# List of spherical symmetry groups

 Polyhedral group, [n,3], (*n32) Involutional symmetryCs, (*)[ ] =  Cyclic symmetryCnv, (*nn)[n] =    Dihedral symmetryDnh, (*n22)[n,2] =      Tetrahedral symmetryTd, (*332)[3,3] =      Octahedral symmetryOh, (*432)[4,3] =      Icosahedral symmetryIh, (*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, 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 (C1), reflection symmetry (Cs), 2-fold rotational symmetry (C2), and central point symmetry (Ci).

Intl Geo

Orb. Schön. Con. Cox. Ord. Fund.
domain
1 1 11 C1 C1 ][
[ ]+
1 2 2 22 D1
= C2
D2
= C2
+ 2 Intl Geo Orb. Schön. Con. Cox. Ord. Fund.
domain
1 22 × Ci
= S2
CC2 [2+,2+] 2 2
= m
1 * Cs
= C1v
= C1h
±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
2 2 22 C2
= D1
C2
= D2
+
[2,1]+
2 mm2 2 *22 C2v
= D1h
CD4
= DD4

[2,1]
4 4 42 2× S4 CC4 [2+,4+] 4 2/m 22 2* C2h
= D1d
±C2
= ±D2
[2,2+]
[2+,2]
4 Intl Geo Orb. Schön. Con. Cox. Ord. Fund.
domain
3
4
5
6
n
3
4
5
6
n
33
44
55
66
nn
C3
C4
C5
C6
Cn
C3
C4
C5
C6
Cn
+
+
+
+
[n]+
3
4
5
6
n 3m
4mm
5m
6mm
-
3
4
5
6
n
*33
*44
*55
*66
*nn
C3v
C4v
C5v
C6v
Cnv
CD6
CD8
CD10
CD12
CD2n




[n]
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
S2n
±C3
CC8
±C5
CC12
CC2n / ±Cn
[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*
C3h
C4h
C5h
C6h
Cnh
CC6
±C4
CC10
±C6
±Cn / CC2n
[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 as 2 or higher. (n may be 1 as a special case)

Intl Geo
Orb. Schön. Con. Cox. Ord. Fund.
domain
222 2.2 222 D2 D4 [2,2]+ 4 42m 42 2*2 D2d DD8 [2+,4] 8 mmm 22 *222 D2h ±D4 [2,2] 8 Intl Geo Orb. Schön. Con. Cox. Ord. Fund.
domain
32
422
52
622
3.2
4.2
5.2
6.2
n.2
223
224
225
226
22n
D3
D4
D5
D6
Dn
D6
D8
D10
D12
D2n
[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
2*3
2*4
2*5
2*6
2*n
D3d
D4d
D5d
D6d
Dnd
±D6
DD16
±D10
DD24
DD4n / ±D2n
[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
D3h
D4h
D5h
D6h
Dnh
DD12
±D8
DD20
±D12
±D2n / DD4n
[2,3]
[2,4]
[2,5]
[2,6]
[2,n]
12
16
20
24
4n ## Polyhedral symmetry

Further information: Polyhedral groups

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. Fund.
domain
23 3.3 332 T T [3,3]+
= [4,3+]+
12 m3 43 3*2 Th ±T [4,3+] 24 43m 33 *332 Td 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 Oh ±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 Ih ±I [5,3] 120 ## Notes

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

## 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, (2015) Chapter 11: Finite symmetry groups