Point group
Encyclopedia
In geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, a point group is a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

of geometric symmetries
Symmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...

(isometries
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

) that keep at least one point fixed. Point groups can exist in a Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

with any dimension, and every point group in dimension d is a subgroup of the orthogonal group
Orthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...

O(d). Point groups can be realized as sets of orthogonal matrices
Orthogonal matrix
In linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....

M that transform point x into point y:

y = M.x

where the origin is the fixed point. Point-group elements can either be rotations (determinant of M = 1) or else reflections, improper rotations, rotation-reflections, or rotoreflections (determinant of M = -1). All point groups of rotations with dimension d are subgroups of the special orthogonal group SO(d).

Discrete point groups in more than one dimension come in infinite families, but from the crystallographic restriction theorem
Crystallographic restriction theorem
The crystallographic restriction theorem in its basic form was based on the observation that the rotational symmetries of a crystal are usually limited to 2-fold, 3-fold, 4-fold, and 6-fold...

and one of Bieberbach's theorems, each number of dimensions has only a finite number of point groups that are symmetric over some lattice
Lattice (group)
In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...

or grid with that number. These are the crystallographic point group
Crystallographic point group
In crystallography, a crystallographic point group is a set of symmetry operations, like rotations or reflections, that leave a central point fixed while moving other directions and faces of the crystal to the positions of features of the same kind...

s.

## One Dimension

There are only two one-dimensional point groups, the identity group and the reflection group.
Group 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...

Coxeter diagram
Coxeter-Dynkin diagram
In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

Order Description
C1 [ ]+ 1 Identity
D1 [ ]
}||2||Reflection group
|}

## Two Dimensions

Point groups in two dimensions
Point groups in two dimensions
In geometry, a two-dimensional point group or rosette group is a group of geometric symmetries that keep at least one point fixed in a plane. Every such group is a subgroup of the orthogonal group O, including O itself...

, sometimes called rosette groups.

They come in two infinite families:
1. Cyclic groups Cn of n-fold rotation groups
2. Dihedral groups Dn of n-fold rotation and reflection groups

Applying the crystallographic restriction theorem restricts n to values 1, 2, 3, 4, and 6 for both families, yielding 10 groups.
Group Intl
Hermann-Mauguin notation
Hermann–Mauguin notation is used to represent the symmetry elements in point groups, plane groups and space groups. It is named after the German crystallographer Carl Hermann and the French mineralogist Charles-Victor Mauguin...

Orbifold
Orbifold
In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold is a generalization of a manifold...

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 Description
Cn n nn [n]+ n Cyclic: n-fold rotations. Abstract group Zn, the group of integers under addition modulo n.
Dn nm *nn [n] 2n Dihedral: cyclic with reflections. Abstract group Dihn, the dihedral group
Dihedral group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.See also: Dihedral symmetry in three...

.

The subset of pure reflectional point groups, defined by 1 or 2 mirror lines, can also be given by their Coxeter group
Coxeter group
In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...

and related polygons. These include 5 crystallographic groups.
Group Coxeter group
Coxeter group
In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...

Coxeter diagram
Coxeter-Dynkin diagram
In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

Order Related polygons
D3 A2 [3] 6 Equilateral triangle
D4 BC2 [4] 8 Square
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...

D5 H2 [5] 10 Regular pentagon
D6 G2 [6] 12 Regular hexagon
Dn I2(n) [n] 2n Regular polygon
Regular polygon
A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...

D2n I2(2n) n=[2n] 4n Regular polygon
Regular polygon
A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...

D2 A12 [2] 4 Rectangle
Rectangle
In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a non-square rectangle...

D1 A1 [ ]
}||2
|Digon
Digon
In geometry, a digon is a polygon with two sides and two vertices. It is degenerate in a Euclidean space, but may be non-degenerate in a spherical space.A digon must be regular because its two edges are the same length...

|}

## Three Dimensions

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...

, sometimes called molecular point groups, after their wide use in studying the symmetries of small molecule
Molecule
A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...

s.

They come in 7 infinite families of axial or prismatic groups, and 7 additional polyhedral or Platonic groups. In Schönflies notation,*
• Axial groups: Cn, S2n, Cnh, Cnv, Dn, Dnd, Dnh
• Polyhedral group
Polyhedral group
In geometry, the polyhedral group is any of the symmetry groups of the Platonic solids. There are three polyhedral groups:*The tetrahedral group of order 12, rotational symmetry group of the regular tetrahedron....

s: T, Td, Th, O, Oh, I, Ih

Applying the crystallographic restriction theorem to these groups yields 32 Crystallographic point group
Crystallographic point group
In crystallography, a crystallographic point group is a set of symmetry operations, like rotations or reflections, that leave a central point fixed while moving other directions and faces of the crystal to the positions of features of the same kind...

s.
Intl
Hermann-Mauguin notation
Hermann–Mauguin notation is used to represent the symmetry elements in point groups, plane groups and space groups. It is named after the German crystallographer Carl Hermann and the French mineralogist Charles-Victor Mauguin...

*
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
1 1 C1 C1 [ ]+ 1
×1 Ci = S2 CC2 [2+,2+] 2
= m 1 *1 Cs = C1v = C1h ±C1 = CD2 [ ] 2
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]+
2
3
4
5
6
n
2mm
3m
4mm
5m
6mm
nmm
nm
2
3
4
5
6
n
*22
*33
*44
*55
*66
*nn
C2v
C3v
C4v
C5v
C6v
Cnv
CD4
CD6
CD8
CD10
CD12
CD2n
[2]
[3]
[4]
[5]
[6]
[n]
4
6
8
10
12
2n
2/m
3/m
4/m
5/m
6/m
n/m
2
2
2
2
2
2
2*
3*
4*
5*
6*
n*
C2h
C3h
C4h
C5h
C6h
Cnh
±C2
CC6
±C4
CC10
±C6
±Cn / CC2n
[2,2+]
[2,3+]
[2,4+]
[2,5+]
[2,6+]
[2,n+]
4
6
8
10
12
2n

S4
S6
S8
S10
S12
S2n
CC4
±C3
CC8
±C5
CC12
CC2n / ±Cn
[2+,4+]
[2+,6+]
[2+,8+]
[2+,10+]
[2+,12+]
[2+,2n+]
4
6
8
10
12
2n
Intl
Hermann-Mauguin notation
Hermann–Mauguin notation is used to represent the symmetry elements in point groups, plane groups and space groups. It is named after the German crystallographer Carl Hermann and the French mineralogist Charles-Victor Mauguin...

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
222
32
422
52
622
n22
n2

222
223
224
225
226
22n
D2
D3
D4
D5
D6
Dn
D4
D6
D8
D10
D12
D2n
[2,2]+
[2,3]+
[2,4]+
[2,5]+
[2,6]+
[2,n]+
4
6
8
10
12
2n
mmm
m2
4/mmm
m2
6/mmm
n/mmm
m2
2 2
3 2
4 2
5 2
6 2
n 2
*222
*223
*224
*225
*226
*22n
D2h
D3h
D4h
D5h
D6h
Dnh
±D4
DD12
±D8
DD20
±D12
±D2n / DD4n
[2,2]
[2,3]
[2,4]
[2,5]
[2,6]
[2,n]
8
12
16
20
24
4n
2m
m
2m
m
2m
2m
m
4
6
8
10
12
n
2*2
2*3
2*4
2*5
2*6
2*n
D2d
D3d
D4d
D5d
D6d
Dnd
±D4
±D6
DD16
±D10
DD24
DD4n / ±D2n
[2+,4]
[2+,6]
[2+,8]
[2+,10]
[2+,12]
[2+,2n]
8
12
16
20
24
4n
23 332 T T [3,3]+ 12
m 4 3*2 Th ±T [3+,4] 24
3m 3 3 *332 Td TO [3,3] 24
432 432 O O [3,4]+ 24
mm 4 3 *432 Oh ±O [3,4] 48
532 532 I I [3,5]+ 60
m 5 3 *532 Ih ±I [3,5] 120
(*) When the Intl entries are duplicated, the first is for even n, the second for odd n.

The subset of pure reflectional point groups, defined by 1 to 3 mirror planes, can also be given by their Coxeter group
Coxeter group
In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...

and related polyhedra. The [3,3] group can be doubled, written as 3,3, mapping the first and last mirrors onto each other, doubling the symmetry to 48, and isomorphic to the [4,3] group.
Schönflies Coxeter group
Coxeter group
In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...

Coxeter diagram
Coxeter-Dynkin diagram
In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

Order Related regular and prismatic polyhedra
Td A3 [3,3] 24 Tetrahedron
Tetrahedron
In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...

Oh BC3 [4,3]
=3,3

48 Cube
Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...

, octahedron
Octahedron
In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....

Stellated octahedron
Ih H3 [5,3] 120 Icosahedron
Icosahedron
In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....

, dodecahedron
D3h A2×A1 [3,2] 12 Triangular prism
Triangular prism
In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides....

D4h BC2×A1 [4,2] 16 Square prism
D5h H2×A1 [5,2] 20 Pentagonal prism
Pentagonal prism
In geometry, the pentagonal prism is a prism with a pentagonal base. It is a type of heptahedron with 7 faces, 15 edges, and 10 vertices.- As a semiregular polyhedron :...

D6h G2×A1 [6,2] 24 Hexagonal prism
Hexagonal prism
In geometry, the hexagonal prism is a prism with hexagonal base. The shape has 8 faces, 18 edges, and 12 vertices.Since it has eight faces, it is an octahedron. However, the term octahedron is primarily used to refer to the regular octahedron, which has eight triangular faces...

Dnh I2(n)×A1 [n,2] 4n n-gonal prism
Prism (geometry)
In geometry, a prism is a polyhedron with an n-sided polygonal base, a translated copy , and n other faces joining corresponding sides of the two bases. All cross-sections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a...

D2h A13 [2,2] 8 Cuboid
Cuboid
In geometry, a cuboid is a solid figure bounded by six faces, forming a convex polyhedron. There are two competing definitions of a cuboid in mathematical literature...

C3v A2×A1 [3] 6 Hosohedron
C4v BC2×A1 [4] 8
C5v H2×A1 [5] 10
C6v G2×A1 [6] 12
Cnv I2(n)×A1 [n] 2n
C2v A12 [2] 4
Cs A1 [ ]
}||2
|}

## Four dimensions

The four-dimensional point groups, limiting to purely reflectional groups, can be listed by their Coxeter group
Coxeter group
In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...

, and like the polyhedral group
Polyhedral group
In geometry, the polyhedral group is any of the symmetry groups of the Platonic solids. There are three polyhedral groups:*The tetrahedral group of order 12, rotational symmetry group of the regular tetrahedron....

s of 3D, can be named by their related convex regular 4-polytope
Convex regular 4-polytope
In mathematics, a convex regular 4-polytope is a 4-dimensional polytope that is both regular and convex. These are the four-dimensional analogs of the Platonic solids and the regular polygons ....

s. Related pure rotational groups exist for each with half the order, defined by an even number of reflections, and can be represented by the bracket 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...

with a '+' exponent, for example [3,3,3]+ has three 3-fold gyration points and symmetry order 60. Front-back symmetric groups like [3,3,3] and [3,4,3] can be doubled, shown as double brackets in Coxeter's notation, for example 3,3,3 with its order doubled to 240.
Coxeter group
Coxeter group
In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...

/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...

Coxeter diagram
Coxeter-Dynkin diagram
In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

Order Related regular/prismatic polytopes
A4 [3,3,3] 120 5-cell
A4×2 3,3,3 240 5-cell dual compound
BC4 [4,3,3] 384 16-cell
16-cell
In four dimensional geometry, a 16-cell or hexadecachoron is a regular convex 4-polytope. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century....

/Tesseract
Tesseract
In geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8...

D4 [31,1,1] 192 Demitesseractic
F4 [3,4,3] 1152 24-cell
F4×2 3,4,3 2304 24-cell dual compound
H4 [5,3,3] 14400 120-cell/600-cell
A3×A1 [3,3,2] 48 Tetrahedral prism
BC3×A1 [4,3,2] 96 Octahedral prism
Octahedral prism
In geometry, a octahedral prism is a convex uniform polychoron . This polychoron has 10 polyhedral cells: 2 octahedra connected by 8 triangular prisms.- Related polytopes :...

H3×A1 [5,3,2] 240 Icosahedral prism
Icosahedral prism
In geometry, an icosahedral prism is a convex uniform polychoron . This polychoron has 22 polyhedral cells: 2 icosahedra connected by 20 triangular prisms. It has 70 faces: 30 squares and 40 triangles...

A2×A2 [3,2,3] 36 Duoprism
Duoprism
In geometry of 4 dimensions or higher, a duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher...

A2×BC2 [3,2,4] 48
A2×H2 [3,2,5] 60
A2×G2 [3,2,6] 72
BC2×BC2 [4,2,4] 64
BC2×H2 [4,2,5] 80
BC2×G2 [4,2,6] 96
H2×H2 [5,2,5] 100
H2×G2 [5,2,6] 120
G2×G2 [6,2,6] 144
I2(p)×I2(q) [p,2,q] 4pq
p,2,p 8p2
A2×A12 [3,2,2] 24
BC2×A12 [4,2,2] 32
H2×A12 [5,2,2] 40
G2×A12 [6,2,2] 48
I2(p)×A12 [p,2,2] 8p
A14 [2,2,2] 16 4-orthotope

## Five dimensions

The five-dimensional point groups, limiting to purely reflectional groups, can be listed by their Coxeter group
Coxeter group
In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...

. Related pure rotational groups exist for each with half the order, defined by an even number of reflections, and can be represented by the bracket 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...

with a '+' exponent, for example [3,3,3,3]+ has four 3-fold gyration points and symmetry order 360.
Coxeter group
Coxeter group
In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...

Coxeter
diagram
Coxeter-Dynkin diagram
In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

Order Related regular/prismatic polytopes
A5 [3,3,3,3] 720 5-simplex
A5×2 3,3,3,3 1440 5-simplex dual compound
BC5 [4,3,3,3] 3840 5-cube, 5-orthoplex
D5 [32,1,1] 1920 5-demicube
A4×A1 [3,3,3,2] 240 5-cell prism
BC4×A1 [4,3,3,2] 768 tesseract
Tesseract
In geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8...

prism
F4×A1 [3,4,3,2] 2304 24-cell prism
H4×A1 [5,3,3,2] 28800 600-cell or 120-cell prism
D4×A1 [31,1,1,2] 384 Demitesseract prism
A3×A2 [3,3,2,3] 144 Duoprism
Duoprism
In geometry of 4 dimensions or higher, a duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher...

A3×BC2 [3,3,2,4] 192
A3×H2 [3,3,2,5] 240
A3×G2 [3,3,2,6] 288
A3×I2(p) [3,3,2,p] 48p
BC3×A2 [4,3,2,3] 288
BC3×BC2 [4,3,2,4] 384
BC3×H2 [4,3,2,5] 480
BC3×G2 [4,3,2,6] 576
BC3×I2(p) [4,3,2,p] 96p
H3×A2 [5,3,2,3] 720
H3×BC2 [5,3,2,4] 960
H3×H2 [5,3,2,5] 1200
H3×G2 [5,3,2,6] 1440
H3×I2(p) [5,3,2,p] 240p
A3×A12 [3,3,2,2] 96
BC3×A12 [4,3,2,2] 192
H3×A12 [5,3,2,2] 480
A22×A1 [3,2,3,2] 72 duoprism prism
A2×BC2×A1 [3,2,4,2] 96
A2×H2×A1 [3,2,5,2] 120
A2×G2×A1 [3,2,6,2] 144
BC22×A1 [4,2,4,2] 128
BC2×H2×A1 [4,2,5,2] 160
BC2×G2×A1 [4,2,6,2] 192
H22×A1 [5,2,5,2] 200
H2×G2×A1 [5,2,6,2] 240
G22×A1 [6,2,6,2] 288
I2(p)×I2(q)×A1 [p,2,q,2] 8pq
A2×A13 [3,2,2,2] 48
BC2×A13 [4,2,2,2] 64
H2×A13 [5,2,2,2] 80
G2×A13 [6,2,2,2] 96
I2(p)×A13 [p,2,2,2] 16p
A15 [2,2,2,2] 32 5-orthotope

## Six dimensions

The six-dimensional point groups, limiting to purely reflectional groups, can be listed by their Coxeter group
Coxeter group
In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...

. Related pure rotational groups exist for each with half the order, defined by an even number of reflections, and can be represented by the bracket 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...

with a '+' exponent, for example [3,3,3,3,3]+ has five 3-fold gyration points and symmetry order 2520.
Coxeter group
Coxeter group
In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...

Coxeter
diagram
Coxeter-Dynkin diagram
In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

Order Related regular/prismatic polytopes
A6 [3,3,3,3,3] 5040 (7!) 6-simplex
A6×2 3,3,3,3,3 10080 (2×7!) 6-simplex dual compound
BC6 [4,3,3,3,3] 46080 (26×6!) 6-cube, 6-orthoplex
D6 [3,3,3,31,1] 23040 (25×6!) 6-demicube
E6 [3,32,2] 51840 (72×6!) 122, 221
A5×A1 [3,3,3,3,2] 1440 (2×6!) 5-simplex prism
BC5×A1 [4,3,3,3,2] 7680 (26×5!) 5-cube prism
D5×A1 [3,3,31,1,2] 3840 (25×5!) 5-demicube prism
A4×I2(p) [3,3,3,2,p] 240p Duoprism
Duoprism
In geometry of 4 dimensions or higher, a duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher...

BC4×I2(p) [4,3,3,2,p] 768p
F4×I2(p) [3,4,3,2,p] 2304p
H4×I2(p) [5,3,3,2,p] 28800p
D4×I2(p) [3,31,1,2,p] 384p
A4×A12 [3,3,3,2,2] 480
BC4×A12 [4,3,3,2,2] 1536
F4×A12 [3,4,3,2,2] 4608
H4×A12 [5,3,3,2,2] 57600
D4×A12 [3,31,1,2,2] 768
A32 [3,3,2,3,3] 576
A3×BC3 [3,3,2,4,3] 1152
A3×H3 [3,3,2,5,3] 2880
BC32 [4,3,2,4,3] 2304
BC3×H3 [4,3,2,5,3] 5760
H32 [5,3,2,5,3] 14400
A3×I2(p)×A1 [3,3,2,p,2] 96p Duoprism prism
BC3×I2(p)×A1 [4,3,2,p,2] 192p
H3×I2(p)×A1 [5,3,2,p,2] 480p
A3×A13 [3,3,2,2,2] 192
BC3×A13 [4,3,2,2,2] 384
H3×A13 [5,3,2,2,2] 960
I2(p)×I2(q)×I2(r) [p,2,q,2,r] 8pqr Triaprism
Triaprism
In geometry of 6 dimensions or higher, a triaprism is a polytope resulting from the Cartesian product of three polytopes, each of two dimensions or higher...

I2(p)×I2(q)×A12 [p,2,q,2,2] 16pq
I2(p)×A14 [p,2,2,2,2] 32p
A16 [2,2,2,2,2] 64 6-orthotope

## Seven dimensions

The seven-dimensional point groups, limiting to purely reflectional groups, can be listed by their Coxeter group
Coxeter group
In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...

. Related pure rotational groups exist for each with half the order, defined by an even number of reflections, and can be represented by the bracket 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...

with a '+' exponent, for example [3,3,3,3,3,3]+ has six 3-fold gyration points and symmetry order 20160.
Coxeter group
Coxeter group
In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...

Coxeter diagram
Coxeter-Dynkin diagram
In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

Order Related polytopes
A7 [3,3,3,3,3,3] 40320 (8!) 7-simplex
A7×2 3,3,3,3,3,3 80640 (2×8!) 7-simplex dual compound
BC7 [4,3,3,3,3,3] 645120 (27×7!) 7-cube, 7-orthoplex
D7 [3,3,3,3,31,1] 322560 (26×7!) 7-demicube
E7 [3,3,3,32,1] 2903040 (8×9!) 321, 231, 132
A6×A1 [3,3,3,3,3,2] 10080 (2×7!)
BC6×A1 [4,3,3,3,3,2] 92160 (27×6!)
D6×A1 [3,3,3,31,1,2] 46080 (26×6!)
E6×A1 [3,3,32,1,2] 103680 (144×6!)
A5×I2(p) [3,3,3,3,2,p] 1440p
BC5×I2(p) [4,3,3,3,2,p] 7680p
D5×I2(p) [3,3,31,1,2,p] 3840p
A5×A12 [3,3,3,3,2,2] 2880
BC5×A12 [4,3,3,3,2,2] 15360
D5×A12 [3,3,31,1,2,2] 7680
A4×A3 [3,3,3,2,3,3] 2880
A4×BC3 [3,3,3,2,4,3] 5760
A4×H3 [3,3,3,2,5,3] 14400
BC4×A3 [4,3,3,2,3,3] 9216
BC4×BC3 [4,3,3,2,4,3] 18432
BC4×H3 [4,3,3,2,5,3] 46080
H4×A3 [5,3,3,2,3,3] 345600
H4×BC3 [5,3,3,2,4,3] 691200
H4×H3 [5,3,3,2,5,3] 1728000
F4×A3 [3,4,3,2,3,3] 27648
F4×BC3 [3,4,3,2,4,3] 55296
F4×H3 [3,4,3,2,5,3] 138240
D4×A3 [31,1,1,2,3,3] 4608
D4×BC3 [3,31,1,2,4,3] 9216
D4×H3 [3,31,1,2,5,3] 23040
A4×I2(p)×A1 [3,3,3,2,p,2] 480p
BC4×I2(p)×A1 [4,3,3,2,p,2] 1536p
D4×I2(p)×A1 [3,31,1,2,p,2] 768p
F4×I2(p)×A1 [3,4,3,2,p,2] 4608p
H4×I2(p)×A1 [5,3,3,2,p,2] 57600p
A4×A13 [3,3,3,2,2,2] 960
BC4×A13 [4,3,3,2,2,2] 3072
F4×A13 [3,4,3,2,2,2] 9216
H4×A13 [5,3,3,2,2,2] 115200
D4×A13 [3,31,1,2,2,2] 1536
A32×A1 [3,3,2,3,3,2] 1152
A3×BC3×A1 [3,3,2,4,3,2] 2304
A3×H3×A1 [3,3,2,5,3,2] 5760
BC32×A1 [4,3,2,4,3,2] 4608
BC3×H3×A1 [4,3,2,5,3,2] 11520
H32×A1 [5,3,2,5,3,2] 28800
A3×I2(p)×I2(q) [3,3,2,p,2,q] 96pq
BC3×I2(p)×I2(q) [4,3,2,p,2,q] 192pq
H3×I2(p)×I2(q) [5,3,2,p,2,q] 480pq
A3×I2(p)×A12 [3,3,2,p,2,2] 192p
BC3×I2(p)×A12 [4,3,2,p,2,2] 384p
H3×I2(p)×A12 [5,3,2,p,2,2] 960p
A3×A14 [3,3,2,2,2,2] 384
BC3×A14 [4,3,2,2,2,2] 768
H3×A14 [5,3,2,2,2,2] 1920
I2(p)×I2(q)×I2(r)×A1 [p,2,q,2,r,2] 16pqr
I2(p)×I2(q)×A13 [p,2,q,2,2,2] 32pq
I2(p)×A15 [p,2,2,2,2,2] 64p
A17 [2,2,2,2,2,2] 128

## Eight dimensions

The eight-dimensional point groups, limiting to purely reflectional groups, can be listed by their Coxeter group
Coxeter group
In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...

. Related pure rotational groups exist for each with half the order, defined by an even number of reflections, and can be represented by the bracket 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...

with a '+' exponent, for example [3,3,3,3,3,3,3]+ has seven 3-fold gyration points and symmetry order 181440.
Coxeter group
Coxeter group
In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...

Coxeter diagram
Coxeter-Dynkin diagram
In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

Order Related polytopes
A8 [3,3,3,3,3,3,3] 362880 (9!) 8-simplex
A8×2 3,3,3,3,3,3,3 725760 (2x9!) 8-simplex dual compound
BC8 [4,3,3,3,3,3,3] 10321920 (288!) 8-cube,8-orthoplex
D8 [3,3,3,3,3,31,1] 5160960 (278!) 8-demicube
E8
E8 (mathematics)
In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8...

[3,3,3,3,32,1] 696729600 421, 241, 142
A7×A1 [3,3,3,3,3,3,2] 80640 7-simplex prism
BC7×A1 [4,3,3,3,3,3,2] 645120 7-cube prism
D7×A1 [3,3,3,3,31,1,2] 322560 7-demicube prism
E7×A1 [3,3,3,32,1,2] 5806080 321 prism, 231 prism, 142 prism
A6×I2(p) [3,3,3,3,3,2,p] 10080p duoprism
BC6×I2(p) [4,3,3,3,3,2,p] 92160p
D6×I2(p) [3,3,3,31,1,2,p] 46080p
E6×I2(p) [3,3,32,1,2,p] 103680p
A6×A12 [3,3,3,3,3,2,2] 20160
BC6×A12 [4,3,3,3,3,2,2] 184320
D6×A12 [33,1,1,2,2] 92160
E6×A12 [3,3,32,1,2,2] 207360
A5×A3 [3,3,3,3,2,3,3] 17280
BC5×A3 [4,3,3,3,2,3,3] 92160
D5×A3 [32,1,1,2,3,3] 46080
A5×BC3 [3,3,3,3,2,4,3] 34560
BC5×BC3 [4,3,3,3,2,4,3] 184320
D5×BC3 [32,1,1,2,4,3] 92160
A5×H3 [3,3,3,3,2,5,3]
BC5×H3 [4,3,3,3,2,5,3]
D5×H3 [32,1,1,2,5,3]
A5×I2(p)×A1 [3,3,3,3,2,p,2]
BC5×I2(p)×A1 [4,3,3,3,2,p,2]
D5×I2(p)×A1 [32,1,1,2,p,2]
A5×A13 [3,3,3,3,2,2,2]
BC5×A13 [4,3,3,3,2,2,2]
D5×A13 [32,1,1,2,2,2]
A4×A4 [3,3,3,2,3,3,3]
BC4×A4 [4,3,3,2,3,3,3]
D4×A4 [31,1,1,2,3,3,3]
F4×A4 [3,4,3,2,3,3,3]
H4×A4 [5,3,3,2,3,3,3]
BC4×BC4 [4,3,3,2,4,3,3]
D4×BC4 [31,1,1,2,4,3,3]
F4×BC4 [3,4,3,2,4,3,3]
H4×BC4 [5,3,3,2,4,3,3]
D4×D4 [31,1,1,2,31,1,1]
F4×D4 [3,4,3,2,31,1,1]
H4×D4 [5,3,3,2,31,1,1]
F4×F4 [3,4,3,2,3,4,3]
H4×F4 [5,3,3,2,3,4,3]
H4×H4 [5,3,3,2,5,3,3]
A4×A3×A1 [3,3,3,2,3,3,2] duoprism prisms
A4×BC3×A1 [3,3,3,2,4,3,2]
A4×H3×A1 [3,3,3,2,5,3,2]
BC4×A3×A1 [4,3,3,2,3,3,2]
BC4×BC3×A1 [4,3,3,2,4,3,2]
BC4×H3×A1 [4,3,3,2,5,3,2]
H4×A3×A1 [5,3,3,2,3,3,2]
H4×BC3×A1 [5,3,3,2,4,3,2]
H4×H3×A1 [5,3,3,2,5,3,2]
F4×A3×A1 [3,4,3,2,3,3,2]
F4×BC3×A1 [3,4,3,2,4,3,2]
F4×H3×A1 [3,4,2,3,5,3,2]
D4×A3×A1 [31,1,1,2,3,3,2]
D4×BC3×A1 [31,1,1,2,4,3,2]
D4×H3×A1 [31,1,1,2,5,3,2]
A4×I2(p)×I2(q) [3,3,3,2,p,2,q] triaprism
BC4×I2(p)×I2(q) [4,3,3,2,p,2,q]
F4×I2(p)×I2(q) [3,4,3,2,p,2,q]
H4×I2(p)×I2(q) [5,3,3,2,p,2,q]
D4×I2(p)×I2(q) [31,1,1,2,p,2,q]
A4×I2(p)×A12 [3,3,3,2,p,2,2]
BC4×I2(p)×A12 [4,3,3,2,p,2,2]
F4×I2(p)×A12 [3,4,3,2,p,2,2]
H4×I2(p)×A12 [5,3,3,2,p,2,2]
D4×I2(p)×A12 [31,1,1,2,p,2,2]
A4×A14 [3,3,3,2,2,2,2]
BC4×A14 [4,3,3,2,2,2,2]
F4×A14 [3,4,3,2,2,2,2]
H4×A14 [5,3,3,2,2,2,2]
D4×A14 [31,1,1,2,2,2,2]
A3×A3×I2(p) [3,3,2,3,3,2,p]
BC3×A3×I2(p) [4,3,2,3,3,2,p]
H3×A3×I2(p) [5,3,2,3,3,2,p]
BC3×BC3×I2(p) [4,3,2,4,3,2,p]
H3×BC3×I2(p) [5,3,2,4,3,2,p]
H3×H3×I2(p) [5,3,2,5,3,2,p]
A3×A3×A12 [3,3,2,3,3,2,2]
BC3×A3×A12 [4,3,2,3,3,2,2]
H3×A3×A12 [5,3,2,3,3,2,2]
BC3×BC3×A12 [4,3,2,4,3,2,2]
H3×BC3×A12 [5,3,2,4,3,2,2]
H3×H3×A12 [5,3,2,5,3,2,2]
A3×I2(p)×I2(q)×A1 [3,3,2,p,2,q,2]
BC3×I2(p)×I2(q)×A1 [4,3,2,p,2,q,2]
H3×I2(p)×I2(q)×A1 [5,3,2,p,2,q,2]
A3×I2(p)×A13 [3,3,2,p,2,2,2]
BC3×I2(p)×A13 [4,3,2,p,2,2,2]
H3×I2(p)×A13 [5,3,2,p,2,2,2]
A3×A15 [3,3,2,2,2,2,2]
BC3×A15 [4,3,2,2,2,2,2]
H3×A15 [5,3,2,2,2,2,2]
I2(p)×I2(q)×I2(r)×I2(s) [p,2,q,2,r,2,s] 16pqrs
I2(p)×I2(q)×I2(r)×A12 [p,2,q,2,r,2,2] 32pqr
I2(p)×I2(q)×A14 [p,2,q,2,2,2,2] 64pq
I2(p)×A16 [p,2,2,2,2,2,2] 128p
A18 [2,2,2,2,2,2,2] 256

• Point groups in two dimensions
Point groups in two dimensions
In geometry, a two-dimensional point group or rosette group is a group of geometric symmetries that keep at least one point fixed in a plane. Every such group is a subgroup of the orthogonal group O, including O itself...

• 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...

• 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...

• Crystallographic point group
Crystallographic point group
In crystallography, a crystallographic point group is a set of symmetry operations, like rotations or reflections, that leave a central point fixed while moving other directions and faces of the crystal to the positions of features of the same kind...

• Molecular symmetry
Molecular symmetry
Molecular symmetry in chemistry describes the symmetry present in molecules and the classification of molecules according to their symmetry. Molecular symmetry is a fundamental concept in chemistry, as it can predict or explain many of a molecule's chemical properties, such as its dipole moment...

• Space group
Space group
In mathematics and geometry, a space group is a symmetry group, usually for three dimensions, that divides space into discrete repeatable domains.In three dimensions, there are 219 unique types, or counted as 230 if chiral copies are considered distinct...

• X-ray diffraction
• Bravais lattice