geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a point group is a mathematical group of

symmetry operation In group theory, geometry, representation theory and molecular symmetry, a symmetry operation is a transformation of an object that leaves an object looking the same after it has been carried out. For example, as transformations of an object in spac ...

s (

isometries In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...

in a

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

) that have a fixed point in common. The

coordinate origin In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter ''O'', used as a fixed point of reference for the geometry of the surrounding space. In physical problems, the choice of origin is often arbitrar ...

of the Euclidean space is conventionally taken to be a fixed point, and every point group in dimension ''d'' is then a subgroup of the

orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...

O(''d''). Point groups are used to describe the symmetries of geometric figures and physical objects such as molecules. Each point group can be represented as sets of

orthogonal matrices In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q^\mathrm Q = Q Q^\mathrm = I, where is the transpose of and is the identity ma ...

''M'' that transform point ''x'' into point ''y'' according to ''y'' = ''Mx''. Each element of a point group is either a

rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...

(

determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...

of ''M'' = 1), or it is a

reflection Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...

improper rotation In geometry, an improper rotation,. also called rotation-reflection, rotoreflection, rotary reflection,. or rotoinversion is an isometry in Euclidean space that is a combination of a rotation about an axis and a reflection in a plane perpendicul ...

(determinant of ''M'' = −1). The geometric symmetries of

crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macros ...

s are described by

space group In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of an object that leave it unchan ...

s, which allow

translations Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transl ...

and contain point groups as subgroups. Discrete point groups in more than one dimension come in infinite families, but from the

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. However, quasicrystals can occur with other diffraction ...

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 may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...

or grid with that number of dimensions. These are the

crystallographic point group In crystallography, a crystallographic point group is a set of symmetry operations, corresponding to one of the point groups in three dimensions, such that each operation (perhaps followed by a translation) would leave the structure of a crystal u ...

Chiral and achiral point groups, reflection groups

Point groups can be classified into ''

chiral Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from ...

'' (or purely rotational) groups and ''achiral'' groups. The chiral groups are subgroups of the special orthogonal group SO(''d''): they contain only orientation-preserving orthogonal transformations, i.e., those of determinant +1. The achiral groups contain also transformations of determinant −1. In an achiral group, the orientation-preserving transformations form a (chiral) subgroup of index 2.

Finite 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 reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...

s or ''reflection groups'' are those point groups that are generated purely by a set of reflectional mirrors passing through the same point. A rank ''n'' Coxeter group has ''n'' mirrors and is represented by a Coxeter-Dynkin diagram.

Coxeter notation In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram ...

offers a bracketed notation equivalent to the Coxeter diagram, with markup symbols for rotational and other subsymmetry point groups. Reflection groups are necessarily achiral (except for the trivial group containing only the identity element).

List of point groups

One dimension

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

Two dimensions

Point groups in two dimensions In geometry, a two-dimensional point group or rosette group is a group of geometric symmetries (isometries) that keep at least one point fixed in a plane. Every such group is a subgroup of the orthogonal group O(2), including O(2) itself. Its ...

, sometimes called rosette groups. They come in two infinite families: #

Cyclic groups In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binary ...

''C''_''n'' of ''n''-fold rotation groups #

Dihedral group In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ge ...

s ''D''_''n'' of ''n''-fold rotation and reflection groups Applying the

restricts ''n'' to values 1, 2, 3, 4, and 6 for both families, yielding 10 groups. Coxeter diagram finite rank2 correspondence

The subset of pure reflectional point groups, defined by 1 or 2 mirrors, can also be given by their

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 reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refl ...

and related polygons. These include 5 crystallographic groups. The symmetry of the reflectional groups can be doubled by an

isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...

, mapping both mirrors onto each other by a bisecting mirror, doubling the symmetry order.

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(3), the group of all isometrie ...

, sometimes called molecular point groups after their wide use in studying symmetries of molecules. They come in 7 infinite families of axial groups (also called prismatic), and 7 additional polyhedral groups (also called Platonic). In

Schönflies notation The Schoenflies (or Schönflies) notation, named after the German mathematician Arthur Moritz Schoenflies, is a notation primarily used to specify point groups in three dimensions. Because a point group alone is completely adequate to describe the ...

, * Axial groups: C_''n'', S_2''n'', C_''n''h, C_''n''v, D_''n'', D_''n''d, D_''n''h *

Polyhedral group In geometry, the polyhedral group is any of the symmetry groups of the Platonic solids. Groups There are three polyhedral groups: *The tetrahedral group of order 12, rotational symmetry group of the regular tetrahedron. It is isomorphic to ''A' ...

s: T, T_d, T_h, O, O_h, I, I_h Applying the

to these groups yields the 32

Reflection groups

Coxeter diagram finite rank3 correspondence

The reflection point groups, defined by 1 to 3 mirror planes, can also be given by their

and related polyhedra. The ,3group can be doubled, written as , mapping the first and last mirrors onto each other, doubling the symmetry to 48, and isomorphic to the ,3group.

Four dimensions

The four-dimensional point groups (chiral as well as achiral) are listed in Conway and Smith, Section 4, Tables 4.1-4.3. Coxeter diagram finite rank4 correspondence

Coxeter diagram finite rank4 correspondence

The following list gives the four-dimensional reflection groups (excluding those that leave a subspace fixed and that are therefore lower-dimensional reflection groups). Each group is specified as a

, and like the

polyhedral group In geometry, the polyhedral group is any of the symmetry groups of the Platonic solids. Groups There are three polyhedral groups: *The tetrahedral group of order 12, rotational symmetry group of the regular tetrahedron. It is isomorphic to ''A' ...

s of 3D, it can be named by its related

convex regular 4-polytope In mathematics, a regular 4-polytope is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions. There are six convex and ten star regu ...

. Related pure rotational groups exist for each with half the order, and can be represented by the bracket

with a '+' exponent, for example ,3,3sup>+ has three 3-fold gyration points and symmetry order 60. Front-back symmetric groups like ,3,3and ,4,3can be doubled, shown as double brackets in Coxeter's notation, for example with its order doubled to 240.

Five dimensions

Coxeter diagram finite rank5 correspondence

The following table gives the five-dimensional reflection groups (excluding those that are lower-dimensional reflection groups), by listing them as

s. Related chiral groups exist for each with half the order, and can be represented by the bracket

with a '+' exponent, for example ,3,3,3sup>+ has four 3-fold gyration points and symmetry order 360.

Six dimensions

Coxeter diagram finite rank6 correspondence

The following table gives the six-dimensional reflection groups (excluding those that are lower-dimensional reflection groups), by listing them as

s. Related pure rotational groups exist for each with half the order, and can be represented by the bracket

with a '+' exponent, for example ,3,3,3,3sup>+ has five 3-fold gyration points and symmetry order 2520.

Seven dimensions

The following table gives the seven-dimensional reflection groups (excluding those that are lower-dimensional reflection groups), by listing them as

s. Related chiral groups exist for each with half the order, defined by an

even number In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 41 ...

of reflections, and can be represented by the bracket

with a '+' exponent, for example ,3,3,3,3,3sup>+ has six 3-fold gyration points and symmetry order 20160.

Eight dimensions

The following table gives the eight-dimensional reflection groups (excluding those that are lower-dimensional reflection groups), by listing them as