Point group

In geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every point group in dimension d is then a subgroup of the orthogonal group O(d). Point groups are used to describe the symmetries of geometric figures and physical objects such as molecules.


The Bauhinia blakeana flower on the Hong Kong region flag has C5 symmetry; the star on each petal has D5 symmetry.
The Yin and Yang symbol has C2 symmetry of geometry with inverted colors

Each point group can be represented as sets of orthogonal matrices M that transform point x into point y according to y = Mx. Each element of a point group is either a rotation (determinant of M = 1), or it is a reflection or improper rotation (determinant of M = −1).

The geometric symmetries of crystals are described by space groups, which allow translations and contain point groups as subgroups. Discrete point groups in more than one dimension come in infinite families, but from the crystallographic restriction theorem and one of Bieberbach's theorems, each number of dimensions has only a finite number of point groups that are symmetric over some lattice or grid with that number of dimensions. These are the crystallographic point groups.

Chiral and achiral point groups, reflection groups

Point groups can be classified into chiral (or purely rotational) groups and achiral groups.[1] The chiral groups are subgroups of the special orthogonal group SO(d): they contain only orientation-preserving orthogonal transformations, i.e., those of determinant +1. The achiral groups contain also transformations of determinant −1. In an achiral group, the orientation-preserving transformations form a (chiral) subgroup of index 2.

Finite Coxeter groups or reflection groups are those point groups that are generated purely by a set of reflectional mirrors passing through the same point. A rank n Coxeter group has n mirrors and is represented by a Coxeter-Dynkin diagram. Coxeter notation offers a bracketed notation equivalent to the Coxeter diagram, with markup symbols for rotational and other subsymmetry point groups. Reflection groups are necessarily achiral (except for the trivial group containing only the identity element).

List of point groups

One dimension

There are only two one-dimensional point groups, the identity group and the reflection group.

Group Coxeter Coxeter diagram Order Description
C1[ ]+1Identity
D1[ ]2Reflection 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
D1A1[ ]2C1[]+1 Digon
D2A12[2]4C2[2]+2 Rectangle
D3A2[3]6C3[3]+3 Equilateral triangle
D4BC2[4]8C4[4]+4 Square
D5H2[5]10C5[5]+5 Regular pentagon
D6G2[6]12C6[6]+6 Regular hexagon
DnI2(n)[n]2nCn[n]+n Regular polygon
D2×2A12×2[[2]] = [4] = 8
D3×2A2×2[[3]] = [6] = 12
D4×2BC2×2[[4]] = [8] = 16
D5×2H2×2[[5]] = [10] = 20
D6×2G2×2[[6]] = [12] = 24
Dn×2I2(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
TdA3[3,3] 24Tetrahedron
Td×Dih1 = OhA3×2 = BC3[[3,3]] = [4,3] = 48Stellated octahedron
OhBC3[4,3] 48Cube, octahedron
IhH3[5,3] 120Icosahedron, dodecahedron
D3hA2×A1[3,2] 12Triangular prism
D3h×Dih1 = D6hA2×A1×2[[3],2] = 24Hexagonal prism
D4hBC2×A1[4,2] 16Square prism
D4h×Dih1 = D8hBC2×A1×2[[4],2] = [8,2] = 32Octagonal prism
D5hH2×A1[5,2] 20Pentagonal prism
D6hG2×A1[6,2] 24Hexagonal prism
DnhI2(n)×A1[n,2] 4nn-gonal prism
Dnh×Dih1 = D2nhI2(n)×A1×2[[n],2] = 8n
D2hA13[2,2] 8 Cuboid
D2h×Dih1A13×2[[2],2] = [4,2] = 16
D2h×Dih3 = OhA13×6[3[2,2]] = [4,3] = 48
C3vA2[1,3] 6 Hosohedron
C4vBC2[1,4] 8
C5vH2[1,5] 10
C6vG2[1,6] 12
CnvI2(n)[1,n] 2n
Cnv×Dih1 = C2nvI2(n)×2[1,[n]] = [1,2n] = 4n
C2vA12[1,2] 4
C2v×Dih1A12×2[1,[2]] = 8
CsA1[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]1205-cell
A4×2[[3,3,3]]2405-cell dual compound
BC4[4,3,3]38416-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]115224-cell
F4×2[[3,4,3]]230424-cell dual compound
H4[5,3,3]14400120-cell/600-cell
A3×A1[3,3,2]48Tetrahedral prism
A3×A1×2[[3,3],2] = [4,3,2]= 96 Octahedral prism
BC3×A1[4,3,2]96
H3×A1[5,3,2]240Icosahedral 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] 7205-simplex
A5×2[[3,3,3,3]] 14405-simplex dual compound
BC5[4,3,3,3] 38405-cube, 5-orthoplex
D5[32,1,1] 19205-demicube
D5×2<[3,3,31,1]> = 3840
A4×A1[3,3,3,2] 2405-cell prism
A4×A1×2[[3,3,3],2] 480
BC4×A1[4,3,3,2] 768tesseract 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] 28800600-cell or 120-cell prism
D4×A1[31,1,1,2] 384Demitesseract 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]646-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]806407-simplex prism
BC7×A1[4,3,3,3,3,3,2]6451207-cube prism
D7×A1[3,3,3,3,31,1,2]3225607-demicube prism
E7×A1[3,3,3,32,1,2]5806080321 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

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

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