Point group

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 (isometries in a Euclidean space) 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(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 M that transform point x into point y according to y = Mx. Each element of a point group is either a rotation (determinant of M = 1), or it is a reflection or improper rotation (determinant of M = −1).

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.

Point groups can be classified into chiral (or purely rotational) groups and achiral groups.^[1] 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 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 (except for the trivial group containing only the identity element).

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

Point groups in two dimensions, sometimes called rosette groups.

They come in two infinite families:

1. Cyclic groups C_n of n-fold rotation groups
2. 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.

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.

Reflective						Rotational			Related polygons
Group	Coxeter group		Coxeter diagram		Order	Subgroup	Coxeter	Order
D1	A1	[ ]			2	C1	[]+	1	digon
D2	A12	[2]			4	C2	[2]+	2	rectangle
D3	A2	[3]			6	C3	[3]+	3	equilateral triangle
D4	BC2	[4]			8	C4	[4]+	4	square
D5	H2	[5]			10	C5	[5]+	5	regular pentagon
D6	G2	[6]			12	C6	[6]+	6	regular hexagon
Dn	I2(n)	[n]			2n	Cn	[n]+	n	regular polygon
D2×2	A12×2	[[2]] = [4]		=	8
D3×2	A2×2	[[3]] = [6]		=	12
D4×2	BC2×2	[[4]] = [8]		=	16
D5×2	H2×2	[[5]] = [10]		=	20
D6×2	G2×2	[[6]] = [12]		=	24
Dn×2	I2(n)×2	[[n]] = [2n]		=	4n

Point groups in 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 (also called prismatic), and 7 additional polyhedral groups (also called Platonic). In Schönflies 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.

Even/odd colored fundamental domains of the reflective groups

C1v Order 2	C2v Order 4	C3v Order 6	C4v Order 8	C5v Order 10	C6v Order 12	...
D1h Order 4	D2h Order 8	D3h Order 12	D4h Order 16	D5h Order 20	D6h Order 24	...
Td Order 24	Oh Order 48	Ih Order 120

The reflection 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.

Schönflies	Coxeter group		Coxeter diagram			Order	Related regular and prismatic polyhedra
Td	A3	[3,3]				24	tetrahedron
Td×Dih1 = Oh	A3×2 = BC3	[[3,3]] = [4,3]			=	48	stellated octahedron
Oh	BC3	[4,3]				48	cube, octahedron
Ih	H3	[5,3]				120	icosahedron, dodecahedron
D3h	A2×A1	[3,2]				12	triangular prism
D3h×Dih1 = D6h	A2×A1×2	[[3],2]			=	24	hexagonal prism
D4h	BC2×A1	[4,2]				16	square prism
D4h×Dih1 = D8h	BC2×A1×2	[[4],2] = [8,2]			=	32	octagonal prism
D5h	H2×A1	[5,2]				20	pentagonal prism
D6h	G2×A1	[6,2]				24	hexagonal prism
Dnh	I2(n)×A1	[n,2]				4n	n-gonal prism
Dnh×Dih1 = D2nh	I2(n)×A1×2	[[n],2]			=	8n
D2h	A13	[2,2]				8	cuboid
D2h×Dih1	A13×2	[[2],2] = [4,2]			=	16
D2h×Dih3 = Oh	A13×6	[3[2,2]] = [4,3]			=	48
C3v	A2	[1,3]				6	hosohedron
C4v	BC2	[1,4]				8
C5v	H2	[1,5]				10
C6v	G2	[1,6]				12
Cnv	I2(n)	[1,n]				2n
Cnv×Dih1 = C2nv	I2(n)×2	[1,[n]] = [1,2n]			=	4n
C2v	A12	[1,2]				4
C2v×Dih1	A12×2	[1,[2]]			=	8
Cs	A1	[1,1]				2

The four-dimensional point groups (chiral as well as achiral) are listed in Conway and Smith,^[1] Section 4, Tables 4.1–4.3.

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

Coxeter group/notation		Coxeter diagram		Order	Related polytopes
A4	[3,3,3]			120	5-cell
A4×2	[[3,3,3]]			240	5-cell dual compound
BC4	[4,3,3]			384	16-cell / tesseract
D4	[31,1,1]			192	demitesseractic
D4×2 = BC4	<[3,31,1]> = [4,3,3]		=	384
D4×6 = F4	[3[31,1,1]] = [3,4,3]		=	1152
F4	[3,4,3]			1152	24-cell
F4×2	[[3,4,3]]			2304	24-cell dual compound
H4	[5,3,3]			14400	120-cell / 600-cell
A3×A1	[3,3,2]			48	tetrahedral prism
A3×A1×2	[[3,3],2] = [4,3,2]		=	96	octahedral prism
BC3×A1	[4,3,2]			96
H3×A1	[5,3,2]			240	icosahedral prism
A2×A2	[3,2,3]			36	duoprism
A2×BC2	[3,2,4]			48
A2×H2	[3,2,5]			60
A2×G2	[3,2,6]			72
BC2×BC2	[4,2,4]			64
BC22×2	[[4,2,4]]			128
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
I2(2p)×I2(q)	[[p],2,q] = [2p,2,q]		=	8pq
I2(2p)×I2(2q)	[[p]],2,[[q]] = [2p,2,2q]		=	16pq
I2(p)2×2	[[p,2,p]]			8p2
I2(2p)2×2	[[[p]],2,[p]]] = [[2p,2,2p]]		=	32p2
A2×A1×A1	[3,2,2]			24
BC2×A1×A1	[4,2,2]			32
H2×A1×A1	[5,2,2]			40
G2×A1×A1	[6,2,2]			48
I2(p)×A1×A1	[p,2,2]			8p
I2(2p)×A1×A1×2	[[p],2,2] = [2p,2,2]		=	16p
I2(p)×A12×2	[p,2,[2]] = [p,2,4]		=	16p
I2(2p)×A12×4	[[p]],2,[[2]] = [2p,2,4]		=	32p
A1×A1×A1×A1	[2,2,2]			16	4-orthotope
A12×A1×A1×2	[[2],2,2] = [4,2,2]		=	32
A12×A12×4	[[2]],2,[[2]] = [4,2,4]		=	64
A13×A1×6	[3[2,2],2] = [4,3,2]		=	96
A14×24	[3,3[2,2,2]] = [4,3,3]		=	384

The following table gives the five-dimensional reflection groups (excluding those that are lower-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 [3,3,3,3]⁺ has four 3-fold gyration points and symmetry order 360.

Coxeter group/notation		Coxeter diagrams		Order	Related regular and prismatic polytopes
A5	[3,3,3,3]			720	5-simplex
A5×2	[[3,3,3,3]]			1440	5-simplex dual compound
BC5	[4,3,3,3]			3840	5-cube, 5-orthoplex
D5	[32,1,1]			1920	5-demicube
D5×2	<[3,3,31,1]>		=	3840
A4×A1	[3,3,3,2]			240	5-cell prism
A4×A1×2	[[3,3,3],2]			480
BC4×A1	[4,3,3,2]			768	tesseract prism
F4×A1	[3,4,3,2]			2304	24-cell prism
F4×A1×2	[[3,4,3],2]			4608
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
A3×A2×2	[[3,3],2,3]			288
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
A15×(2!)	[[2],2,2,2]		=	64
A15×(2!×2!)	[[2]],2,[2],2]		=	128
A15×(3!)	[3[2,2],2,2]		=	192
A15×(3!×2!)	[3[2,2],2,[[2]]		=	384
A15×(4!)	[3,3[2,2,2],2]]		=	768
A15×(5!)	[3,3,3[2,2,2,2]]		=	3840

The following table gives the six-dimensional reflection groups (excluding those that are lower-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 [3,3,3,3,3]⁺ has five 3-fold gyration points and symmetry order 2520.

Coxeter group		Coxeter diagram	Order	Related regular and prismatic polytopes
A6	[3,3,3,3,3]		5040 (7!)	6-simplex
A6×2	[[3,3,3,3,3]]		10080 (2×7!)	6-simplex dual compound
BC6	[4,3,3,3,3]		46080 (26×6!)	6-cube, 6-orthoplex
D6	[3,3,3,31,1]		23040 (25×6!)	6-demicube
E6	[3,32,2]		51840 (72×6!)	122, 221
A5×A1	[3,3,3,3,2]		1440 (2×6!)	5-simplex prism
BC5×A1	[4,3,3,3,2]		7680 (26×5!)	5-cube prism
D5×A1	[3,3,31,1,2]		3840 (25×5!)	5-demicube prism
A4×I2(p)	[3,3,3,2,p]		240p	duoprism
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
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