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Wikipedia

Point group

April 04, 2023


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.

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 Cn of n-fold rotation groups
  2. Dihedral groups Dn 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: Cn, S2n, Cnh, Cnv, Dn, Dnd, Dnh
  • Polyhedral groups: T, Td, Th, O, Oh, I, Ih

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]
April 04, 2023
point, group, group, geometric, symmetries, with, least, fixed, point, bauhinia, blakeana, flower, hong, kong, region, flag, symmetry, star, each, petal, symmetry, yang, symbol, symmetry, geometry, with, inverted, colors, geometry, point, group, mathematical, . Group of geometric symmetries with at least one fixed point 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 8722 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 Contents 1 Chiral and achiral point groups reflection groups 2 List of point groups 2 1 One dimension 2 2 Two dimensions 2 3 Three dimensions 2 3 1 Reflection groups 2 4 Four dimensions 2 5 Five dimensions 2 6 Six dimensions 2 7 Seven dimensions 2 8 Eight dimensions 3 See also 4 References 5 Further reading 6 External links Chiral and achiral point groups reflection groups Edit Point groups can be classified into chiral or purely rotational groups and achiral groups 91 1 93 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 Edit One dimension Edit There are only two one dimensional point groups the identity group and the reflection group Group Coxeter Coxeter diagram Order Description C1 160 1 Identity D1 160 2 Reflection group Two dimensions Edit Point groups in two dimensions sometimes called rosette groups They come in two infinite families Cyclic groups Cn of n fold rotation groups Dihedral groups Dn 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 160 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 Edit 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 Schonflies notation Axial groups Cn S2n Cnh Cnv Dn Dnd Dnh Polyhedral groups T Td Th O Oh I Ih Applying the crystallographic restriction theorem to these groups yields the 32 crystallographic point groups Even odd colored fundamental domains of the reflective groups C1vOrder 2 C2vOrder 4 C3vOrder 6 C4vOrder 8 C5vOrder 10 C6vOrder 12 D1hOrder 4 D2hOrder 8 D3hOrder 12 D4hOrder 16 D5hOrder 20 D6hOrder 24 TdOrder 24 OhOrder 48 IhOrder 120 Intl Geo 91 2 93 Orbifold Schonflies Coxeter Order 1 1 1 C1 160 1 1 22 1 Ci S2 2 2 2 2 m 1 1 Cs C1v C1h 160 2 23456n 2 3 4 5 6 n 2233445566nn C2C3C4C5C6Cn 2 3 4 5 6 n 23456n mm23m4mm5m6mmnmmnm 23456n 22 33 44 55 66 nn C2vC3vC4vC5vC6vCnv 2 3 4 5 6 n 46810122n 2 m6 4 m10 6 mn m2n 2 23 24 25 26 2n 2 2 3 4 5 6 n C2hC3hC4hC5hC6hCnh 2 2 2 3 2 4 2 5 2 6 2 n 46810122n 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 S4S6S8S10S12S2n 2 4 2 6 2 8 2 10 2 12 2 2n 46810122n Intl Geo Orbifold Schonflies Coxeter Order 2223242252622n22n2 2 2 3 2 4 2 5 2 6 2 n 2 22222322422522622n D2D3D4D5D6Dn 2 2 2 3 2 4 2 5 2 6 2 n 46810122n mmm6 m24 mmm10 m26 mmmn mmm2n m2 2 23 24 25 26 2n 2 222 223 224 225 226 22n D2hD3hD4hD5hD6hDnh 2 2 2 3 2 4 2 5 2 6 2 n 8121620244n 4 2m3 m8 2m5 m12 2m2n 2mn m 4 2 6 2 8 2 10 2 12 2 n 2 2 22 32 42 52 62 n D2dD3dD4dD5dD6dDnd 2 4 2 6 2 8 2 10 2 12 2 2n 8121620244n 23 3 3 332 T 3 3 12 m3 4 3 3 2 Th 3 4 24 4 3m 3 3 332 Td 3 3 24 432 4 3 432 O 3 4 24 m3 m 4 3 432 Oh 3 4 48 532 5 3 532 I 3 5 60 5 3 m 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 Edit 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 Schonflies 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 Edit Main article Point groups in four dimensions The four dimensional point groups chiral as well as achiral are listed in Conway and Smith 91 1 93 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 lt 3 31 1 gt 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 Edit 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 Coxeterdiagrams 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 lt 3 3 31 1 gt 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 Edit 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 Coxeterdiagram 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 Edit 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 Edit 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 img, wikipedia, wiki, book, books, library,

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