List of spherical symmetry groups

# List of spherical symmetry groups

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Encyclopedia
Spherical symmetry groups are also called point groups in three dimensions
Point groups in three dimensions
In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O, the group of all isometries that leave the origin fixed, or correspondingly, the group...

There are five fundamental symmetry classes which have triangular fundamental domains: dihedral, cyclic, tetrahedral
Tetrahedral symmetry
150px|right|thumb|A regular [[tetrahedron]], an example of a solid with full tetrahedral symmetryA regular tetrahedron has 12 rotational symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.The group of all symmetries is isomorphic to the group...

, octahedral
Octahedral symmetry
150px|thumb|right|The [[cube]] is the most common shape with octahedral symmetryA regular octahedron has 24 rotational symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation...

, and icosahedral
Icosahedral symmetry
A regular icosahedron has 60 rotational symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation...

symmetry.

This article lists the groups by Schoenflies notation
Schoenflies notation
The Schoenflies notation or Schönflies notation, named after the German mathematician Arthur Moritz Schoenflies, is one of two conventions commonly used to describe Point groups. This notation is used in spectroscopy. The other convention is the Hermann–Mauguin notation, also known as the...

, Coxeter notation
Coxeter notation
In geometry, Coxeter notation is a system of classifying symmetry groups, describing the angles between with fundamental reflections of a Coxeter group. It uses a bracketed notation, with modifiers to indicate certain subgroups. The notation is named after H. S. M...

, orbifold notation, and order. John Conway
John Horton Conway
John Horton Conway is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory...

uses a variation of the Schoenflies notation, named 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.

Hermann–Mauguin notation (International notation) is also given. The crystallography
Crystallography
Crystallography is the experimental science of the arrangement of atoms in solids. The word "crystallography" derives from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and grapho = write.Before the development of...

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
Reflection symmetry
Reflection symmetry, reflectional symmetry, line symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is symmetry with respect to reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.In 2D there is a line of symmetry, in 3D a...

, 2-fold rotational symmetry, and central point symmetry.
Intl Geo
Orbifold Schönflies Conway
John Horton Conway
John Horton Conway is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory...

Coxeter
Coxeter notation
In geometry, Coxeter notation is a system of classifying symmetry groups, describing the angles between with fundamental reflections of a Coxeter group. It uses a bracketed notation, with modifiers to indicate certain subgroups. The notation is named after H. S. M...

Order Fundamental
domain
Fundamental domain
In geometry, the fundamental domain of a symmetry group of an object is a part or pattern, as small or irredundant as possible, which determines the whole object based on the symmetry. More rigorously, given a topological space and a group acting on it, the images of a single point under the group...

1 1 C1 C1 [ ]+ 1
2 22 D1
= C2
D2
= C2
[2]+ 2
Intl Geo Orbifold Schönflies Conway Coxeter Order Fundamental
domain
× Ci
= S2
CC2 [2+,2+] 2

= m
1 * Cs
= C1v
= C1h
±C1
= CD2
[ ] 2

## Cyclic symmetry

There are four infinite cyclic symmetry
Cyclic symmetries
This article deals with the four infinite series of point groups in three dimensions with n-fold rotational symmetry about one axis , and no other rotational symmetry :Chiral:*Cn of order n - n-fold rotational symmetry...

families, with n=2 or higher. (n may be 1 as a special case)
Intl Geo Orbifold Schönflies Conway Coxeter Order Fundamental
domain
2 22 C2
= D1
C2
= D2
[2]+ 2
mm2 2 *22 C2v
= D1h
CD4
= DD4
[2] 4
S4 CC4 [2+,4+] 4
2/m 2 2* C2h
= D1d
±C2
= ±D2
[2,2+] 4
Intl Geo Orbifold Schönflies Conway Coxeter Order Fundamental
domain
3
4
5
6
n

33
44
55
66
nn
C3
C4
C5
C6
Cn
C3
C4
C5
C6
Cn
[3]+
[4]+
[5]+
[6]+
[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
[3]
[4]
[5]
[6]
[n]
6
8
10
12
2n

-

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
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
[2,3+]
[2,4+]
[2,5+]
[2,6+]
[2,n+]
6
8
10
12
2n

## Dihedral symmetry

There are three infinite dihedral symmetry
Dihedral symmetry in three dimensions
This article deals with three infinite sequences of point groups in three dimensions which have a symmetry group that as abstract group is a dihedral group Dihn .See also point groups in two dimensions.Chiral:...

families, with n as 2 or higher. (n may be 1 as a special case)
Intl Geo Orbifold Schönflies Conway Coxeter Order Fundamental
domain
222 . 222 D2 D4 [2,2]+ 4
2m 4 2*2 D2d DD8 [2+,4] 8
mmm 22 *222 D2h ±D4 [2,2] 8
Intl Geo Orbifold Schönflies Conway Coxeter Order Fundamental
domain
32
422
52
622
.
.
.
.
.
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
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
[2+,6]
[2+,8]
[2+,10]
[2+,12]
[2+,2n]
12
16
20
24
4n
m2
4/mmm
m2
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

There are three types of polyhedral symmetry: tetrahedral symmetry
Tetrahedral symmetry
150px|right|thumb|A regular [[tetrahedron]], an example of a solid with full tetrahedral symmetryA regular tetrahedron has 12 rotational symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.The group of all symmetries is isomorphic to the group...

, octahedral symmetry
Octahedral symmetry
150px|thumb|right|The [[cube]] is the most common shape with octahedral symmetryA regular octahedron has 24 rotational symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation...

, and icosahedral symmetry
Icosahedral symmetry
A regular icosahedron has 60 rotational symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation...

, named after the triangle-faced regular polyhedra with these symmetries.
[3,3]
Intl Geo Orbifold Schönflies Conway Coxeter Order Fundamental
domain
23 . 332 T T [3,3]+
= [3+,4,1+]
12
m 4 3*2 Th ±T [3+,4]
= /nowiki>3,3]+]
24
3m 33 *332 Td TO [3,3]
= [3,4,1+]
24
[3,4]
Intl Geo Orbifold Schönflies Conway Coxeter Order Fundamental
domain
432 . 432 O O [3,4]+
= 3,3+
24
mm 43 *432 Oh ±O [3,4]
= 3,3
48
[3,5]
Intl Geo Orbifold Schönflies Conway Coxeter Order Fundamental
domain
532 . 532 I I [3,5]+ 60
2/m 53 *532 Ih ±I [3,5] 120