Point group
Encyclopedia
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 | [ ] |
|}
Two Dimensions
Point groups in two dimensionsPoint 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:
- Cyclic groups Cn of n-fold rotation groups
- 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) | 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 | [ ] |
|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 dimensionsPoint 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 groupPolyhedral groupIn 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.
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| (*) 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
| 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] = |
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 | [ ] |
|}
Four dimensions
The four-dimensional point groups, limiting to purely reflectional groups, can be listed by their Coxeter groupCoxeter 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
| 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 | 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 | 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 | ||
| 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 groupCoxeter 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 | |
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 groupCoxeter 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 | |
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 groupCoxeter 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 | |
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 groupCoxeter 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 | |
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 | ||
See also
- Point groups in two dimensionsPoint groups in two dimensionsIn 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 dimensionsPoint groups in three dimensionsIn 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...
- CrystallographyCrystallographyCrystallography 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 groupCrystallographic point groupIn 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 symmetryMolecular symmetryMolecular 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 groupSpace groupIn 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