Point group

File:Flag of Hong Kong.svg The Bauhinia blakeana flower on the Hong Kong region flag has C5 symmetry; the star on each petal has D5 symmetry.

File:Yin and Yang.svg 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 Template:Nowrap. Each element of a point group is either a rotation (determinant of Template:Nowrap), or it is a reflection or improper rotation (determinant of Template:Nowrap).

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 (or purely rotational) groups and achiral groups.Template:Refn 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).

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	[ ]	Template:CDD	2	reflection group

Two dimensions

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.

Group	Intl	Orbifold	Coxeter	Order	Description
Cn	n	n•	[n]+	n	cyclic: n-fold rotations; abstract group Zn, the group of integers under addition modulo n
Dn	nm	*n•	[n]	2n	dihedral: cyclic with reflections; abstract group Dihn, the dihedral group

File:Coxeter diagram finite rank2 correspondence.png

Finite isomorphism and correspondences

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	[ ]	Template:CDD	Template:CDD	2	C1	[]+	1	digon
D2	A12	[2]	Template:CDD	Template:CDD	4	C2	[2]+	2	rectangle
D3	A2	[3]	Template:CDD	Template:CDD	6	C3	[3]+	3	equilateral triangle
D4	BC2	[4]	Template:CDD	Template:CDD	8	C4	[4]+	4	square
D5	H2	[5]	Template:CDD	Template:CDD	10	C5	[5]+	5	regular pentagon
D6	G2	[6]	Template:CDD	Template:CDD	12	C6	[6]+	6	regular hexagon
Dn	I2(n)	[n]	Template:CDD	Template:CDD	2n	Cn	[n]+	n	regular polygon
D2×2	A12×2	Template:Brackets = [4]	Template:CDD	Template:CDD = Template:CDD	8
D3×2	A2×2	Template:Brackets = [6]	Template:CDD	Template:CDD = Template:CDD	12
D4×2	BC2×2	Template:Brackets = [8]	Template:CDD	Template:CDD = Template:CDD	16
D5×2	H2×2	Template:Brackets = [10]	Template:CDD	Template:CDD = Template:CDD	20
D6×2	G2×2	Template:Brackets = [12]	Template:CDD	Template:CDD = Template:CDD	24
Dn×2	I2(n)×2	Template:Brackets = [2n]	Template:CDD	Template:CDD = Template:CDD	4n

Three dimensions

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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 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.

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	...
File:Spherical digonal hosohedron2.png	File:Spherical square hosohedron2.png	File:Spherical hexagonal hosohedron2.png	File:Spherical octagonal hosohedron2.png	File:Spherical decagonal hosohedron2.png	File:Spherical dodecagonal hosohedron2.png
D1h Order 4	D2h Order 8	D3h Order 12	D4h Order 16	D5h Order 20	D6h Order 24	...
File:Spherical digonal bipyramid2.svg	File:Spherical square bipyramid2.svg	File:Spherical hexagonal bipyramid2.png	File:Spherical octagonal bipyramid2.png	File:Spherical decagonal bipyramid2.png	File:Spherical dodecagonal bipyramid2.png
Td Order 24	Oh Order 48	Ih Order 120
File:Tetrahedral reflection domains.png	File:Octahedral reflection domains.png	File:Icosahedral reflection domains.png

Intl* Geo Template:Refn Orbifold Schoenflies Coxeter Order 1 Template:Overline 1 C1 [ ]+ 1 Template:Overline Template:Overline ×1 Ci = S2 [2+,2+] 2 Template:Overline = m 1 *1 Cs = C1v = C1h [ ] 2 2 3 4 5 6 n Template:Overline Template:Overline Template:Overline Template:Overline Template:Overline Template:Overline 22 33 44 55 66 nn C2 C3 C4 C5 C6 Cn [2]+ [3]+ [4]+ [5]+ [6]+ [n]+ 2 3 4 5 6 n mm2 3m 4mm 5m 6mm nmm nm 2 3 4 5 6 n *22 *33 *44 *55 *66 *nn C2v C3v C4v C5v C6v Cnv [2] [3] [4] [5] [6] [n] 4 6 8 10 12 2n 2/m Template:Overline 4/m Template:Overline 6/m n/m Template:Overline Template:Overline 2 Template:Overline 2 Template:Overline 2 Template:Overline 2 Template:Overline 2 Template:Overline 2 2* 3* 4* 5* 6* n* C2h C3h C4h C5h C6h Cnh [2,2+] [2,3+] [2,4+] [2,5+] [2,6+] [2,n+] 4 6 8 10 12 2n Template:Overline Template:Overline Template:Overline Template:Overline Template:Overline Template:Overline Template:Overline Template:Overline Template:Overline Template:Overline Template:Overline Template:Overline Template:Overline 2× 3× 4× 5× 6× n× S4 S6 S8 S10 S12 S2n [2+,4+] [2+,6+] [2+,8+] [2+,10+] [2+,12+] [2+,2n+] 4 6 8 10 12 2n

Intl Geo Orbifold Schoenflies Coxeter Order 222 32 422 52 622 n22 n2 Template:Overline Template:Overline Template:Overline Template:Overline Template:Overline Template:Overline Template:Overline Template:Overline Template:Overline Template:Overline Template:Overline Template:Overline 222 223 224 225 226 22n D2 D3 D4 D5 D6 Dn [2,2]+ [2,3]+ [2,4]+ [2,5]+ [2,6]+ [2,n]+ 4 6 8 10 12 2n mmm Template:Overlinem2 4/mmm Template:Overlinem2 6/mmm n/mmm Template:Overlinem2 2 2 3 2 4 2 5 2 6 2 n 2 *222 *223 *224 *225 *226 *22n D2h D3h D4h D5h D6h Dnh [2,2] [2,3] [2,4] [2,5] [2,6] [2,n] 8 12 16 20 24 4n Template:Overline2m Template:Overlinem Template:Overline2m Template:Overlinem Template:Overline2m Template:Overline2m Template:Overlinem 4 Template:Overline 6 Template:Overline 8 Template:Overline 10 Template:Overline 12 Template:Overline n Template:Overline 2*2 2*3 2*4 2*5 2*6 2*n D2d D3d D4d D5d D6d Dnd [2+,4] [2+,6] [2+,8] [2+,10] [2+,12] [2+,2n] 8 12 16 20 24 4n 23 Template:Overline Template:Overline 332 T [3,3]+ 12 mTemplate:Overline 4 Template:Overline 3*2 Th [3+,4] 24 Template:Overline3m 3 3 *332 Td [3,3] 24 432 Template:Overline Template:Overline 432 O [3,4]+ 24 mTemplate:Overlinem 4 3 *432 Oh [3,4] 48 532 Template:Overline Template:Overline 532 I [3,5]+ 60 Template:OverlineTemplate:Overlinem 5 3 *532 Ih [3,5] 120

(*) When the Intl entries are duplicated, the first is for even n, the second for odd n.

Reflection groups

File:Coxeter diagram finite rank3 correspondence.png

Finite isomorphism and correspondences

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 Template:Brackets, mapping the first and last mirrors onto each other, doubling the symmetry to 48, and isomorphic to the [4,3] group.

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

Four dimensions

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The four-dimensional point groups (chiral as well as achiral) are listed in Conway and Smith,<ref name="Conway-Smith"/> Section 4, Tables 4.1–4.3.

File:Coxeter diagram finite rank4 correspondence.png

Finite isomorphism and correspondences

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 Template:Brackets with its order doubled to 240.

Coxeter group/notation		Coxeter diagram		Order	Related polytopes
A4	[3,3,3]	Template:CDD		120	5-cell
A4×2	Template:Brackets	Template:CDD		240	5-cell dual compound
BC4	[4,3,3]	Template:CDD		384	16-cell / tesseract
D4	[31,1,1]	Template:CDD		192	demitesseractic
D4×2 = BC4	<[3,31,1]> = [4,3,3]	Template:CDD	= Template:CDD	384
D4×6 = F4	[3[31,1,1]] = [3,4,3]	Template:CDD	= Template:CDD	1152
F4	[3,4,3]	Template:CDD		1152	24-cell
F4×2	Template:Brackets	Template:CDD		2304	24-cell dual compound
H4	[5,3,3]	Template:CDD		14400	120-cell / 600-cell
A3×A1	[3,3,2]	Template:CDD		48	tetrahedral prism
A3×A1×2	[[3,3],2] = [4,3,2]	Template:CDD	= Template:CDD	96	octahedral prism
BC3×A1	[4,3,2]	Template:CDD		96
H3×A1	[5,3,2]	Template:CDD		240	icosahedral prism
A2×A2	[3,2,3]	Template:CDD		36	duoprism
A2×BC2	[3,2,4]	Template:CDD		48
A2×H2	[3,2,5]	Template:CDD		60
A2×G2	[3,2,6]	Template:CDD		72
BC2×BC2	[4,2,4]	Template:CDD		64
BC22×2	Template:Brackets	Template:CDD		128
BC2×H2	[4,2,5]	Template:CDD		80
BC2×G2	[4,2,6]	Template:CDD		96
H2×H2	[5,2,5]	Template:CDD		100
H2×G2	[5,2,6]	Template:CDD		120
G2×G2	[6,2,6]	Template:CDD		144
I2(p)×I2(q)	[p,2,q]	Template:CDD		4pq
I2(2p)×I2(q)	[[p],2,q] = [2p,2,q]	Template:CDD	= Template:CDD	8pq
I2(2p)×I2(2q)	Template:Brackets,2,Template:Brackets = [2p,2,2q]	Template:CDD	= Template:CDD	16pq
I2(p)2×2	Template:Brackets	Template:CDD		8p2
I2(2p)2×2	[[[p]],2,[p]]] = Template:Brackets	Template:CDD	= Template:CDD	32p2
A2×A1×A1	[3,2,2]	Template:CDD		24
BC2×A1×A1	[4,2,2]	Template:CDD		32
H2×A1×A1	[5,2,2]	Template:CDD		40
G2×A1×A1	[6,2,2]	Template:CDD		48
I2(p)×A1×A1	[p,2,2]	Template:CDD		8p
I2(2p)×A1×A1×2	[[p],2,2] = [2p,2,2]	Template:CDD	= Template:CDD	16p
I2(p)×A12×2	[p,2,[2]] = [p,2,4]	Template:CDD	= Template:CDD	16p
I2(2p)×A12×4	Template:Brackets,2,Template:Brackets = [2p,2,4]	Template:CDD	= Template:CDD	32p
A1×A1×A1×A1	[2,2,2]	Template:CDD		16	4-orthotope
A12×A1×A1×2	[[2],2,2] = [4,2,2]	Template:CDD	= Template:CDD	32
A12×A12×4	Template:Brackets,2,Template:Brackets = [4,2,4]	Template:CDD	= Template:CDD	64
A13×A1×6	[3[2,2],2] = [4,3,2]	Template:CDD	= Template:CDD	96
A14×24	[3,3[2,2,2]] = [4,3,3]	Template:CDD	= Template:CDD	384

Five dimensions

File:Coxeter diagram finite rank5 correspondence.png

Finite isomorphism and correspondences

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]	Template:CDD	Template:CDD	720	5-simplex
A5×2	Template:Brackets	Template:CDD	Template:CDD	1440	5-simplex dual compound
BC5	[4,3,3,3]	Template:CDD	Template:CDD	3840	5-cube, 5-orthoplex
D5	[32,1,1]	Template:CDD	Template:CDD	1920	5-demicube
D5×2	<[3,3,31,1]>	Template:CDD	Template:CDD = Template:CDD	3840
A4×A1	[3,3,3,2]	Template:CDD	Template:CDD	240	5-cell prism
A4×A1×2	[[3,3,3],2]	Template:CDD	Template:CDD	480
BC4×A1	[4,3,3,2]	Template:CDD	Template:CDD	768	tesseract prism
F4×A1	[3,4,3,2]	Template:CDD	Template:CDD	2304	24-cell prism
F4×A1×2	[[3,4,3],2]	Template:CDD	Template:CDD	4608
H4×A1	[5,3,3,2]	Template:CDD	Template:CDD	28800	600-cell or 120-cell prism
D4×A1	[31,1,1,2]	Template:CDD	Template:CDD	384	demitesseract prism
A3×A2	[3,3,2,3]	Template:CDD	Template:CDD	144	duoprism
A3×A2×2	[[3,3],2,3]	Template:CDD	Template:CDD	288
A3×BC2	[3,3,2,4]	Template:CDD	Template:CDD	192
A3×H2	[3,3,2,5]	Template:CDD	Template:CDD	240
A3×G2	[3,3,2,6]	Template:CDD	Template:CDD	288
A3×I2(p)	[3,3,2,p]	Template:CDD	Template:CDD	48p
BC3×A2	[4,3,2,3]	Template:CDD	Template:CDD	288
BC3×BC2	[4,3,2,4]	Template:CDD	Template:CDD	384
BC3×H2	[4,3,2,5]	Template:CDD	Template:CDD	480
BC3×G2	[4,3,2,6]	Template:CDD	Template:CDD	576
BC3×I2(p)	[4,3,2,p]	Template:CDD	Template:CDD	96p
H3×A2	[5,3,2,3]	Template:CDD	Template:CDD	720
H3×BC2	[5,3,2,4]	Template:CDD	Template:CDD	960
H3×H2	[5,3,2,5]	Template:CDD	Template:CDD	1200
H3×G2	[5,3,2,6]	Template:CDD	Template:CDD	1440
H3×I2(p)	[5,3,2,p]	Template:CDD		240p
A3×A12	[3,3,2,2]	Template:CDD		96
BC3×A12	[4,3,2,2]	Template:CDD		192
H3×A12	[5,3,2,2]	Template:CDD		480
A22×A1	[3,2,3,2]	Template:CDD		72	duoprism prism
A2×BC2×A1	[3,2,4,2]	Template:CDD		96
A2×H2×A1	[3,2,5,2]	Template:CDD		120
A2×G2×A1	[3,2,6,2]	Template:CDD		144
BC22×A1	[4,2,4,2]	Template:CDD		128
BC2×H2×A1	[4,2,5,2]	Template:CDD		160
BC2×G2×A1	[4,2,6,2]	Template:CDD		192
H22×A1	[5,2,5,2]	Template:CDD		200
H2×G2×A1	[5,2,6,2]	Template:CDD		240
G22×A1	[6,2,6,2]	Template:CDD		288
I2(p)×I2(q)×A1	[p,2,q,2]	Template:CDD		8pq
A2×A13	[3,2,2,2]	Template:CDD		48
BC2×A13	[4,2,2,2]	Template:CDD		64
H2×A13	[5,2,2,2]	Template:CDD		80
G2×A13	[6,2,2,2]	Template:CDD		96
I2(p)×A13	[p,2,2,2]	Template:CDD		16p
A15	[2,2,2,2]	Template:CDD	Template:CDD	32	5-orthotope
A15×(2!)	[[2],2,2,2]	Template:CDD	Template:CDD = Template:CDD	64
A15×(2!×2!)	Template:Brackets,2,[2],2]	Template:CDD	Template:CDD = Template:CDD	128
A15×(3!)	[3[2,2],2,2]	Template:CDD	Template:CDD = Template:CDD	192
A15×(3!×2!)	[3[2,2],2,Template:Brackets	Template:CDD	Template:CDD = Template:CDD	384
A15×(4!)	[3,3[2,2,2],2]]	Template:CDD	Template:CDD = Template:CDD	768
A15×(5!)	[3,3,3[2,2,2,2]]	Template:CDD	Template:CDD = Template:CDD	3840

Six dimensions

File:Coxeter diagram finite rank6 correspondence.png

Finite isomorphism and correspondences

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]	Template:CDD	5040 (7!)	6-simplex
A6×2	Template:Brackets	Template:CDD	10080 (2×7!)	6-simplex dual compound
BC6	[4,3,3,3,3]	Template:CDD	46080 (26×6!)	6-cube, 6-orthoplex
D6	[3,3,3,31,1]	Template:CDD	23040 (25×6!)	6-demicube
E6	[3,32,2]	Template:CDD	51840 (72×6!)	122, 221
A5×A1	[3,3,3,3,2]	Template:CDD	1440 (2×6!)	5-simplex prism
BC5×A1	[4,3,3,3,2]	Template:CDD	7680 (26×5!)	5-cube prism
D5×A1	[3,3,31,1,2]	Template:CDD	3840 (25×5!)	5-demicube prism
A4×I2(p)	[3,3,3,2,p]	Template:CDD	240p	duoprism
BC4×I2(p)	[4,3,3,2,p]	Template:CDD	768p
F4×I2(p)	[3,4,3,2,p]	Template:CDD	2304p
H4×I2(p)	[5,3,3,2,p]	Template:CDD	28800p
D4×I2(p)	[3,31,1,2,p]	Template:CDD	384p
A4×A12	[3,3,3,2,2]	Template:CDD	480
BC4×A12	[4,3,3,2,2]	Template:CDD	1536
F4×A12	[3,4,3,2,2]	Template:CDD	4608
H4×A12	[5,3,3,2,2]	Template:CDD	57600
D4×A12	[3,31,1,2,2]	Template:CDD	768
A32	[3,3,2,3,3]	Template:CDD	576
A3×BC3	[3,3,2,4,3]	Template:CDD	1152
A3×H3	[3,3,2,5,3]	Template:CDD	2880
BC32	[4,3,2,4,3]	Template:CDD	2304
BC3×H3	[4,3,2,5,3]	Template:CDD	5760
H32	[5,3,2,5,3]	Template:CDD	14400
A3×I2(p)×A1	[3,3,2,p,2]	Template:CDD	96p	duoprism prism
BC3×I2(p)×A1	[4,3,2,p,2]	Template:CDD	192p
H3×I2(p)×A1	[5,3,2,p,2]	Template:CDD	480p
A3×A13	[3,3,2,2,2]	Template:CDD	192
BC3×A13	[4,3,2,2,2]	Template:CDD	384
H3×A13	[5,3,2,2,2]	Template:CDD	960
I2(p)×I2(q)×I2(r)	[p,2,q,2,r]	Template:CDD	8pqr	triaprism
I2(p)×I2(q)×A12	[p,2,q,2,2]	Template:CDD	16pq
I2(p)×A14	[p,2,2,2,2]	Template:CDD	32p
A16	[2,2,2,2,2]	Template:CDD	64	6-orthotope

Seven dimensions

The following table gives the seven-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, 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 3-fold gyration points and symmetry order 20160.

Coxeter group		Coxeter diagram	Order	Related polytopes
A7	[3,3,3,3,3,3]	Template:CDD	40320 (8!)	7-simplex
A7×2	Template:Brackets	Template:CDD	80640 (2×8!)	7-simplex dual compound
BC7	[4,3,3,3,3,3]	Template:CDD	645120 (27×7!)	7-cube, 7-orthoplex
D7	[3,3,3,3,31,1]	Template:CDD	322560 (26×7!)	7-demicube
E7	[3,3,3,32,1]	Template:CDD	2903040 (8×9!)	321, 231, 132
A6×A1	[3,3,3,3,3,2]	Template:CDD	10080 (2×7!)
BC6×A1	[4,3,3,3,3,2]	Template:CDD	92160 (27×6!)
D6×A1	[3,3,3,31,1,2]	Template:CDD	46080 (26×6!)
E6×A1	[3,3,32,1,2]	Template:CDD	103680 (144×6!)
A5×I2(p)	[3,3,3,3,2,p]	Template:CDD	1440p
BC5×I2(p)	[4,3,3,3,2,p]	Template:CDD	7680p
D5×I2(p)	[3,3,31,1,2,p]	Template:CDD	3840p
A5×A12	[3,3,3,3,2,2]	Template:CDD	2880
BC5×A12	[4,3,3,3,2,2]	Template:CDD	15360
D5×A12	[3,3,31,1,2,2]	Template:CDD	7680
A4×A3	[3,3,3,2,3,3]	Template:CDD	2880
A4×BC3	[3,3,3,2,4,3]	Template:CDD	5760
A4×H3	[3,3,3,2,5,3]	Template:CDD	14400
BC4×A3	[4,3,3,2,3,3]	Template:CDD	9216
BC4×BC3	[4,3,3,2,4,3]	Template:CDD	18432
BC4×H3	[4,3,3,2,5,3]	Template:CDD	46080
H4×A3	[5,3,3,2,3,3]	Template:CDD	345600
H4×BC3	[5,3,3,2,4,3]	Template:CDD	691200
H4×H3	[5,3,3,2,5,3]	Template:CDD	1728000
F4×A3	[3,4,3,2,3,3]	Template:CDD	27648
F4×BC3	[3,4,3,2,4,3]	Template:CDD	55296
F4×H3	[3,4,3,2,5,3]	Template:CDD	138240
D4×A3	[31,1,1,2,3,3]	Template:CDD	4608
D4×BC3	[3,31,1,2,4,3]	Template:CDD	9216
D4×H3	[3,31,1,2,5,3]	Template:CDD	23040
A4×I2(p)×A1	[3,3,3,2,p,2]	Template:CDD	480p
BC4×I2(p)×A1	[4,3,3,2,p,2]	Template:CDD	1536p
D4×I2(p)×A1	[3,31,1,2,p,2]	Template:CDD	768p
F4×I2(p)×A1	[3,4,3,2,p,2]	Template:CDD	4608p
H4×I2(p)×A1	[5,3,3,2,p,2]	Template:CDD	57600p
A4×A13	[3,3,3,2,2,2]	Template:CDD	960
BC4×A13	[4,3,3,2,2,2]	Template:CDD	3072
F4×A13	[3,4,3,2,2,2]	Template:CDD	9216
H4×A13	[5,3,3,2,2,2]	Template:CDD	115200
D4×A13	[3,31,1,2,2,2]	Template:CDD	1536
A32×A1	[3,3,2,3,3,2]	Template:CDD	1152
A3×BC3×A1	[3,3,2,4,3,2]	Template:CDD	2304
A3×H3×A1	[3,3,2,5,3,2]	Template:CDD	5760
BC32×A1	[4,3,2,4,3,2]	Template:CDD	4608
BC3×H3×A1	[4,3,2,5,3,2]	Template:CDD	11520
H32×A1	[5,3,2,5,3,2]	Template:CDD	28800
A3×I2(p)×I2(q)	[3,3,2,p,2,q]	Template:CDD	96pq
BC3×I2(p)×I2(q)	[4,3,2,p,2,q]	Template:CDD	192pq
H3×I2(p)×I2(q)	[5,3,2,p,2,q]	Template:CDD	480pq
A3×I2(p)×A12	[3,3,2,p,2,2]	Template:CDD	192p
BC3×I2(p)×A12	[4,3,2,p,2,2]	Template:CDD	384p
H3×I2(p)×A12	[5,3,2,p,2,2]	Template:CDD	960p
A3×A14	[3,3,2,2,2,2]	Template:CDD	384
BC3×A14	[4,3,2,2,2,2]	Template:CDD	768
H3×A14	[5,3,2,2,2,2]	Template:CDD	1920
I2(p)×I2(q)×I2(r)×A1	[p,2,q,2,r,2]	Template:CDD	16pqr
I2(p)×I2(q)×A13	[p,2,q,2,2,2]	Template:CDD	32pq
I2(p)×A15	[p,2,2,2,2,2]	Template:CDD	64p
A17	[2,2,2,2,2,2]	Template:CDD	128

Eight dimensions

The following table gives the eight-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, 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 3-fold gyration points and symmetry order 181440.

Coxeter group		Coxeter diagram	Order	Related polytopes
A8	[3,3,3,3,3,3,3]	Template:CDD	362880 (9!)	8-simplex
A8×2	Template:Brackets	Template:CDD	725760 (2×9!)	8-simplex dual compound
BC8	[4,3,3,3,3,3,3]	Template:CDD	10321920 (288!)	8-cube, 8-orthoplex
D8	[3,3,3,3,3,31,1]	Template:CDD	5160960 (278!)	8-demicube
E8	[3,3,3,3,32,1]	Template:CDD	696729600 (192×10!)	421, 241, 142
A7×A1	[3,3,3,3,3,3,2]	Template:CDD	80640	7-simplex prism
BC7×A1	[4,3,3,3,3,3,2]	Template:CDD	645120	7-cube prism
D7×A1	[3,3,3,3,31,1,2]	Template:CDD	322560	7-demicube prism
E7×A1	[3,3,3,32,1,2]	Template:CDD	5806080	321 prism, 231 prism, 142 prism
A6×I2(p)	[3,3,3,3,3,2,p]	Template:CDD	10080p	duoprism
BC6×I2(p)	[4,3,3,3,3,2,p]	Template:CDD	92160p
D6×I2(p)	[3,3,3,31,1,2,p]	Template:CDD	46080p
E6×I2(p)	[3,3,32,1,2,p]	Template:CDD	103680p
A6×A12	[3,3,3,3,3,2,2]	Template:CDD	20160
BC6×A12	[4,3,3,3,3,2,2]	Template:CDD	184320
D6×A12	[33,1,1,2,2]	Template:CDD	92160
E6×A12	[3,3,32,1,2,2]	Template:CDD	207360
A5×A3	[3,3,3,3,2,3,3]	Template:CDD	17280
BC5×A3	[4,3,3,3,2,3,3]	Template:CDD	92160
D5×A3	[32,1,1,2,3,3]	Template:CDD	46080
A5×BC3	[3,3,3,3,2,4,3]	Template:CDD	34560
BC5×BC3	[4,3,3,3,2,4,3]	Template:CDD	184320
D5×BC3	[32,1,1,2,4,3]	Template:CDD	92160
A5×H3	[3,3,3,3,2,5,3]	Template:CDD
BC5×H3	[4,3,3,3,2,5,3]	Template:CDD
D5×H3	[32,1,1,2,5,3]	Template:CDD
A5×I2(p)×A1	[3,3,3,3,2,p,2]	Template:CDD
BC5×I2(p)×A1	[4,3,3,3,2,p,2]	Template:CDD
D5×I2(p)×A1	[32,1,1,2,p,2]	Template:CDD
A5×A13	[3,3,3,3,2,2,2]	Template:CDD
BC5×A13	[4,3,3,3,2,2,2]	Template:CDD
D5×A13	[32,1,1,2,2,2]	Template:CDD
A4×A4	[3,3,3,2,3,3,3]	Template:CDD
BC4×A4	[4,3,3,2,3,3,3]	Template:CDD
D4×A4	[31,1,1,2,3,3,3]	Template:CDD
F4×A4	[3,4,3,2,3,3,3]	Template:CDD
H4×A4	[5,3,3,2,3,3,3]	Template:CDD
BC4×BC4	[4,3,3,2,4,3,3]	Template:CDD
D4×BC4	[31,1,1,2,4,3,3]	Template:CDD
F4×BC4	[3,4,3,2,4,3,3]	Template:CDD
H4×BC4	[5,3,3,2,4,3,3]	Template:CDD
D4×D4	[31,1,1,2,31,1,1]	Template:CDD
F4×D4	[3,4,3,2,31,1,1]	Template:CDD
H4×D4	[5,3,3,2,31,1,1]	Template:CDD
F4×F4	[3,4,3,2,3,4,3]	Template:CDD
H4×F4	[5,3,3,2,3,4,3]	Template:CDD
H4×H4	[5,3,3,2,5,3,3]	Template:CDD
A4×A3×A1	[3,3,3,2,3,3,2]	Template:CDD		duoprism prisms
A4×BC3×A1	[3,3,3,2,4,3,2]	Template:CDD
A4×H3×A1	[3,3,3,2,5,3,2]	Template:CDD
BC4×A3×A1	[4,3,3,2,3,3,2]	Template:CDD
BC4×BC3×A1	[4,3,3,2,4,3,2]	Template:CDD
BC4×H3×A1	[4,3,3,2,5,3,2]	Template:CDD
H4×A3×A1	[5,3,3,2,3,3,2]	Template:CDD
H4×BC3×A1	[5,3,3,2,4,3,2]	Template:CDD
H4×H3×A1	[5,3,3,2,5,3,2]	Template:CDD
F4×A3×A1	[3,4,3,2,3,3,2]	Template:CDD
F4×BC3×A1	[3,4,3,2,4,3,2]	Template:CDD
F4×H3×A1	[3,4,2,3,5,3,2]	Template:CDD
D4×A3×A1	[31,1,1,2,3,3,2]	Template:CDD
D4×BC3×A1	[31,1,1,2,4,3,2]	Template:CDD
D4×H3×A1	[31,1,1,2,5,3,2]	Template:CDD
A4×I2(p)×I2(q)	[3,3,3,2,p,2,q]	Template:CDD		triaprism
BC4×I2(p)×I2(q)	[4,3,3,2,p,2,q]	Template:CDD
F4×I2(p)×I2(q)	[3,4,3,2,p,2,q]	Template:CDD
H4×I2(p)×I2(q)	[5,3,3,2,p,2,q]	Template:CDD
D4×I2(p)×I2(q)	[31,1,1,2,p,2,q]	Template:CDD
A4×I2(p)×A12	[3,3,3,2,p,2,2]	Template:CDD
BC4×I2(p)×A12	[4,3,3,2,p,2,2]	Template:CDD
F4×I2(p)×A12	[3,4,3,2,p,2,2]	Template:CDD
H4×I2(p)×A12	[5,3,3,2,p,2,2]	Template:CDD
D4×I2(p)×A12	[31,1,1,2,p,2,2]	Template:CDD
A4×A14	[3,3,3,2,2,2,2]	Template:CDD
BC4×A14	[4,3,3,2,2,2,2]	Template:CDD
F4×A14	[3,4,3,2,2,2,2]	Template:CDD
H4×A14	[5,3,3,2,2,2,2]	Template:CDD
D4×A14	[31,1,1,2,2,2,2]	Template:CDD
A3×A3×I2(p)	[3,3,2,3,3,2,p]	Template:CDD
BC3×A3×I2(p)	[4,3,2,3,3,2,p]	Template:CDD
H3×A3×I2(p)	[5,3,2,3,3,2,p]	Template:CDD
BC3×BC3×I2(p)	[4,3,2,4,3,2,p]	Template:CDD
H3×BC3×I2(p)	[5,3,2,4,3,2,p]	Template:CDD
H3×H3×I2(p)	[5,3,2,5,3,2,p]	Template:CDD
A3×A3×A12	[3,3,2,3,3,2,2]	Template:CDD
BC3×A3×A12	[4,3,2,3,3,2,2]	Template:CDD
H3×A3×A12	[5,3,2,3,3,2,2]	Template:CDD
BC3×BC3×A12	[4,3,2,4,3,2,2]	Template:CDD
H3×BC3×A12	[5,3,2,4,3,2,2]	Template:CDD
H3×H3×A12	[5,3,2,5,3,2,2]	Template:CDD
A3×I2(p)×I2(q)×A1	[3,3,2,p,2,q,2]	Template:CDD
BC3×I2(p)×I2(q)×A1	[4,3,2,p,2,q,2]	Template:CDD
H3×I2(p)×I2(q)×A1	[5,3,2,p,2,q,2]	Template:CDD
A3×I2(p)×A13	[3,3,2,p,2,2,2]	Template:CDD
BC3×I2(p)×A13	[4,3,2,p,2,2,2]	Template:CDD
H3×I2(p)×A13	[5,3,2,p,2,2,2]	Template:CDD
A3×A15	[3,3,2,2,2,2,2]	Template:CDD
BC3×A15	[4,3,2,2,2,2,2]	Template:CDD
H3×A15	[5,3,2,2,2,2,2]	Template:CDD
I2(p)×I2(q)×I2(r)×I2(s)	[p,2,q,2,r,2,s]	Template:CDD	16pqrs
I2(p)×I2(q)×I2(r)×A12	[p,2,q,2,r,2,2]	Template:CDD	32pqr
I2(p)×I2(q)×A14	[p,2,q,2,2,2,2]	Template:CDD	64pq
I2(p)×A16	[p,2,2,2,2,2,2]	Template:CDD	128p
A18	[2,2,2,2,2,2,2]	Template:CDD	256

See also

• Point groups in two dimensions
• Point groups in three dimensions
• Point groups in four dimensions
• Crystallography
• Crystallographic point group
• Molecular symmetry
• Space group
• X-ray diffraction
• Bravais lattice
• Infrared spectroscopy of metal carbonyls

Further reading

• Template:Citation
  • (Paper 23) H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559–591]
• Template:Citation
• Template:Citation

External links

• Web-based point group tutorial (needs Java and Flash)
• Subgroup enumeration (needs Java)
• The Geometry Center: 2.1 Formulas for Symmetries in Cartesian Coordinates (two dimensions)
• The Geometry Center: 10.1 Formulas for Symmetries in Cartesian Coordinates (three dimensions)