List of finite spherical symmetry groups

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.

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] =

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 [5]	Orb.	Schön.	Con.	Cox.	Ord.
1	1	11	C₁	C₁	][ [ ]⁺	1
2	2	22	D₁ = C₂	D₂ = C₂	[2]⁺	2
1	22	×	C_i = S₂	CC₂	[2⁺,2⁺]	2
2 = m	1	*	C_s = C_1v = C_1h	±C₁ = CD₂	[ ]	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	Orb.	Schön.	Con.	Cox.	Ord.	Fund. domain
4	42	2×	S₄	CC₄	[2⁺,4⁺]	4
2/m	22	2*	C_2h = D_1d	±C₂ = ±D₂	[2,2⁺] [2⁺,2]	4

Intl	Geo	Orb.	Schön.	Con.	Cox.	Ord.
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
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
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
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

Dihedral symmetry

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

Intl	Geo	Orb.	Schön.	Con.	Cox.	Ord.
222	2.2	222	D₂	D₄	[2,2]⁺	4
42m	42	2*2	D_2d	DD₈	[2⁺,4]	8
mmm	22	*222	D_2h	±D₄	[2,2]	8

Intl	Geo	Orb.	Schön.	Con.	Cox.	Ord.
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
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
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

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	Orb.	Schön.	Con.	Cox.	Ord.
23	3.3	332	T	T	[3,3]⁺ = [4,3⁺]⁺	12
m3	43	3*2	T_h	±T	[4,3⁺]	24
43m	33	*332	T_d	TO	[3,3] = [1⁺,4,3]	24

Octahedral symmetry
Intl	Geo	Orb.	Schön.	Con.	Cox.	Ord.	Fund. domain
432	4.3	432	O	O	[4,3]⁺ = [[3,3]]⁺	24
m3m	43	*432	O_h	±O	[4,3] = [[3,3]]	48

Icosahedral symmetry
Intl	Geo	Orb.	Schön.	Con.	Cox.	Ord.	Fund. domain
532	5.3	532	I	I	[5,3]⁺	60
532/m	53	*532	I_h	±I	[5,3]	120

Notes

Johnson, 2015
Conway, 2008
Conway, 2003
Sands, 1993
The Crystallographic Space groups in Geometric algebra, D. Hestenes and J. Holt, Journal of Mathematical Physics. 48, 023514 (2007) (22 pages) PDF

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 0-486-67839-3.
On Quaternions and Octonions, 2003, John Horton Conway and Derek A. Smith ISBN 978-1-56881-134-5
The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
- (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 Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559–591]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
N.W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite symmetry groups, Table 11.4 Finite Groups of Isometries in 3-space

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

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[1] Johnson, 2015

[2] Conway, 2008

[3] Conway, 2003

[4] Sands, 1993

[5] The Crystallographic Space groups in Geometric algebra, D. Hestenes and J. Holt, Journal of Mathematical Physics. 48, 023514 (2007) (22 pages) PDF