# Point group

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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](/source/Geometry), a **point group** is a [mathematical group](/source/Group_(mathematics)) of [symmetry operations](/source/Symmetry_operation) ([isometries](/source/Isometry) in a [Euclidean space](/source/Euclidean_space)) that have a [fixed point](/source/Fixed_point_(mathematics)) in common. The [coordinate origin](/source/Origin_(mathematics)) 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](/source/Orthogonal_group) O(*d*). Point groups are used to describe the [symmetries](/source/Symmetry_(geometry)) of geometric figures and physical objects such as [molecules](/source/Molecular_symmetry).

Each point group can be [represented](/source/Group_representation) as sets of [orthogonal matrices](/source/Orthogonal_matrix) *M* that transform point *x* into point *y* according to *y* = *Mx*. Each element of a point group is either a [rotation](/source/Rotation_(mathematics)) ([determinant](/source/Determinant) of *M* = 1), or it is a [reflection](/source/Reflection_(mathematics)) or [improper rotation](/source/Improper_rotation) (determinant of *M* = −1).

The geometric symmetries of [crystals](/source/Crystal) are described by [space groups](/source/Space_group), which allow [translations](/source/Translation_(geometry)) and contain point groups as subgroups. Discrete point groups in more than one dimension come in infinite families, but from the [crystallographic restriction theorem](/source/Crystallographic_restriction_theorem) and [one of Bieberbach's theorems](/source/Space_group#Bieberbach.27s_theorems), each number of dimensions has only a finite number of point groups that are symmetric over some [lattice](/source/Lattice_(group)) or grid with that number of dimensions. These are the [crystallographic point groups](/source/Crystallographic_point_group).

## Chiral and achiral point groups, reflection groups

Point groups can be classified into *[chiral](/source/Chiral)* (or purely rotational) groups and *achiral* groups.[1] The chiral groups are subgroups of the [special orthogonal group](/source/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](/source/Finite_Coxeter_group) 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](/source/Coxeter%E2%80%93Dynkin_diagram). [Coxeter notation](/source/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](/source/Point_groups_in_two_dimensions), sometimes called **rosette groups**.

They come in two infinite families:

1. [Cyclic groups](/source/Cyclic_group) C*n* of *n*-fold rotation groups

1. [Dihedral groups](/source/Dihedral_group) D*n* of *n*-fold rotation and reflection groups

Applying the [crystallographic restriction theorem](/source/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](/source/Coxeter_group) and related polygons. These include 5 crystallographic groups. The symmetry of the reflectional groups can be doubled by an [isomorphism](/source/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](/source/Point_groups_in_three_dimensions)

[Point groups in three dimensions](/source/Point_groups_in_three_dimensions), sometimes called **molecular point groups** after their wide use in studying [symmetries of molecules](/source/Molecular_symmetry).

They come in 7 infinite families of axial groups (also called prismatic), and 7 additional polyhedral groups (also called Platonic). In [Schoenflies notation](/source/Schoenflies_notation),

- Axial groups: C*n*, S2*n*, C*n*h, C*n*v, D*n*, D*n*d, D*n*h

- [Polyhedral groups](/source/Polyhedral_group): T, Td, Th, O, Oh, I, Ih

Applying the [crystallographic restriction theorem](/source/Crystallographic_restriction_theorem) to these groups yields the 32 [crystallographic point groups](/source/Crystallographic_point_group).

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 Schoenflies 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 Schoenflies 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](/source/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.

Schoenflies 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](/source/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](/source/Coxeter_group), and like the [polyhedral groups](/source/Polyhedral_group) of 3D, it can be named by its related [convex regular 4-polytope](/source/Convex_regular_4-polytope). Related pure rotational groups exist for each with half the order, and can be represented by the bracket [Coxeter notation](/source/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](/source/Coxeter_group). Related chiral groups exist for each with half the order, and can be represented by the bracket [Coxeter notation](/source/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](/source/Coxeter_group). Related pure rotational groups exist for each with half the order, and can be represented by the bracket [Coxeter notation](/source/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](/source/Coxeter_group). Related chiral groups exist for each with half the order, defined by an [even number](/source/Even_number) of reflections, and can be represented by the bracket [Coxeter notation](/source/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](/source/Coxeter_group). Related chiral groups exist for each with half the order, defined by an [even number](/source/Even_number) of reflections, and can be represented by the bracket [Coxeter notation](/source/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

- [Bravais lattice](/source/Bravais_lattice)

- [Crystallographic point group](/source/Crystallographic_point_group)

- [Crystallography](/source/Crystallography)

- [Infrared spectroscopy of metal carbonyls](/source/Infrared_spectroscopy_of_metal_carbonyls)

- [Molecular symmetry](/source/Molecular_symmetry)

- [Point groups in two dimensions](/source/Point_groups_in_two_dimensions)

- [Point groups in three dimensions](/source/Point_groups_in_three_dimensions)

- [Point groups in four dimensions](/source/Point_groups_in_four_dimensions)

- [Space group](/source/Space_group)

- [X-ray diffraction](/source/X-ray_diffraction)

## References

1. ^ [***a***](#cite_ref-Conway-Smith_1-0) [***b***](#cite_ref-Conway-Smith_1-1) [Conway, John H.](/source/John_H_Conway); Smith, Derek A. (2003). *On quaternions and octonions: their geometry, arithmetic, and symmetry*. A K Peters. [ISBN](/source/ISBN_(identifier)) [978-1-56881-134-5](https://en.wikipedia.org/wiki/Special:BookSources/978-1-56881-134-5).

1. **[^](#cite_ref-2)** [The Crystallographic Space groups in Geometric algebra](https://davidhestenes.net/geocalc/pdf/CrystalGA.pdf), [D. Hestenes](/source/David_Hestenes) and J. Holt, Journal of Mathematical Physics. 48, 023514 (2007) (22 pages)

## Further reading

- [H. S. M. Coxeter](/source/Harold_Scott_MacDonald_Coxeter) (1995), F. Arthur Sherk; Peter McMullen; Anthony C. Thompson; Asia Ivic Weiss (eds.), [*Kaleidoscopes: Selected Writings of H. S. M. Coxeter*](http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html), Wiley-Interscience Publication, [ISBN](/source/ISBN_(identifier)) [978-0-471-01003-6](https://en.wikipedia.org/wiki/Special:BookSources/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](/source/Harold_Scott_MacDonald_Coxeter); W. O. J. Moser (1980), *Generators and Relations for Discrete Groups* (4th ed.), New York: Springer-Verlag

- [N. W. Johnson](/source/Norman_Johnson_(mathematician)) (2018), "Chapter 11: Finite symmetry groups", *Geometries and Transformations*

## External links

- [Web-based point group tutorial](http://www.reciprocalnet.org/edumodules/symmetry/index.html) (needs Java and Flash)

- [Subgroup enumeration](http://plus.swap-zt.com/2010/06/sieve) (needs Java)

- [The Geometry Center: 2.1 Formulas for Symmetries in Cartesian Coordinates (two dimensions)](http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node9.html)

- [The Geometry Center: 10.1 Formulas for Symmetries in Cartesian Coordinates (three dimensions)](http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node45.html)

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