Point group

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The Bauhinia blakeana flower on the Hong Kong region flag has C₅ symmetry; the star on each petal has D₅ symmetry.

The Yin and Yang symbol has C₂ symmetry of geometry with inverted colors

In geometry, a point group is a mathematical group of symmetry operations that have a fixed point in common. The coordinate origin 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 O. 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 M that transform point x into point y according to. Each element of a point group is either a rotation, or it is a reflection or improper rotation.
The geometric symmetries of crystals are described by space groups, which allow translations and contain point groups as subgroups. 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 of dimensions. These are the crystallographic point groups.

Chiral and achiral point groups, reflection groups

Point groups can be classified into chiral groups and achiral groups.
The chiral groups are subgroups of the special orthogonal group SO: 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 subgroup of index 2.
Finite Coxeter groups 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 offers a bracketed notation equivalent to the Coxeter diagram, with markup symbols for rotational and other subsymmetry point groups. Reflection groups are necessarily achiral.

List of point groups

One dimension

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

Group	Coxeter	Coxeter diagram	Order	Description
C₁	⁺		1	identity
D₁			2	reflection group

Two dimensions

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

Cyclic groups C_n of n-fold rotation groups
Dihedral groups D_n 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	Orbifold	Coxeter	Order	Description
C_n	n	n•	⁺	n	cyclic: n-fold rotations; abstract group Z_n, the group of integers under addition modulo n
D_n	nm	*n•		2n	dihedral: cyclic with reflections; abstract group Dih_n, the dihedral group

The subset of pure reflectional point groups, defined by 1 or 2 mirrors, can also be given by their Coxeter group and related polygons. These include 5 crystallographic groups. The symmetry of the reflectional groups can be doubled by an isomorphism, mapping both mirrors onto each other by a bisecting mirror, doubling the symmetry order.

Three dimensions

, sometimes called molecular point groups after their wide use in studying symmetries of molecules.
They come in 7 infinite families of axial groups, and 7 additional polyhedral groups. In Schoenflies notation,

Axial groups: C_n, S_2n, C_nh, C_nv, D_n, D_nd, D_nh
Polyhedral groups: T, T_d, T_h, O, O_h, I, I_h

Applying the crystallographic restriction theorem to these groups yields the 32 crystallographic point groups.

C_1v Order 2	C_2v Order 4	C_3v Order 6	C_4v Order 8	C_5v Order 10	C_6v Order 12	...
80px	80px	80px	80px	80px	80px	-
D_1h Order 4	D_2h Order 8	D_3h Order 12	D_4h Order 16	D_5h Order 20	D_6h Order 24	...
80px	80px	80px	80px	80px	80px	-
T_d Order 24	O_h Order 48	I_h Order 120	-	-	-	-
80px	80px	80px	-	-	-	-

Reflection groups

The reflection point groups, defined by 1 to 3 mirror planes, can also be given by their Coxeter group and related polyhedra. The group can be doubled, written as, mapping the first and last mirrors onto each other, doubling the symmetry to 48, and isomorphic to the group.

Four dimensions

The four-dimensional point groups are listed in Conway and Smith, Section 4, Tables 4.1–4.3.
The following list gives the four-dimensional reflection groups. Each group is specified as a Coxeter group, and like the polyhedral groups of 3D, it can be named by its related convex regular 4-polytope. Related pure rotational groups exist for each with half the order, and can be represented by the bracket Coxeter notation with a '+' exponent, for example ⁺ has three 3-fold gyration points and symmetry order 60. Front-back symmetric groups like and can be doubled, shown as double brackets in Coxeter's notation, for example with its order doubled to 240.

Five dimensions

The following table gives the five-dimensional reflection groups, by listing them as Coxeter groups. Related chiral groups exist for each with half the order, and can be represented by the bracket Coxeter notation with a '+' exponent, for example ⁺ has four 3-fold gyration points and symmetry order 360.

Six dimensions

The following table gives the six-dimensional reflection groups, by listing them as Coxeter groups. Related pure rotational groups exist for each with half the order, and can be represented by the bracket Coxeter notation with a '+' exponent, for example ⁺ has five 3-fold gyration points and symmetry order 2520.

Seven dimensions

The following table gives the seven-dimensional reflection groups, by listing them as Coxeter groups. Related chiral 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 ⁺ has six 3-fold gyration points and symmetry order 20160.

Eight dimensions

The following table gives the eight-dimensional reflection groups, by listing them as Coxeter groups. Related chiral 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 ⁺ has seven 3-fold gyration points and symmetry order 181440.