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.
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 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, reflection symmetry, 2-fold rotational symmetry, and central point symmetry.
IntlGeo
Orb.Schön.Con.Cox.Ord.Fund.
domain
111C1C1]+1
222D1
= C2		D2
= C2		+2
×Ci
= S2		CC22

= m		1*Cs
= C1v
= C1h		±C1
= CD2		2

Cyclic symmetry

There are four infinite cyclic symmetry families, with n = 2 or higher.
IntlGeo
Orb.Schön.Con.Cox.Ord.Fund.
domain
2×S4CC44
2/m22*C2h
= D1d		±C2
= ±D2
4

IntlGeo
Orb.Schön.Con.Cox.Ord.Fund.
domain
2
3
4
5
6
n
22
33
44
55
66
nn		C2
C3
C4
C5
C6
Cn		C2
C3
C4
C5
C6
Cn		+
+
+
+
+
+
2
3
4
5
6
n
2mm
3m
4mm
5m
6mm
nm
nmm 		2
3
4
5
6
n		*22
  • 33
  • 44
  • 55
  • 66
  • nn
C2vC3v
C4v
C5v
C6v
Cnv		CD4
CD6
CD8
CD10
CD12
CD2n
4
6
8
10
12
2n

-
3×
4×
5×
6×
n×		S6
S8
S10
S12
S2n		±C3
CC8
±C5
CC12
CC2n / ±Cn
6
8
10
12
2n
3/m=
4/m
5/m=
6/m
n/m		2
2
2
2
2		3*
4*
5*
6*
n*		C3h
C4h
C5h
C6h
Cnh		CC6
±C4
CC10
±C6
±Cn / CC2n
6
8
10
12
2n

Dihedral symmetry

There are three infinite dihedral symmetry families, with n = 2 or higher.
IntlGeo
Orb.Schön.Con.Cox.Ord.Fund.
domain
222.222D2D4+4
2m42*2D2dDD88
mmm22*222D2h±D48

IntlGeo
Orb.Schön.Con.Cox.Ord.Fund.
domain
32
422
52
622		.
.
.
.
.		223
224
225
226
22n		D3
D4
D5
D6
Dn		D6
D8
D10
D12
D2n		+
+
+
+
+		6
8
10
12
2n
m
2m
m
.2m
6
8
10.
12.
n
2*3
2*4
2*5
2*6
2*n		D3d
D4d
D5d
D6d
Dnd		±D6
DD16
±D10
DD24
DD4n / ±D2n
12
16
20
24
4n
m2
4/mmm
m2
6/mmm		32
42
52
62
n2		*223
  • 224
  • 225
  • 226
  • 22n
D3hD4h
D5h
D6h
Dnh		DD12
±D8
DD20
±D12
±D2n / DD4n
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.
IntlGeo
Orb.Schön.Con.Cox.Ord.Fund.
domain
23.332TT+
= +		12
m43*2Th±T24
3m33*332TdTO
= 		24

IntlGeoOrb.Schön.Con.Cox.Ord.Fund.
domain
432.432OO+
= 3,3+		24
mm43*432Oh±O
= 3,3		48

IntlGeoOrb.Schön.Con.Cox.Ord.Fund.
domain
532.532II+60
2/m53*532Ih±I120