Point group

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.

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

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.

Chiral and achiral point groups, reflection 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).

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

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

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	[ ]			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

Three dimensions

Main article: Point groups in three dimensions

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

Intl* Geo [2] Orbifold Schönflies Coxeter Order 1 1 1 C1 [ ]+ 1 1 22 ×1 Ci = S2 [2+,2+] 2 2 = m 1 *1 Cs = C1v = C1h [ ] 2 2 3 4 5 6 n 2 3 4 5 6 n 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 6 4/m 10 6/m n/m 2n 2 2 3 2 4 2 5 2 6 2 n 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 4 3 8 5 12 2n n 4 2 6 2 8 2 10 2 12 2 2n 2 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 Schönflies Coxeter Order 222 32 422 52 622 n22 n2 2 2 3 2 4 2 5 2 6 2 n 2 222 223 224 225 226 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 6m2 4/mmm 10m2 6/mmm n/mmm 2nm2 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 42m 3m 82m 5m 122m 2n2m nm 4 2 6 2 8 2 10 2 12 2 n 2 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 3 3 332 T [3,3]+ 12 m3 4 3 3*2 Th [3+,4] 24 43m 3 3 *332 Td [3,3] 24 432 4 3 432 O [3,4]+ 24 m3m 4 3 *432 Oh [3,4] 48 532 5 3 532 I [3,5]+ 60 53m 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

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 [[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

Four dimensions

Main article: Point groups in four dimensions

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

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 [[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

Five dimensions

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]			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

Six dimensions

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]		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

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]		40320 (8!)	7-simplex
A7×2	[[3,3,3,3,3,3]]		80640 (2×8!)	7-simplex dual compound
BC7	[4,3,3,3,3,3]		645120 (27×7!)	7-cube, 7-orthoplex
D7	[3,3,3,3,31,1]		322560 (26×7!)	7-demicube
E7	[3,3,3,32,1]		2903040 (8×9!)	321, 231, 132
A6×A1	[3,3,3,3,3,2]		10080 (2×7!)
BC6×A1	[4,3,3,3,3,2]		92160 (27×6!)
D6×A1	[3,3,3,31,1,2]		46080 (26×6!)
E6×A1	[3,3,32,1,2]		103680 (144×6!)
A5×I2(p)	[3,3,3,3,2,p]		1440p
BC5×I2(p)	[4,3,3,3,2,p]		7680p
D5×I2(p)	[3,3,31,1,2,p]		3840p
A5×A12	[3,3,3,3,2,2]		2880
BC5×A12	[4,3,3,3,2,2]		15360
D5×A12	[3,3,31,1,2,2]		7680
A4×A3	[3,3,3,2,3,3]		2880
A4×BC3	[3,3,3,2,4,3]		5760
A4×H3	[3,3,3,2,5,3]		14400
BC4×A3	[4,3,3,2,3,3]		9216
BC4×BC3	[4,3,3,2,4,3]		18432
BC4×H3	[4,3,3,2,5,3]		46080
H4×A3	[5,3,3,2,3,3]		345600
H4×BC3	[5,3,3,2,4,3]		691200
H4×H3	[5,3,3,2,5,3]		1728000
F4×A3	[3,4,3,2,3,3]		27648
F4×BC3	[3,4,3,2,4,3]		55296
F4×H3	[3,4,3,2,5,3]		138240
D4×A3	[31,1,1,2,3,3]		4608
D4×BC3	[3,31,1,2,4,3]		9216
D4×H3	[3,31,1,2,5,3]		23040
A4×I2(p)×A1	[3,3,3,2,p,2]		480p
BC4×I2(p)×A1	[4,3,3,2,p,2]		1536p
D4×I2(p)×A1	[3,31,1,2,p,2]		768p
F4×I2(p)×A1	[3,4,3,2,p,2]		4608p
H4×I2(p)×A1	[5,3,3,2,p,2]		57600p
A4×A13	[3,3,3,2,2,2]		960
BC4×A13	[4,3,3,2,2,2]		3072
F4×A13	[3,4,3,2,2,2]		9216
H4×A13	[5,3,3,2,2,2]		115200
D4×A13	[3,31,1,2,2,2]		1536
A32×A1	[3,3,2,3,3,2]		1152
A3×BC3×A1	[3,3,2,4,3,2]		2304
A3×H3×A1	[3,3,2,5,3,2]		5760
BC32×A1	[4,3,2,4,3,2]		4608
BC3×H3×A1	[4,3,2,5,3,2]		11520
H32×A1	[5,3,2,5,3,2]		28800
A3×I2(p)×I2(q)	[3,3,2,p,2,q]		96pq
BC3×I2(p)×I2(q)	[4,3,2,p,2,q]		192pq
H3×I2(p)×I2(q)	[5,3,2,p,2,q]		480pq
A3×I2(p)×A12	[3,3,2,p,2,2]		192p
BC3×I2(p)×A12	[4,3,2,p,2,2]		384p
H3×I2(p)×A12	[5,3,2,p,2,2]		960p
A3×A14	[3,3,2,2,2,2]		384
BC3×A14	[4,3,2,2,2,2]		768
H3×A14	[5,3,2,2,2,2]		1920
I2(p)×I2(q)×I2(r)×A1	[p,2,q,2,r,2]		16pqr
I2(p)×I2(q)×A13	[p,2,q,2,2,2]		32pq
I2(p)×A15	[p,2,2,2,2,2]		64p
A17	[2,2,2,2,2,2]		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]		362880 (9!)	8-simplex
A8×2	[[3,3,3,3,3,3,3]]		725760 (2×9!)	8-simplex dual compound
BC8	[4,3,3,3,3,3,3]		10321920 (288!)	8-cube, 8-orthoplex
D8	[3,3,3,3,3,31,1]		5160960 (278!)	8-demicube
E8	[3,3,3,3,32,1]		696729600 (192×10!)	421, 241, 142
A7×A1	[3,3,3,3,3,3,2]		80640	7-simplex prism
BC7×A1	[4,3,3,3,3,3,2]		645120	7-cube prism
D7×A1	[3,3,3,3,31,1,2]		322560	7-demicube prism
E7×A1	[3,3,3,32,1,2]		5806080	321 prism, 231 prism, 142 prism
A6×I2(p)	[3,3,3,3,3,2,p]		10080p	duoprism
BC6×I2(p)	[4,3,3,3,3,2,p]		92160p
D6×I2(p)	[3,3,3,31,1,2,p]		46080p
E6×I2(p)	[3,3,32,1,2,p]		103680p
A6×A12	[3,3,3,3,3,2,2]		20160
BC6×A12	[4,3,3,3,3,2,2]		184320
D6×A12	[33,1,1,2,2]		92160
E6×A12	[3,3,32,1,2,2]		207360
A5×A3	[3,3,3,3,2,3,3]		17280
BC5×A3	[4,3,3,3,2,3,3]		92160
D5×A3	[32,1,1,2,3,3]		46080
A5×BC3	[3,3,3,3,2,4,3]		34560
BC5×BC3	[4,3,3,3,2,4,3]		184320
D5×BC3	[32,1,1,2,4,3]		92160
A5×H3	[3,3,3,3,2,5,3]
BC5×H3	[4,3,3,3,2,5,3]
D5×H3	[32,1,1,2,5,3]
A5×I2(p)×A1	[3,3,3,3,2,p,2]
BC5×I2(p)×A1	[4,3,3,3,2,p,2]
D5×I2(p)×A1	[32,1,1,2,p,2]
A5×A13	[3,3,3,3,2,2,2]
BC5×A13	[4,3,3,3,2,2,2]
D5×A13	[32,1,1,2,2,2]
A4×A4	[3,3,3,2,3,3,3]
BC4×A4	[4,3,3,2,3,3,3]
D4×A4	[31,1,1,2,3,3,3]
F4×A4	[3,4,3,2,3,3,3]
H4×A4	[5,3,3,2,3,3,3]
BC4×BC4	[4,3,3,2,4,3,3]
D4×BC4	[31,1,1,2,4,3,3]
F4×BC4	[3,4,3,2,4,3,3]
H4×BC4	[5,3,3,2,4,3,3]
D4×D4	[31,1,1,2,31,1,1]
F4×D4	[3,4,3,2,31,1,1]
H4×D4	[5,3,3,2,31,1,1]
F4×F4	[3,4,3,2,3,4,3]
H4×F4	[5,3,3,2,3,4,3]
H4×H4	[5,3,3,2,5,3,3]
A4×A3×A1	[3,3,3,2,3,3,2]			duoprism prisms
A4×BC3×A1	[3,3,3,2,4,3,2]
A4×H3×A1	[3,3,3,2,5,3,2]
BC4×A3×A1	[4,3,3,2,3,3,2]
BC4×BC3×A1	[4,3,3,2,4,3,2]
BC4×H3×A1	[4,3,3,2,5,3,2]
H4×A3×A1	[5,3,3,2,3,3,2]
H4×BC3×A1	[5,3,3,2,4,3,2]
H4×H3×A1	[5,3,3,2,5,3,2]
F4×A3×A1	[3,4,3,2,3,3,2]
F4×BC3×A1	[3,4,3,2,4,3,2]
F4×H3×A1	[3,4,2,3,5,3,2]
D4×A3×A1	[31,1,1,2,3,3,2]
D4×BC3×A1	[31,1,1,2,4,3,2]
D4×H3×A1	[31,1,1,2,5,3,2]
A4×I2(p)×I2(q)	[3,3,3,2,p,2,q]			triaprism
BC4×I2(p)×I2(q)	[4,3,3,2,p,2,q]
F4×I2(p)×I2(q)	[3,4,3,2,p,2,q]
H4×I2(p)×I2(q)	[5,3,3,2,p,2,q]
D4×I2(p)×I2(q)	[31,1,1,2,p,2,q]
A4×I2(p)×A12	[3,3,3,2,p,2,2]
BC4×I2(p)×A12	[4,3,3,2,p,2,2]
F4×I2(p)×A12	[3,4,3,2,p,2,2]
H4×I2(p)×A12	[5,3,3,2,p,2,2]
D4×I2(p)×A12	[31,1,1,2,p,2,2]
A4×A14	[3,3,3,2,2,2,2]
BC4×A14	[4,3,3,2,2,2,2]
F4×A14	[3,4,3,2,2,2,2]
H4×A14	[5,3,3,2,2,2,2]
D4×A14	[31,1,1,2,2,2,2]
A3×A3×I2(p)	[3,3,2,3,3,2,p]
BC3×A3×I2(p)	[4,3,2,3,3,2,p]
H3×A3×I2(p)	[5,3,2,3,3,2,p]
BC3×BC3×I2(p)	[4,3,2,4,3,2,p]
H3×BC3×I2(p)	[5,3,2,4,3,2,p]
H3×H3×I2(p)	[5,3,2,5,3,2,p]
A3×A3×A12	[3,3,2,3,3,2,2]
BC3×A3×A12	[4,3,2,3,3,2,2]
H3×A3×A12	[5,3,2,3,3,2,2]
BC3×BC3×A12	[4,3,2,4,3,2,2]
H3×BC3×A12	[5,3,2,4,3,2,2]
H3×H3×A12	[5,3,2,5,3,2,2]
A3×I2(p)×I2(q)×A1	[3,3,2,p,2,q,2]
BC3×I2(p)×I2(q)×A1	[4,3,2,p,2,q,2]
H3×I2(p)×I2(q)×A1	[5,3,2,p,2,q,2]
A3×I2(p)×A13	[3,3,2,p,2,2,2]
BC3×I2(p)×A13	[4,3,2,p,2,2,2]
H3×I2(p)×A13	[5,3,2,p,2,2,2]
A3×A15	[3,3,2,2,2,2,2]
BC3×A15	[4,3,2,2,2,2,2]
H3×A15	[5,3,2,2,2,2,2]
I2(p)×I2(q)×I2(r)×I2(s)	[p,2,q,2,r,2,s]		16pqrs
I2(p)×I2(q)×I2(r)×A12	[p,2,q,2,r,2,2]		32pqr
I2(p)×I2(q)×A14	[p,2,q,2,2,2,2]		64pq
I2(p)×A16	[p,2,2,2,2,2,2]		128p
A18	[2,2,2,2,2,2,2]		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

References

1. Conway, John H.; Smith, Derek A. (2003). On quaternions and octonions: their geometry, arithmetic, and symmetry. A K Peters. ISBN 978-1-56881-134-5.
2. The Crystallographic Space groups in Geometric algebra, D. Hestenes and J. Holt, Journal of Mathematical Physics. 48, 023514 (2007) (22 pages)

Further reading

• H. S. M. Coxeter (1995), F. Arthur Sherk; Peter McMullen; Anthony C. Thompson; Asia Ivic Weiss (eds.), Kaleidoscopes: Selected Writings of H. S. M. Coxeter, Wiley-Interscience Publication, ISBN 978-0-471-01003-6
  • (Paper 23) H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559–591]
• H. S. M. Coxeter; W. O. J. Moser (1980), Generators and Relations for Discrete Groups (4th ed.), New York: Springer-Verlag
• N. W. Johnson (2018), "Chapter 11: Finite symmetry groups", Geometries and Transformations

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)