Point group

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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.

Contents

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 OrderDescription
C1[ ]+1Identity
D1[ ]CDel node.png2Reflection 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 OrderDescription
Cnnn•[n]+nCyclic: n-fold rotations. Abstract group Zn, the group of integers under addition modulo n.
Dnnm*n•[n]2nDihedral: cyclic with reflections. Abstract group Dihn, the dihedral group.
Finite isomorphism and correspondences 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.

ReflectiveRotationalRelated
polygons
Group Coxeter group Coxeter diagram OrderSubgroupCoxeterOrder
D1A1[ ]CDel node.pngCDel node c1.png2C1[]+1 Digon
D2A12[2]CDel node.pngCDel 2.pngCDel node.pngCDel node c1.pngCDel 2.pngCDel node c2.png4C2[2]+2 Rectangle
D3A2[3]CDel node.pngCDel 3.pngCDel node.pngCDel node c1.pngCDel 3.pngCDel node c2.png6C3[3]+3 Equilateral triangle
D4BC2[4]CDel node.pngCDel 4.pngCDel node.pngCDel node c1.pngCDel 4.pngCDel node c2.png8C4[4]+4 Square
D5H2[5]CDel node.pngCDel 5.pngCDel node.pngCDel node c1.pngCDel 5.pngCDel node c2.png10C5[5]+5 Regular pentagon
D6G2[6]CDel node.pngCDel 6.pngCDel node.pngCDel node c1.pngCDel 6.pngCDel node c2.png12C6[6]+6 Regular hexagon
DnI2(n)[n]CDel node.pngCDel n.pngCDel node.pngCDel node c1.pngCDel n.pngCDel node c2.png2nCn[n]+n Regular polygon
D2×2A12×2[[2]] = [4]CDel node.pngCDel 2.pngCDel node.pngCDel node c1.pngCDel 2.pngCDel node c1.png = CDel node c1.pngCDel 4.pngCDel node.png8
D3×2A2×2[[3]] = [6]CDel node.pngCDel 3.pngCDel node.pngCDel node c1.pngCDel 3.pngCDel node c1.png = CDel node c1.pngCDel 6.pngCDel node.png12
D4×2BC2×2[[4]] = [8]CDel node.pngCDel 4.pngCDel node.pngCDel node c1.pngCDel 4.pngCDel node c1.png = CDel node c1.pngCDel 8.pngCDel node.png16
D5×2H2×2[[5]] = [10]CDel node.pngCDel 5.pngCDel node.pngCDel node c1.pngCDel 5.pngCDel node c1.png = CDel node c1.pngCDel 10.pngCDel node.png20
D6×2G2×2[[6]] = [12]CDel node.pngCDel 6.pngCDel node.pngCDel node c1.pngCDel 6.pngCDel node c1.png = CDel node c1.pngCDel 12.pngCDel node.png24
Dn×2I2(n)×2[[n]] = [2n]CDel node.pngCDel n.pngCDel node.pngCDel node c1.pngCDel n.pngCDel node c1.png = CDel node c1.pngCDel 2x.pngCDel n.pngCDel node.png4n

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 or prismatic groups, and 7 additional polyhedral or Platonic groups. In Schönflies notation,

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
...
Spherical digonal hosohedron2.png Spherical square hosohedron2.png Spherical hexagonal hosohedron2.png Spherical octagonal hosohedron2.png Spherical decagonal hosohedron2.png Spherical dodecagonal hosohedron2.png
D1h
Order 4
D2h
Order 8
D3h
Order 12
D4h
Order 16
D5h
Order 20
D6h
Order 24
...
Spherical digonal bipyramid2.svg Spherical square bipyramid2.svg Spherical hexagonal bipyramid2.png Spherical octagonal bipyramid2.png Spherical decagonal bipyramid2.png Spherical dodecagonal bipyramid2.png
Td
Order 24
Oh
Order 48
Ih
Order 120
Tetrahedral reflection domains.png Octahedral reflection domains.png Icosahedral reflection domains.png
Intl *Geo
[2]
Orbifold Schönflies Coxeter Order
111C1[ ]+1
122×1Ci = S2[2+,2+]2
2 = m1*1Cs = 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





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
22
32
42
52
62
n2
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
2333332T[3,3]+12
m34 33*2Th[3+,4]24
43m3 3*332Td[3,3]24
43243432O[3,4]+24
m3m4 3*432Oh[3,4]48
53253532I[3,5]+60
53m5 3*532Ih[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 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 [[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 OrderRelated regular and
prismatic polyhedra
TdA3[3,3]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.png24 Tetrahedron
Td×Dih1 = OhA3×2 = BC3[[3,3]] = [4,3]CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c1.png= CDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.png48 Stellated octahedron
OhBC3[4,3]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c3.png48 Cube, octahedron
IhH3[5,3]CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel node c1.pngCDel 5.pngCDel node c2.pngCDel 3.pngCDel node c3.png120 Icosahedron, dodecahedron
D3hA2×A1[3,2]CDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 2.pngCDel node c3.png12 Triangular prism
D3h×Dih1 = D6hA2×A1×2[[3],2]CDel node c1.pngCDel 3.pngCDel node c1.pngCDel 2.pngCDel node c2.png= CDel node c1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node c2.png24 Hexagonal prism
D4hBC2×A1[4,2]CDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel node c1.pngCDel 4.pngCDel node c2.pngCDel 2.pngCDel node c3.png16 Square prism
D4h×Dih1 = D8hBC2×A1×2[[4],2] = [8,2]CDel node c1.pngCDel 4.pngCDel node c1.pngCDel 2.pngCDel node c2.png= CDel node c1.pngCDel 8.pngCDel node.pngCDel 2.pngCDel node c2.png32 Octagonal prism
D5hH2×A1[5,2]CDel node.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node.pngCDel node c1.pngCDel 5.pngCDel node c2.pngCDel 2.pngCDel node c3.png20 Pentagonal prism
D6hG2×A1[6,2]CDel node.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.pngCDel node c1.pngCDel 6.pngCDel node c2.pngCDel 2.pngCDel node c3.png24 Hexagonal prism
DnhI2(n)×A1[n,2]CDel node.pngCDel n.pngCDel node.pngCDel 2.pngCDel node.pngCDel node c1.pngCDel n.pngCDel node c2.pngCDel 2.pngCDel node c3.png4nn-gonal prism
Dnh×Dih1 = D2nhI2(n)×A1×2[[n],2]CDel node c1.pngCDel n.pngCDel node c1.pngCDel 2.pngCDel node c2.png= CDel node c1.pngCDel 2x.pngCDel n.pngCDel node.pngCDel 2.pngCDel node c2.png8n
D2hA13[2,2]CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel node c1.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c3.png8 Cuboid
D2h×Dih1A13×2[[2],2] = [4,2]CDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c2.png= CDel node c1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node c2.png16
D2h×Dih3 = OhA13×6[3[2,2]] = [4,3]CDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c1.png= CDel node c1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png48
C3vA2[1,3]CDel node.pngCDel 3.pngCDel node.pngCDel node c1.pngCDel 3.pngCDel node c2.png6 Hosohedron
C4vBC2[1,4]CDel node.pngCDel 4.pngCDel node.pngCDel node c1.pngCDel 4.pngCDel node c2.png8
C5vH2[1,5]CDel node.pngCDel 5.pngCDel node.pngCDel node c1.pngCDel 5.pngCDel node c2.png10
C6vG2[1,6]CDel node.pngCDel 6.pngCDel node.pngCDel node c1.pngCDel 6.pngCDel node c2.png12
CnvI2(n)[1,n]CDel node.pngCDel n.pngCDel node.pngCDel node c1.pngCDel n.pngCDel node c2.png2n
Cnv×Dih1 = C2nvI2(n)×2[1,[n]] = [1,2n]CDel node c1.pngCDel n.pngCDel node c1.png= CDel node c1.pngCDel 2x.pngCDel n.pngCDel node.png4n
C2vA12[1,2]CDel node.pngCDel 2.pngCDel node.pngCDel node c1.pngCDel 2.pngCDel node c2.png4
C2v×Dih1A12×2[1,[2]]CDel node c1.pngCDel 2.pngCDel node c1.png= CDel node c1.pngCDel 4.pngCDel node.png8
CsA1[1,1]CDel node.pngCDel node c1.png2

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 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 [[3,3,3]] with its order doubled to 240.

Coxeter group/notation Coxeter diagram OrderRelated polytopes
A4[3,3,3]CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c4.png120 5-cell
A4×2[[3,3,3]]CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c1.png2405-cell dual compound
BC4[4,3,3]CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c4.png384 16-cell/Tesseract
D4[31,1,1]CDel nodeab c1-2.pngCDel split2.pngCDel node c3.pngCDel 3.pngCDel node c4.png192 Demitesseractic
D4×2 = BC4<[3,31,1]> = [4,3,3]CDel nodeab c1.pngCDel split2.pngCDel node c2.pngCDel 3.pngCDel node c3.png= CDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.png384
D4×6 = F4[3[31,1,1]] = [3,4,3]CDel nodeab c1.pngCDel split2.pngCDel node c2.pngCDel 3.pngCDel node c1.png= CDel node c2.pngCDel 3.pngCDel node c1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png1152
F4[3,4,3]CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 4.pngCDel node c3.pngCDel 3.pngCDel node c4.png1152 24-cell
F4×2[[3,4,3]]CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c1.png230424-cell dual compound
H4[5,3,3]CDel node c1.pngCDel 5.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c4.png14400 120-cell/600-cell
A3×A1[3,3,2]CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.png48 Tetrahedral prism
A3×A1×2[[3,3],2] = [4,3,2]CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c1.pngCDel 2.pngCDel node c3.png= CDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 2.pngCDel node c3.png96 Octahedral prism
BC3×A1[4,3,2]CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.png96
H3×A1[5,3,2]CDel node c1.pngCDel 5.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.png240 Icosahedral prism
A2×A2[3,2,3]CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 3.pngCDel node c4.png36 Duoprism
A2×BC2[3,2,4]CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 4.pngCDel node c4.png48
A2×H2[3,2,5]CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 5.pngCDel node c4.png60
A2×G2[3,2,6]CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 6.pngCDel node c4.png72
BC2×BC2[4,2,4]CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 4.pngCDel node c4.png64
BC22×2[[4,2,4]]CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 2.pngCDel node c2.pngCDel 4.pngCDel node c1.png128
BC2×H2[4,2,5]CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 5.pngCDel node c4.png80
BC2×G2[4,2,6]CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 6.pngCDel node c4.png96
H2×H2[5,2,5]CDel node c1.pngCDel 5.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 5.pngCDel node c4.png100
H2×G2[5,2,6]CDel node c1.pngCDel 5.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 6.pngCDel node c4.png120
G2×G2[6,2,6]CDel node c1.pngCDel 6.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 6.pngCDel node c4.png144
I2(p)×I2(q)[p,2,q]CDel node c1.pngCDel p.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel q.pngCDel node c4.png4pq
I2(2p)×I2(q)[[p],2,q] = [2p,2,q]CDel node c1.pngCDel p.pngCDel node c1.pngCDel 2.pngCDel node c2.pngCDel q.pngCDel node c3.png= CDel node c1.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 2.pngCDel node c2.pngCDel q.pngCDel node c3.png8pq
I2(2p)×I2(2q)[[p]],2,[[q]] = [2p,2,2q]CDel node c1.pngCDel p.pngCDel node c1.pngCDel 2.pngCDel node c2.pngCDel q.pngCDel node c2.png= CDel node c1.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 2.pngCDel node c2.pngCDel 2x.pngCDel q.pngCDel node.png16pq
I2(p)2×2[[p,2,p]]CDel node c1.pngCDel p.pngCDel node c2.pngCDel 2.pngCDel node c2.pngCDel p.pngCDel node c1.png8p2
I2(2p)2×2[[[p],2,[p]]] = [[2p,2,2p]]CDel node c1.pngCDel p.pngCDel node c1.pngCDel 2.pngCDel node c1.pngCDel p.pngCDel node c1.png= CDel node c1.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 2.pngCDel node c1.pngCDel 2x.pngCDel p.pngCDel node.png32p2
A2×A1×A1[3,2,2]CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 2.pngCDel node c4.png24
BC2×A1×A1[4,2,2]CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 2.pngCDel node c4.png32
H2×A1×A1[5,2,2]CDel node c1.pngCDel 5.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 2.pngCDel node c4.png40
G2×A1×A1[6,2,2]CDel node c1.pngCDel 6.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 2.pngCDel node c4.png48
I2(p)×A1×A1[p,2,2]CDel node c1.pngCDel p.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 2.pngCDel node c4.png8p
I2(2p)×A1×A1×2[[p],2,2] = [2p,2,2]CDel node c1.pngCDel p.pngCDel node c1.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c3.png= CDel node c1.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c3.png16p
I2(p)×A12×2[p,2,[2]] = [p,2,4]CDel node c1.pngCDel p.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 2.pngCDel node c3.png= CDel node c1.pngCDel p.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 4.pngCDel node.png16p
I2(2p)×A12×4[[p]],2,[[2]] = [2p,2,4]CDel node c1.pngCDel p.pngCDel node c1.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c2.png= CDel node c1.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 2.pngCDel node c2.pngCDel 4.pngCDel node.png32p
A1×A1×A1×A1[2,2,2]CDel node c1.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 2.pngCDel node c4.png164-orthotope
A12×A1×A1×2[[2],2,2] = [4,2,2]CDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c3.png= CDel node c1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c3.png32
A12×A12×4[[2]],2,[[2]] = [4,2,4]CDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c2.png= CDel node c1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node c2.pngCDel 4.pngCDel node.png64
A13×A1×6[3[2,2],2] = [4,3,2]CDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c2.png= CDel node c1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node c2.png96
A14×24[3,3[2,2,2]] = [4,3,3]CDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c1.png= CDel node c1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png384

Five dimensions

Finite isomorphism and correspondences 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
OrderRelated regular and
prismatic polytopes
A5[3,3,3,3]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c4.pngCDel 3.pngCDel node c5.png720 5-simplex
A5×2[[3,3,3,3]]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c1.png1440 5-simplex dual compound
BC5[4,3,3,3]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c4.pngCDel 3.pngCDel node c5.png3840 5-cube, 5-orthoplex
D5[32,1,1]CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel nodeab c1-2.pngCDel split2.pngCDel node c3.pngCDel 3.pngCDel node c4.pngCDel 3.pngCDel node c5.png1920 5-demicube
D5×2<[3,3,31,1]>CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel nodeab c1.pngCDel split2.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c4.png = CDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c4.png3840
A4×A1[3,3,3,2]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c4.pngCDel 2.pngCDel node c5.png240 5-cell prism
A4×A1×2[[3,3,3],2]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c1.pngCDel 2.pngCDel node c3.png480
BC4×A1[4,3,3,2]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c4.pngCDel 2.pngCDel node c5.png768 tesseract prism
F4×A1[3,4,3,2]CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 4.pngCDel node c3.pngCDel 3.pngCDel node c4.pngCDel 2.pngCDel node c5.png2304 24-cell prism
F4×A1×2[[3,4,3],2]CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c1.pngCDel 2.pngCDel node c3.png4608
H4×A1[5,3,3,2]CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel node c1.pngCDel 5.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c4.pngCDel 2.pngCDel node c5.png28800 600-cell or 120-cell prism
D4×A1[31,1,1,2]CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel nodeab c1-2.pngCDel split2.pngCDel node c3.pngCDel 3.pngCDel node c4.pngCDel 2.pngCDel node c5.png384Demitesseract prism
A3×A2[3,3,2,3]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.pngCDel 3.pngCDel node c5.png144 Duoprism
A3×A2×2[[3,3],2,3]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c1.pngCDel 2.pngCDel node c3.pngCDel 3.pngCDel node c4.png288
A3×BC2[3,3,2,4]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.pngCDel 4.pngCDel node c5.png192
A3×H2[3,3,2,5]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.pngCDel 5.pngCDel node c5.png240
A3×G2[3,3,2,6]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.pngCDel 6.pngCDel node c5.png288
A3×I2(p)[3,3,2,p]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.pngCDel p.pngCDel node c5.png48p
BC3×A2[4,3,2,3]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.pngCDel 3.pngCDel node c5.png288
BC3×BC2[4,3,2,4]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.pngCDel 4.pngCDel node c5.png384
BC3×H2[4,3,2,5]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.pngCDel 5.pngCDel node c5.png480
BC3×G2[4,3,2,6]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.pngCDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.pngCDel 6.pngCDel node c5.png576
BC3×I2(p)[4,3,2,p]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.pngCDel p.pngCDel node c5.png96p
H3×A2[5,3,2,3]CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel node c1.pngCDel 5.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.pngCDel 3.pngCDel node c5.png720
H3×BC2[5,3,2,4]CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel node c1.pngCDel 5.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.pngCDel 4.pngCDel node c5.png960
H3×H2[5,3,2,5]CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel node c1.pngCDel 5.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.pngCDel 5.pngCDel node c5.png1200
H3×G2[5,3,2,6]CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.pngCDel node c1.pngCDel 5.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 2.pngCDel node c4.pngCDel 6.pngCDel node c5.png1440
H3×I2(p)[5,3,2,p]CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png240p
A3×A12[3,3,2,2]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png96
BC3×A12[4,3,2,2]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png192
H3×A12[5,3,2,2]CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png480
A22×A1[3,2,3,2]CDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png72duoprism prism
A2×BC2×A1[3,2,4,2]CDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.png96
A2×H2×A1[3,2,5,2]CDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node.png120
A2×G2×A1[3,2,6,2]CDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.png144
BC22×A1[4,2,4,2]CDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.png128
BC2×H2×A1[4,2,5,2]CDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node.png160
BC2×G2×A1[4,2,6,2]CDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.png192
H22×A1[5,2,5,2]CDel node.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node.png200
H2×G2×A1[5,2,6,2]CDel node.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.png240
G22×A1[6,2,6,2]CDel node.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.png288
I2(p)×I2(q)×A1[p,2,q,2]CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.pngCDel 2.pngCDel node.png8pq
A2×A13[3,2,2,2]CDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png48
BC2×A13[4,2,2,2]CDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png64
H2×A13[5,2,2,2]CDel node.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png80
G2×A13[6,2,2,2]CDel node.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png96
I2(p)×A13[p,2,2,2]CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png16p
A15[2,2,2,2]CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel node c1.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 2.pngCDel node c4.pngCDel 2.pngCDel node c5.png325-orthotope
A15×(2!)[[2],2,2,2]CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 2.pngCDel node c4.png = CDel node c1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 2.pngCDel node c4.png64
A15×(2!×2!)[[2]],2,[2],2]CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c3.png = CDel node c1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node c2.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node c3.png128
A15×(3!)[3[2,2],2,2]CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c3.png = CDel node c1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c3.png192
A15×(3!×2!)[3[2,2],2,[[2]]CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c2.pngCDel 2.pngCDel node c2.png = CDel node c1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node c2.pngCDel 4.pngCDel node.png384
A15×(4!)[3,3[2,2,2],2]]CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c2.png = CDel node c1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node c2.png768
A15×(5!)[3,3,3[2,2,2,2]]CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c1.png = CDel node c1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png3840

Six dimensions

Finite isomorphism and correspondences 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
OrderRelated regular and
prismatic polytopes
A6[3,3,3,3,3]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png5040 (7!) 6-simplex
A6×2[[3,3,3,3,3]]CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png10080 (2×7!) 6-simplex dual compound
BC6[4,3,3,3,3]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png46080 (26×6!) 6-cube, 6-orthoplex
D6[3,3,3,31,1]CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png23040 (25×6!) 6-demicube
E6 [3,32,2]CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png51840 (72×6!) 122, 221
A5×A1[3,3,3,3,2]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png1440 (2×6!)5-simplex prism
BC5×A1[4,3,3,3,2]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png7680 (26×5!)5-cube prism
D5×A1[3,3,31,1,2]CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png3840 (25×5!)5-demicube prism
A4×I2(p)[3,3,3,2,p]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png240p Duoprism
BC4×I2(p)[4,3,3,2,p]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png768p
F4×I2(p)[3,4,3,2,p]CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png2304p
H4×I2(p)[5,3,3,2,p]CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png28800p
D4×I2(p)[3,31,1,2,p]CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png384p
A4×A12[3,3,3,2,2]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png480
BC4×A12[4,3,3,2,2]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png1536
F4×A12[3,4,3,2,2]CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png4608
H4×A12[5,3,3,2,2]CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png57600
D4×A12[3,31,1,2,2]CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png768
A32[3,3,2,3,3]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png576
A3×BC3[3,3,2,4,3]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png1152
A3×H3[3,3,2,5,3]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png2880
BC32[4,3,2,4,3]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png2304
BC3×H3[4,3,2,5,3]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png5760
H32[5,3,2,5,3]CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png14400
A3×I2(p)×A1[3,3,2,p,2]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.png96pDuoprism prism
BC3×I2(p)×A1[4,3,2,p,2]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.png192p
H3×I2(p)×A1[5,3,2,p,2]CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.png480p
A3×A13[3,3,2,2,2]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png192
BC3×A13[4,3,2,2,2]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png384
H3×A13[5,3,2,2,2]CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png960
I2(p)×I2(q)×I2(r)[p,2,q,2,r]CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.pngCDel 2.pngCDel node.pngCDel r.pngCDel node.png8pqr Triaprism
I2(p)×I2(q)×A12[p,2,q,2,2]CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png16pq
I2(p)×A14[p,2,2,2,2]CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png32p
A16[2,2,2,2,2]CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png646-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 OrderRelated polytopes
A7[3,3,3,3,3,3]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png40320 (8!) 7-simplex
A7×2[[3,3,3,3,3,3]]CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png80640 (2×8!) 7-simplex dual compound
BC7[4,3,3,3,3,3]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png645120 (27×7!) 7-cube, 7-orthoplex
D7[3,3,3,3,31,1]CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png322560 (26×7!) 7-demicube
E7 [3,3,3,32,1]CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png2903040 (8×9!) 321, 231, 132
A6×A1[3,3,3,3,3,2]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png10080 (2×7!)
BC6×A1[4,3,3,3,3,2]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png92160 (27×6!)
D6×A1[3,3,3,31,1,2]CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png46080 (26×6!)
E6×A1[3,3,32,1,2]CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea.png103680 (144×6!)
A5×I2(p)[3,3,3,3,2,p]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png1440p
BC5×I2(p)[4,3,3,3,2,p]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png7680p
D5×I2(p)[3,3,31,1,2,p]CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png3840p
A5×A12[3,3,3,3,2,2]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png2880
BC5×A12[4,3,3,3,2,2]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png15360
D5×A12[3,3,31,1,2,2]CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png7680
A4×A3[3,3,3,2,3,3]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png2880
A4×BC3[3,3,3,2,4,3]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png5760
A4×H3[3,3,3,2,5,3]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png14400
BC4×A3[4,3,3,2,3,3]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png9216
BC4×BC3[4,3,3,2,4,3]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png18432
BC4×H3[4,3,3,2,5,3]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png46080
H4×A3[5,3,3,2,3,3]CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png345600
H4×BC3[5,3,3,2,4,3]CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png691200
H4×H3[5,3,3,2,5,3]CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png1728000
F4×A3[3,4,3,2,3,3]CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png27648
F4×BC3[3,4,3,2,4,3]CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png55296
F4×H3[3,4,3,2,5,3]CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png138240
D4×A3[31,1,1,2,3,3]CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png4608
D4×BC3[3,31,1,2,4,3]CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png9216
D4×H3[3,31,1,2,5,3]CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png23040
A4×I2(p)×A1[3,3,3,2,p,2]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.png480p
BC4×I2(p)×A1[4,3,3,2,p,2]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.png1536p
D4×I2(p)×A1[3,31,1,2,p,2]CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.png768p
F4×I2(p)×A1[3,4,3,2,p,2]CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.png4608p
H4×I2(p)×A1[5,3,3,2,p,2]CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.png57600p
A4×A13[3,3,3,2,2,2]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png960
BC4×A13[4,3,3,2,2,2]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png3072
F4×A13[3,4,3,2,2,2]CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png9216
H4×A13[5,3,3,2,2,2]CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png115200
D4×A13[3,31,1,2,2,2]CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png1536
A32×A1[3,3,2,3,3,2]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png1152
A3×BC3×A1[3,3,2,4,3,2]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.png