Point group
Encyclopedia
In geometry
, a point group is a group
of geometric symmetries
(isometries
) that keep at least one point fixed. Point groups can exist in a Euclidean space
with any dimension, and every point group in dimension d is a subgroup of the orthogonal group
O(d). Point groups can be realized as sets of orthogonal matrices
M that transform point x into point y:
y = M.x
where the origin is the fixed point. Pointgroup elements can either be rotations (determinant of M = 1) or else reflections, improper rotations, rotationreflections, 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
and one of Bieberbach's theorems, each number of dimensions has only a finite number of point groups that are symmetric over some lattice
or grid with that number. These are the crystallographic point group
s.
}2Reflection group
}
, sometimes called rosette groups.
They come in two infinite families:
Applying the crystallographic restriction theorem restricts n to values 1, 2, 3, 4, and 6 for both families, yielding 10 groups.
The subset of pure reflectional point groups, defined by 1 or 2 mirror lines, can also be given by their Coxeter group
and related polygons. These include 5 crystallographic groups.
}2
Digon
}
, sometimes called molecular point groups, after their wide use in studying the symmetries of small molecule
s.
They come in 7 infinite families of axial or prismatic groups, and 7 additional polyhedral or Platonic groups. In Schönflies notation,*
Applying the crystallographic restriction theorem to these groups yields 32 Crystallographic point group
s.
The subset of pure reflectional point groups, defined by 1 to 3 mirror planes, can also be given by their Coxeter group
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.
}2
}
, and like the polyhedral group
s of 3D, can be named by their related convex regular 4polytope
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
with a '+' exponent, for example [3,3,3]^{+} has three 3fold gyration points and symmetry order 60. Frontback 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.
. 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
with a '+' exponent, for example [3,3,3,3]^{+} has four 3fold gyration points and symmetry order 360.
. 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
with a '+' exponent, for example [3,3,3,3,3]^{+} has five 3fold gyration points and symmetry order 2520.
. 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
with a '+' exponent, for example [3,3,3,3,3,3]^{+} has six 3fold gyration points and symmetry order 20160.
. 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
with a '+' exponent, for example [3,3,3,3,3,3,3]^{+} has seven 3fold gyration points and symmetry order 181440.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of premodern 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 distancepreserving 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 threedimensional 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. Pointgroup elements can either be rotations (determinant of M = 1) or else reflections, improper rotations, rotationreflections, 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 2fold, 3fold, 4fold, and 6fold...
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 onedimensional 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 CoxeterDynkin 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 

C_{1}  [ ]^{+}  1  Identity  
D_{1}  [ ] 
}
Two Dimensions
Point groups in two dimensionsPoint groups in two dimensions
In geometry, a twodimensional 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 C_{n} of nfold rotation groups
 Dihedral groups D_{n} of nfold 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 HermannMauguin 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 CharlesVictor 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 

C_{n}  n  nn  [n]^{+}  n  Cyclic: nfold rotations. Abstract group Z_{n}, the group of integers under addition modulo n. 
D_{n}  nm  *nn  [n]  2n  Dihedral: cyclic with reflections. Abstract group Dih_{n}, 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 CoxeterDynkin 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  

D_{3}  A_{2}  [3]  6  Equilateral triangle  
D_{4}  BC_{2}  [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... 

D_{5}  H_{2}  [5]  10  Regular pentagon  
D_{6}  G_{2}  [6]  12  Regular hexagon  
D_{n}  I_{2}(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:... 

D_{2n}  I_{2}(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:... 

D_{2}  A_{1}^{2}  [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 nonsquare rectangle... 

D_{1}  A_{1}  [ ] 
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 nondegenerate 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: C_{n}, S_{2n}, C_{nh}, C_{nv}, D_{n}, D_{nd}, D_{nh}
 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, T_{d}, T_{h}, O, O_{h}, I, I_{h}
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.



(*) 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 CoxeterDynkin 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  

T_{d}  A_{3}  [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... 

O_{h}  BC_{3}  [4,3] = 
48  Cube Cube In geometry, a cube is a threedimensional 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 

I_{h}  H_{3}  [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 

D_{3h}  A_{2}×A_{1}  [3,2]  12  Triangular prism Triangular prism In geometry, a triangular prism is a threesided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides.... 

D_{4h}  BC_{2}×A_{1}  [4,2]  16  Square prism  
D_{5h}  H_{2}×A_{1}  [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 :... 

D_{6h}  G_{2}×A_{1}  [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... 

D_{nh}  I_{2}(n)×A_{1}  [n,2]  4n  ngonal prism Prism (geometry) In geometry, a prism is a polyhedron with an nsided polygonal base, a translated copy , and n other faces joining corresponding sides of the two bases. All crosssections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a... 

D_{2h}  A_{1}^{3}  [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... 

C_{3v}  A_{2}×A_{1}  [3]  6  Hosohedron  
C_{4v}  BC_{2}×A_{1}  [4]  8  
C_{5v}  H_{2}×A_{1}  [5]  10  
C_{6v}  G_{2}×A_{1}  [6]  12  
C_{nv}  I_{2}(n)×A_{1}  [n]  2n  
C_{2v}  A_{1}^{2}  [2]  4  
C_{s}  A_{1}  [ ] 
}
Four dimensions
The fourdimensional 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 4polytope
Convex regular 4polytope
In mathematics, a convex regular 4polytope is a 4dimensional polytope that is both regular and convex. These are the fourdimensional 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 3fold gyration points and symmetry order 60. Frontback 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 CoxeterDynkin 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  

A_{4}  [3,3,3]  120  5cell  
A_{4}×2  240  5cell dual compound  
BC_{4}  [4,3,3]  384  16cell 16cell In four dimensional geometry, a 16cell or hexadecachoron is a regular convex 4polytope. It is one of the six regular convex 4polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid19th century.... /Tesseract Tesseract In geometry, the tesseract, also called an 8cell or regular octachoron or cubic prism, is the fourdimensional 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... 

D_{4}  [3^{1,1,1}]  192  Demitesseractic  
F_{4}  [3,4,3]  1152  24cell  
F_{4}×2  2304  24cell dual compound  
H_{4}  [5,3,3]  14400  120cell/600cell  
A_{3}×A_{1}  [3,3,2]  48  Tetrahedral prism  
BC_{3}×A_{1}  [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 :... 

H_{3}×A_{1}  [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... 

A_{2}×A_{2}  [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... 

A_{2}×BC_{2}  [3,2,4]  48  
A_{2}×H_{2}  [3,2,5]  60  
A_{2}×G_{2}  [3,2,6]  72  
BC_{2}×BC_{2}  [4,2,4]  64  
BC_{2}×H_{2}  [4,2,5]  80  
BC_{2}×G_{2}  [4,2,6]  96  
H_{2}×H_{2}  [5,2,5]  100  
H_{2}×G_{2}  [5,2,6]  120  
G_{2}×G_{2}  [6,2,6]  144  
I_{2}(p)×I_{2}(q)  [p,2,q]  4pq  
8p^{2}  
A_{2}×A_{1}^{2}  [3,2,2]  24  
BC_{2}×A_{1}^{2}  [4,2,2]  32  
H_{2}×A_{1}^{2}  [5,2,2]  40  
G_{2}×A_{1}^{2}  [6,2,2]  48  
I_{2}(p)×A_{1}^{2}  [p,2,2]  8p  
A_{1}^{4}  [2,2,2]  16  4orthotope 
Five dimensions
The fivedimensional 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 3fold 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 CoxeterDynkin 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  

A_{5}  [3,3,3,3]  720  5simplex  
A_{5}×2  
1440  5simplex dual compound  
BC_{5}  [4,3,3,3]  3840  5cube, 5orthoplex  
D_{5}  [3^{2,1,1}]  1920  5demicube  
A_{4}×A_{1}  [3,3,3,2]  240  5cell prism  
BC_{4}×A_{1}  [4,3,3,2]  768  tesseract Tesseract In geometry, the tesseract, also called an 8cell or regular octachoron or cubic prism, is the fourdimensional 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 

F_{4}×A_{1}  [3,4,3,2]  2304  24cell prism  
H_{4}×A_{1}  [5,3,3,2]  28800  600cell or 120cell prism  
D_{4}×A_{1}  [3^{1,1,1},2]  384  Demitesseract prism  
A_{3}×A_{2}  [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... 

A_{3}×BC_{2}  [3,3,2,4]  192  
A_{3}×H_{2}  [3,3,2,5]  240  
A_{3}×G_{2}  [3,3,2,6]  288  
A_{3}×I_{2}(p)  [3,3,2,p]  48p  
BC_{3}×A_{2}  [4,3,2,3]  288  
BC_{3}×BC_{2}  [4,3,2,4]  384  
BC_{3}×H_{2}  [4,3,2,5]  480  
BC_{3}×G_{2}  [4,3,2,6]  576  
BC_{3}×I_{2}(p)  [4,3,2,p]  96p  
H_{3}×A_{2}  [5,3,2,3]  720  
H_{3}×BC_{2}  [5,3,2,4]  960  
H_{3}×H_{2}  [5,3,2,5]  1200  
H_{3}×G_{2}  [5,3,2,6]  1440  
H_{3}×I_{2}(p)  [5,3,2,p]  240p  
A_{3}×A_{1}^{2}  [3,3,2,2]  96  
BC_{3}×A_{1}^{2}  [4,3,2,2]  192  
H_{3}×A_{1}^{2}  [5,3,2,2]  480  
A_{2}^{2}×A_{1}  [3,2,3,2]  72  duoprism prism  
A_{2}×BC_{2}×A_{1}  [3,2,4,2]  96  
A_{2}×H_{2}×A_{1}  [3,2,5,2]  120  
A_{2}×G_{2}×A_{1}  [3,2,6,2]  144  
BC_{2}^{2}×A_{1}  [4,2,4,2]  128  
BC_{2}×H_{2}×A_{1}  [4,2,5,2]  160  
BC_{2}×G_{2}×A_{1}  [4,2,6,2]  192  
H_{2}^{2}×A_{1}  [5,2,5,2]  200  
H_{2}×G_{2}×A_{1}  [5,2,6,2]  240  
G_{2}^{2}×A_{1}  [6,2,6,2]  288  
I_{2}(p)×I_{2}(q)×A_{1}  [p,2,q,2]  8pq  
A_{2}×A_{1}^{3}  [3,2,2,2]  48  
BC_{2}×A_{1}^{3}  [4,2,2,2]  64  
H_{2}×A_{1}^{3}  [5,2,2,2]  80  
G_{2}×A_{1}^{3}  [6,2,2,2]  96  
I_{2}(p)×A_{1}^{3}  [p,2,2,2]  16p  
A_{1}^{5}  [2,2,2,2]  32  5orthotope 
Six dimensions
The sixdimensional 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 3fold 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 CoxeterDynkin 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  

A_{6}  [3,3,3,3,3]  5040 (7!)  6simplex  
A_{6}×2  
10080 (2×7!)  6simplex dual compound  
BC_{6}  [4,3,3,3,3]  46080 (2^{6}×6!)  6cube, 6orthoplex  
D_{6}  [3,3,3,3^{1,1}]  23040 (2^{5}×6!)  6demicube  
E_{6}  [3,3^{2,2}]  51840 (72×6!)  1_{22}, 2_{21}  
A_{5}×A_{1}  [3,3,3,3,2]  1440 (2×6!)  5simplex prism  
BC_{5}×A_{1}  [4,3,3,3,2]  7680 (2^{6}×5!)  5cube prism  
D_{5}×A_{1}  [3,3,3^{1,1},2]  3840 (2^{5}×5!)  5demicube prism  
A_{4}×I_{2}(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... 

BC_{4}×I_{2}(p)  [4,3,3,2,p]  768p  
F_{4}×I_{2}(p)  [3,4,3,2,p]  2304p  
H_{4}×I_{2}(p)  [5,3,3,2,p]  28800p  
D_{4}×I_{2}(p)  [3,3^{1,1},2,p]  384p  
A_{4}×A_{1}^{2}  [3,3,3,2,2]  480  
BC_{4}×A_{1}^{2}  [4,3,3,2,2]  1536  
F_{4}×A_{1}^{2}  [3,4,3,2,2]  4608  
H_{4}×A_{1}^{2}  [5,3,3,2,2]  57600  
D_{4}×A_{1}^{2}  [3,3^{1,1},2,2]  768  
A_{3}^{2}  [3,3,2,3,3]  576  
A_{3}×BC_{3}  [3,3,2,4,3]  1152  
A_{3}×H_{3}  [3,3,2,5,3]  2880  
BC_{3}^{2}  [4,3,2,4,3]  2304  
BC_{3}×H_{3}  [4,3,2,5,3]  5760  
H_{3}^{2}  [5,3,2,5,3]  14400  
A_{3}×I_{2}(p)×A_{1}  [3,3,2,p,2]  96p  Duoprism prism  
BC_{3}×I_{2}(p)×A_{1}  [4,3,2,p,2]  192p  
H_{3}×I_{2}(p)×A_{1}  [5,3,2,p,2]  480p  
A_{3}×A_{1}^{3}  [3,3,2,2,2]  192  
BC_{3}×A_{1}^{3}  [4,3,2,2,2]  384  
H_{3}×A_{1}^{3}  [5,3,2,2,2]  960  
I_{2}(p)×I_{2}(q)×I_{2}(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... 

I_{2}(p)×I_{2}(q)×A_{1}^{2}  [p,2,q,2,2]  16pq  
I_{2}(p)×A_{1}^{4}  [p,2,2,2,2]  32p  
A_{1}^{6}  [2,2,2,2,2]  64  6orthotope 
Seven dimensions
The sevendimensional 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 3fold 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 CoxeterDynkin 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  

A_{7}  [3,3,3,3,3,3]  40320 (8!)  7simplex  
A_{7}×2  
80640 (2×8!)  7simplex dual compound  
BC_{7}  [4,3,3,3,3,3]  645120 (2^{7}×7!)  7cube, 7orthoplex  
D_{7}  [3,3,3,3,3^{1,1}]  322560 (2^{6}×7!)  7demicube  
E_{7}  [3,3,3,3^{2,1}]  2903040 (8×9!)  3_{21}, 2_{31}, 1_{32}  
A_{6}×A_{1}  [3,3,3,3,3,2]  10080 (2×7!)  
BC_{6}×A_{1}  [4,3,3,3,3,2]  92160 (2^{7}×6!)  
D_{6}×A_{1}  [3,3,3,3^{1,1},2]  46080 (2^{6}×6!)  
E_{6}×A_{1}  [3,3,3^{2,1},2]  103680 (144×6!)  
A_{5}×I_{2}(p)  [3,3,3,3,2,p]  1440p  
BC_{5}×I_{2}(p)  [4,3,3,3,2,p]  7680p  
D_{5}×I_{2}(p)  [3,3,3^{1,1},2,p]  3840p  
A_{5}×A_{1}^{2}  [3,3,3,3,2,2]  2880  
BC_{5}×A_{1}^{2}  [4,3,3,3,2,2]  15360  
D_{5}×A_{1}^{2}  [3,3,3^{1,1},2,2]  7680  
A_{4}×A_{3}  [3,3,3,2,3,3]  2880  
A_{4}×BC_{3}  [3,3,3,2,4,3]  5760  
A_{4}×H_{3}  [3,3,3,2,5,3]  14400  
BC_{4}×A_{3}  [4,3,3,2,3,3]  9216  
BC_{4}×BC_{3}  [4,3,3,2,4,3]  18432  
BC_{4}×H_{3}  [4,3,3,2,5,3]  46080  
H_{4}×A_{3}  [5,3,3,2,3,3]  345600  
H_{4}×BC_{3}  [5,3,3,2,4,3]  691200  
H_{4}×H_{3}  [5,3,3,2,5,3]  1728000  
F_{4}×A_{3}  [3,4,3,2,3,3]  27648  
F_{4}×BC_{3}  [3,4,3,2,4,3]  55296  
F_{4}×H_{3}  [3,4,3,2,5,3]  138240  
D_{4}×A_{3}  [3^{1,1,1},2,3,3]  4608  
D_{4}×BC_{3}  [3,3^{1,1},2,4,3]  9216  
D_{4}×H_{3}  [3,3^{1,1},2,5,3]  23040  
A_{4}×I_{2}(p)×A_{1}  [3,3,3,2,p,2]  480p  
BC_{4}×I_{2}(p)×A_{1}  [4,3,3,2,p,2]  1536p  
D_{4}×I_{2}(p)×A_{1}  [3,3^{1,1},2,p,2]  768p  
F_{4}×I_{2}(p)×A_{1}  [3,4,3,2,p,2]  4608p  
H_{4}×I_{2}(p)×A_{1}  [5,3,3,2,p,2]  57600p  
A_{4}×A_{1}^{3}  [3,3,3,2,2,2]  960  
BC_{4}×A_{1}^{3}  [4,3,3,2,2,2]  3072  
F_{4}×A_{1}^{3}  [3,4,3,2,2,2]  9216  
H_{4}×A_{1}^{3}  [5,3,3,2,2,2]  115200  
D_{4}×A_{1}^{3}  [3,3^{1,1},2,2,2]  1536  
A_{3}^{2}×A_{1}  [3,3,2,3,3,2]  1152  
A_{3}×BC_{3}×A_{1}  [3,3,2,4,3,2]  2304  
A_{3}×H_{3}×A_{1}  [3,3,2,5,3,2]  5760  
BC_{3}^{2}×A_{1}  [4,3,2,4,3,2]  4608  
BC_{3}×H_{3}×A_{1}  [4,3,2,5,3,2]  11520  
H_{3}^{2}×A_{1}  [5,3,2,5,3,2]  28800  
A_{3}×I_{2}(p)×I_{2}(q)  [3,3,2,p,2,q]  96pq  
BC_{3}×I_{2}(p)×I_{2}(q)  [4,3,2,p,2,q]  192pq  
H_{3}×I_{2}(p)×I_{2}(q)  [5,3,2,p,2,q]  480pq  
A_{3}×I_{2}(p)×A_{1}^{2}  [3,3,2,p,2,2]  192p  
BC_{3}×I_{2}(p)×A_{1}^{2}  [4,3,2,p,2,2]  384p  
H_{3}×I_{2}(p)×A_{1}^{2}  [5,3,2,p,2,2]  960p  
A_{3}×A_{1}^{4}  [3,3,2,2,2,2]  384  
BC_{3}×A_{1}^{4}  [4,3,2,2,2,2]  768  
H_{3}×A_{1}^{4}  [5,3,2,2,2,2]  1920  
I_{2}(p)×I_{2}(q)×I_{2}(r)×A_{1}  [p,2,q,2,r,2]  16pqr  
I_{2}(p)×I_{2}(q)×A_{1}^{3}  [p,2,q,2,2,2]  32pq  
I_{2}(p)×A_{1}^{5}  [p,2,2,2,2,2]  64p  
A_{1}^{7}  [2,2,2,2,2,2]  128 
Eight dimensions
The eightdimensional 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 3fold 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 CoxeterDynkin 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  

A_{8}  [3,3,3,3,3,3,3]  362880 (9!)  8simplex  
A_{8}×2  
725760 (2x9!)  8simplex dual compound  
BC_{8}  [4,3,3,3,3,3,3]  10321920 (2^{8}8!)  8cube,8orthoplex  
D_{8}  [3,3,3,3,3,3^{1,1}]  5160960 (2^{7}8!)  8demicube  
E_{8} 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... 
[3,3,3,3,3^{2,1}]  696729600  4_{21}, 2_{41}, 1_{42}  
A_{7}×A_{1}  [3,3,3,3,3,3,2]  80640  7simplex prism  
BC_{7}×A_{1}  [4,3,3,3,3,3,2]  645120  7cube prism  
D_{7}×A_{1}  [3,3,3,3,3^{1,1},2]  322560  7demicube prism  
E_{7}×A_{1}  [3,3,3,3^{2,1},2]  5806080  3_{21} prism, 2_{31} prism, 1_{42} prism  
A_{6}×I_{2}(p)  [3,3,3,3,3,2,p]  10080p  duoprism  
BC_{6}×I_{2}(p)  [4,3,3,3,3,2,p]  92160p  
D_{6}×I_{2}(p)  [3,3,3,3^{1,1},2,p]  46080p  
E_{6}×I_{2}(p)  [3,3,3^{2,1},2,p]  103680p  
A_{6}×A_{1}^{2}  [3,3,3,3,3,2,2]  20160  
BC_{6}×A_{1}^{2}  [4,3,3,3,3,2,2]  184320  
D_{6}×A_{1}^{2}  [3^{3,1,1},2,2]  92160  
E_{6}×A_{1}^{2}  [3,3,3^{2,1},2,2]  207360  
A_{5}×A_{3}  [3,3,3,3,2,3,3]  17280  
BC_{5}×A_{3}  [4,3,3,3,2,3,3]  92160  
D_{5}×A_{3}  [3^{2,1,1},2,3,3]  46080  
A_{5}×BC_{3}  [3,3,3,3,2,4,3]  34560  
BC_{5}×BC_{3}  [4,3,3,3,2,4,3]  184320  
D_{5}×BC_{3}  [3^{2,1,1},2,4,3]  92160  
A_{5}×H_{3}  [3,3,3,3,2,5,3]  
BC_{5}×H_{3}  [4,3,3,3,2,5,3]  
D_{5}×H_{3}  [3^{2,1,1},2,5,3]  
A_{5}×I_{2}(p)×A_{1}  [3,3,3,3,2,p,2]  
BC_{5}×I_{2}(p)×A_{1}  [4,3,3,3,2,p,2]  
D_{5}×I_{2}(p)×A_{1}  [3^{2,1,1},2,p,2]  
A_{5}×A_{1}^{3}  [3,3,3,3,2,2,2]  
BC_{5}×A_{1}^{3}  [4,3,3,3,2,2,2]  
D_{5}×A_{1}^{3}  [3^{2,1,1},2,2,2]  
A_{4}×A_{4}  [3,3,3,2,3,3,3]  
BC_{4}×A_{4}  [4,3,3,2,3,3,3]  
D_{4}×A_{4}  [3^{1,1,1},2,3,3,3]  
F_{4}×A_{4}  [3,4,3,2,3,3,3]  
H_{4}×A_{4}  [5,3,3,2,3,3,3]  
BC_{4}×BC_{4}  [4,3,3,2,4,3,3]  
D_{4}×BC_{4}  [3^{1,1,1},2,4,3,3]  
F_{4}×BC_{4}  [3,4,3,2,4,3,3]  
H_{4}×BC_{4}  [5,3,3,2,4,3,3]  
D_{4}×D_{4}  [3^{1,1,1},2,3^{1,1,1}]  
F_{4}×D_{4}  [3,4,3,2,3^{1,1,1}]  
H_{4}×D_{4}  [5,3,3,2,3^{1,1,1}]  
F_{4}×F_{4}  [3,4,3,2,3,4,3]  
H_{4}×F_{4}  [5,3,3,2,3,4,3]  
H_{4}×H_{4}  [5,3,3,2,5,3,3]  
A_{4}×A_{3}×A_{1}  [3,3,3,2,3,3,2]  duoprism prisms  
A_{4}×BC_{3}×A_{1}  [3,3,3,2,4,3,2]  
A_{4}×H_{3}×A_{1}  [3,3,3,2,5,3,2]  
BC_{4}×A_{3}×A_{1}  [4,3,3,2,3,3,2]  
BC_{4}×BC_{3}×A_{1}  [4,3,3,2,4,3,2]  
BC_{4}×H_{3}×A_{1}  [4,3,3,2,5,3,2]  
H_{4}×A_{3}×A_{1}  [5,3,3,2,3,3,2]  
H_{4}×BC_{3}×A_{1}  [5,3,3,2,4,3,2]  
H_{4}×H_{3}×A_{1}  [5,3,3,2,5,3,2]  
F_{4}×A_{3}×A_{1}  [3,4,3,2,3,3,2]  
F_{4}×BC_{3}×A_{1}  [3,4,3,2,4,3,2]  
F_{4}×H_{3}×A_{1}  [3,4,2,3,5,3,2]  
D_{4}×A_{3}×A_{1}  [3^{1,1,1},2,3,3,2]  
D_{4}×BC_{3}×A_{1}  [3^{1,1,1},2,4,3,2]  
D_{4}×H_{3}×A_{1}  [3^{1,1,1},2,5,3,2]  
A_{4}×I_{2}(p)×I_{2}(q)  [3,3,3,2,p,2,q]  triaprism  
BC_{4}×I_{2}(p)×I_{2}(q)  [4,3,3,2,p,2,q]  
F_{4}×I_{2}(p)×I_{2}(q)  [3,4,3,2,p,2,q]  
H_{4}×I_{2}(p)×I_{2}(q)  [5,3,3,2,p,2,q]  
D_{4}×I_{2}(p)×I_{2}(q)  [3^{1,1,1},2,p,2,q]  
A_{4}×I_{2}(p)×A_{1}^{2}  [3,3,3,2,p,2,2]  
BC_{4}×I_{2}(p)×A_{1}^{2}  [4,3,3,2,p,2,2]  
F_{4}×I_{2}(p)×A_{1}^{2}  [3,4,3,2,p,2,2]  
H_{4}×I_{2}(p)×A_{1}^{2}  [5,3,3,2,p,2,2]  
D_{4}×I_{2}(p)×A_{1}^{2}  [3^{1,1,1},2,p,2,2]  
A_{4}×A_{1}^{4}  [3,3,3,2,2,2,2]  
BC_{4}×A_{1}^{4}  [4,3,3,2,2,2,2]  
F_{4}×A_{1}^{4}  [3,4,3,2,2,2,2]  
H_{4}×A_{1}^{4}  [5,3,3,2,2,2,2]  
D_{4}×A_{1}^{4}  [3^{1,1,1},2,2,2,2]  
A_{3}×A_{3}×I_{2}(p)  [3,3,2,3,3,2,p]  
BC_{3}×A_{3}×I_{2}(p)  [4,3,2,3,3,2,p]  
H_{3}×A_{3}×I_{2}(p)  [5,3,2,3,3,2,p]  
BC_{3}×BC_{3}×I_{2}(p)  [4,3,2,4,3,2,p]  
H_{3}×BC_{3}×I_{2}(p)  [5,3,2,4,3,2,p]  
H_{3}×H_{3}×I_{2}(p)  [5,3,2,5,3,2,p]  
A_{3}×A_{3}×A_{1}^{2}  [3,3,2,3,3,2,2]  
BC_{3}×A_{3}×A_{1}^{2}  [4,3,2,3,3,2,2]  
H_{3}×A_{3}×A_{1}^{2}  [5,3,2,3,3,2,2]  
BC_{3}×BC_{3}×A_{1}^{2}  [4,3,2,4,3,2,2]  
H_{3}×BC_{3}×A_{1}^{2}  [5,3,2,4,3,2,2]  
H_{3}×H_{3}×A_{1}^{2}  [5,3,2,5,3,2,2]  
A_{3}×I_{2}(p)×I_{2}(q)×A_{1}  [3,3,2,p,2,q,2]  
BC_{3}×I_{2}(p)×I_{2}(q)×A_{1}  [4,3,2,p,2,q,2]  
H_{3}×I_{2}(p)×I_{2}(q)×A_{1}  [5,3,2,p,2,q,2]  
A_{3}×I_{2}(p)×A_{1}^{3}  [3,3,2,p,2,2,2]  
BC_{3}×I_{2}(p)×A_{1}^{3}  [4,3,2,p,2,2,2]  
H_{3}×I_{2}(p)×A_{1}^{3}  [5,3,2,p,2,2,2]  
A_{3}×A_{1}^{5}  [3,3,2,2,2,2,2]  
BC_{3}×A_{1}^{5}  [4,3,2,2,2,2,2]  
H_{3}×A_{1}^{5}  [5,3,2,2,2,2,2]  
I_{2}(p)×I_{2}(q)×I_{2}(r)×I_{2}(s)  [p,2,q,2,r,2,s]  16pqrs  
I_{2}(p)×I_{2}(q)×I_{2}(r)×A_{1}^{2}  [p,2,q,2,r,2,2]  32pqr  
I_{2}(p)×I_{2}(q)×A_{1}^{4}  [p,2,q,2,2,2,2]  64pq  
I_{2}(p)×A_{1}^{6}  [p,2,2,2,2,2,2]  128p  
A_{1}^{8}  [2,2,2,2,2,2,2]  256 
See also
 Point groups in two dimensionsPoint groups in two dimensionsIn geometry, a twodimensional 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...
 Xray diffraction
 Bravais lattice