order	Abstract	Fund. domain
432	4.3	432	O	O	[4,3]⁺	24	S₄
m3m	43	*432	O_h	±O	[4,3]	48	2×S₄

Icosahedral symmetry
Intl	Geo	Orbifold	Schönflies	Conway	Coxeter	Order	Abstract	Fund. domain
532	5.3	532	I	I	[5,3]⁺	60	A₅
532/m	53	*532	I_h	±I	[5,3]	120	2×A₅

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, Dih₁. SO(1) is just the identity. Half turns, C₂, 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)⋊C₂	Dih_∞×Dih₁ Dih_∞⋊C₂	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_∞×Dih₁	Rotational symmetry with reflection	[p⁺,2] = [p]⁺×[ ] =
SO(2)⋊C₂	C_∞⋊C₂	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]⁺ =

References

^ Johnson, 2015
^ Conway, John H. (2008). The symmetries of things. Wellesley, Mass: A.K. Peters. ISBN 978-1-56881-220-5. OCLC 181862605.
^ Conway, John; Smith, Derek A. (2003). On quaternions and octonions: their geometry, arithmetic, and symmetry. Natick, Mass: A.K. Peters. ISBN 978-1-56881-134-5. OCLC 560284450.
^ Sands, "Introduction to Crystallography", 1993

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

[1] Johnson, 2015

[Conway_2008-2] Conway, John H. (2008). The symmetries of things. Wellesley, Mass: A.K. Peters. ISBN 978-1-56881-220-5. OCLC 181862605.

[Conway_2003-3] Conway, John; Smith, Derek A. (2003). On quaternions and octonions: their geometry, arithmetic, and symmetry. Natick, Mass: A.K. Peters. ISBN 978-1-56881-134-5. OCLC 560284450.

[4] Sands, "Introduction to Crystallography", 1993

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