 Point group

In geometry, a point group is a group of geometric symmetries (isometries) that keep at least one point fixed. Point groups can exist in a Euclidean space with any dimension, and every point group in dimension d is a subgroup of the orthogonal group O(d). Point groups can be realized as sets of orthogonal matrices M that transform point x into point y:
y = M.x
where the origin is the fixed point. Pointgroup elements can either be rotations (determinant of M = 1) or else reflections, improper rotations, rotationreflections, or rotoreflections (determinant of M = 1). All point groups of rotations with dimension d are subgroups of the special orthogonal group SO(d).
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. These are the crystallographic point groups.
Contents
One Dimension
There are only two onedimensional point groups, the identity group and the reflection group.
Group Coxeter Coxeter diagram Order Description C_{1} [ ]^{+} 1 Identity D_{1} [ ] 2 Reflection group Two Dimensions
Point groups in two dimensions, sometimes called rosette groups.
They come in two infinite families:
 Cyclic groups C_{n} of nfold rotation groups
 Dihedral groups D_{n} of nfold 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 C_{n} n nn [n]^{+} n Cyclic: nfold rotations. Abstract group Z_{n}, the group of integers under addition modulo n. D_{n} nm *nn [n] 2n Dihedral: cyclic with reflections. Abstract group Dih_{n}, the dihedral group. The subset of pure reflectional point groups, defined by 1 or 2 mirror lines, can also be given by their Coxeter group and related polygons. These include 5 crystallographic groups.
Group Coxeter group Coxeter diagram Order Related polygons D_{3} A_{2} [3] 6 Equilateral triangle D_{4} BC_{2} [4] 8 Square D_{5} H_{2} [5] 10 Regular pentagon D_{6} G_{2} [6] 12 Regular hexagon D_{n} I_{2}(n) [n] 2n Regular polygon D_{2n} I_{2}(2n) [[n]]=[2n] 4n Regular polygon D_{2} A_{1}^{2} [2] 4 Rectangle D_{1} A_{1} [ ] 2 Digon Three Dimensions
Point groups in three dimensions, sometimes called molecular point groups, after their wide use in studying the symmetries of small molecules.
They come in 7 infinite families of axial or prismatic groups, and 7 additional polyhedral or Platonic groups. In Schönflies notation,*
 Axial groups: C_{n}, S_{2n}, C_{nh}, C_{nv}, D_{n}, D_{nd}, D_{nh}
 Polyhedral groups: T, T_{d}, T_{h}, O, O_{h}, I, I_{h}
Applying the crystallographic restriction theorem to these groups yields 32 Crystallographic point groups.
Intl^{*} Geo
^{[1]}Orbifold Schönflies Conway Coxeter Order 1 1 1 C_{1} C_{1} [ ]^{+} 1 1 22 ×1 C_{i} = S_{2} CC_{2} [2^{+},2^{+}] 2 2 = m 1 *1 C_{s} = C_{1v} = C_{1h} ±C_{1} = CD_{2} [ ] 2 2
3
4
5
6
n2
3
4
5
6
n22
33
44
55
66
nnC_{2}
C_{3}
C_{4}
C_{5}
C_{6}
C_{n}C_{2}
C_{3}
C_{4}
C_{5}
C_{6}
C_{n}[2]^{+}
[3]^{+}
[4]^{+}
[5]^{+}
[6]^{+}
[n]^{+}2
3
4
5
6
n2mm
3m
4mm
5m
6mm
nmm
nm2
3
4
5
6
n*22
*33
*44
*55
*66
*nnC_{2v}
C_{3v}
C_{4v}
C_{5v}
C_{6v}
C_{nv}CD_{4}
CD_{6}
CD_{8}
CD_{10}
CD_{12}
CD_{2n}[2]
[3]
[4]
[5]
[6]
[n]4
6
8
10
12
2n2/m
3/m
4/m
5/m
6/m
n/m2 2
3 2
4 2
5 2
6 2
n 22*
3*
4*
5*
6*
n*C_{2h}
C_{3h}
C_{4h}
C_{5h}
C_{6h}
C_{nh}±C_{2}
CC_{6}
±C_{4}
CC_{10}
±C_{6}
±C_{n} / CC_{2n}[2,2^{+}]
[2,3^{+}]
[2,4^{+}]
[2,5^{+}]
[2,6^{+}]
[2,n^{+}]4
6
8
10
12
2n4
3
8
5
12
2n
n4 2
6 2
8 2
10 2
12 2
2n 22×
3×
4×
5×
6×
n×S_{4}
S_{6}
S_{8}
S_{10}
S_{12}
S_{2n}CC_{4}
±C_{3}
CC_{8}
±C_{5}
CC_{12}
CC_{2n} / ±C_{n}[2^{+},4^{+}]
[2^{+},6^{+}]
[2^{+},8^{+}]
[2^{+},10^{+}]
[2^{+},12^{+}]
[2^{+},2n^{+}]4
6
8
10
12
2nIntl Geo Orbifold Schönflies Conway Coxeter Order 222
32
422
52
622
n22
n22 2
3 2
4 2
5 2
6 2
n 2222
223
224
225
226
22nD_{2}
D_{3}
D_{4}
D_{5}
D_{6}
D_{n}D_{4}
D_{6}
D_{8}
D_{10}
D_{12}
D_{2n}[2,2]^{+}
[2,3]^{+}
[2,4]^{+}
[2,5]^{+}
[2,6]^{+}
[2,n]^{+}4
6
8
10
12
2nmmm
6m2
4/mmm
10m2
6/mmm
n/mmm
2nm22 2
3 2
4 2
5 2
6 2
n 2*222
*223
*224
*225
*226
*22nD_{2h}
D_{3h}
D_{4h}
D_{5h}
D_{6h}
D_{nh}±D_{4}
DD_{12}
±D_{8}
DD_{20}
±D_{12}
±D_{2n} / DD_{4n}[2,2]
[2,3]
[2,4]
[2,5]
[2,6]
[2,n]8
12
16
20
24
4n42m
3m
82m
5m
122m
2n2m
nm4 2
6 2
8 2
10 2
12 2
n 22*2
2*3
2*4
2*5
2*6
2*nD_{2d}
D_{3d}
D_{4d}
D_{5d}
D_{6d}
D_{nd}±D_{4}
±D_{6}
DD_{16}
±D_{10}
DD_{24}
DD_{4n} / ±D_{2n}[2^{+},4]
[2^{+},6]
[2^{+},8]
[2^{+},10]
[2^{+},12]
[2^{+},2n]8
12
16
20
24
4n23 3 3 332 T T [3,3]^{+} 12 m3 4 3 3*2 T_{h} ±T [3^{+},4] 24 43m 3 3 *332 T_{d} TO [3,3] 24 432 4 3 432 O O [3,4]^{+} 24 m3m 4 3 *432 O_{h} ±O [3,4] 48 532 5 3 532 I I [3,5]^{+} 60 53m 5 3 *532 I_{h} ±I [3,5] 120 (*) When the Intl entries are duplicated, the first is for even n, the second for odd n. The subset of pure reflectional 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 T_{d} A_{3} [3,3] 24 Tetrahedron O_{h} BC_{3} [4,3]
=[[3,3]]
48 Cube, octahedron
Stellated octahedronI_{h} H_{3} [5,3] 120 Icosahedron, dodecahedron D_{3h} A_{2}×A_{1} [3,2] 12 Triangular prism D_{4h} BC_{2}×A_{1} [4,2] 16 Square prism D_{5h} H_{2}×A_{1} [5,2] 20 Pentagonal prism D_{6h} G_{2}×A_{1} [6,2] 24 Hexagonal prism D_{nh} I_{2}(n)×A_{1} [n,2] 4n ngonal prism D_{2h} A_{1}^{3} [2,2] 8 Cuboid C_{3v} A_{2}×A_{1} [3] 6 Hosohedron C_{4v} BC_{2}×A_{1} [4] 8 C_{5v} H_{2}×A_{1} [5] 10 C_{6v} G_{2}×A_{1} [6] 12 C_{nv} I_{2}(n)×A_{1} [n] 2n C_{2v} A_{1}^{2} [2] 4 C_{s} A_{1} [ ] 2 Four dimensions
The fourdimensional point groups, limiting to purely reflectional groups, can be listed by their Coxeter group, and like the polyhedral groups of 3D, can be named by their related convex regular 4polytopes. Related pure rotational 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]^{+} has three 3fold gyration points and symmetry order 60. Frontback 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 regular/prismatic polytopes A_{4} [3,3,3] 120 5cell A_{4}×2 [[3,3,3]] 240 5cell dual compound BC_{4} [4,3,3] 384 16cell/Tesseract D_{4} [3^{1,1,1}] 192 Demitesseractic F_{4} [3,4,3] 1152 24cell F_{4}×2 [[3,4,3]] 2304 24cell dual compound H_{4} [5,3,3] 14400 120cell/600cell A_{3}×A_{1} [3,3,2] 48 Tetrahedral prism BC_{3}×A_{1} [4,3,2] 96 Octahedral prism H_{3}×A_{1} [5,3,2] 240 Icosahedral prism A_{2}×A_{2} [3,2,3] 36 Duoprism A_{2}×BC_{2} [3,2,4] 48 A_{2}×H_{2} [3,2,5] 60 A_{2}×G_{2} [3,2,6] 72 BC_{2}×BC_{2} [4,2,4] 64 BC_{2}×H_{2} [4,2,5] 80 BC_{2}×G_{2} [4,2,6] 96 H_{2}×H_{2} [5,2,5] 100 H_{2}×G_{2} [5,2,6] 120 G_{2}×G_{2} [6,2,6] 144 I_{2}(p)×I_{2}(q) [p,2,q] 4pq [[p,2,p]] 8p^{2} A_{2}×A_{1}^{2} [3,2,2] 24 BC_{2}×A_{1}^{2} [4,2,2] 32 H_{2}×A_{1}^{2} [5,2,2] 40 G_{2}×A_{1}^{2} [6,2,2] 48 I_{2}(p)×A_{1}^{2} [p,2,2] 8p A_{1}^{4} [2,2,2] 16 4orthotope Five dimensions
The fivedimensional point groups, limiting to purely reflectional groups, can be listed by their Coxeter group. Related pure rotational 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]^{+} has four 3fold gyration points and symmetry order 360.
Coxeter group Coxeter
diagramOrder Related regular/prismatic polytopes A_{5} [3,3,3,3] 720 5simplex A_{5}×2 [[3,3,3,3]] 1440 5simplex dual compound BC_{5} [4,3,3,3] 3840 5cube, 5orthoplex D_{5} [3^{2,1,1}] 1920 5demicube A_{4}×A_{1} [3,3,3,2] 240 5cell prism BC_{4}×A_{1} [4,3,3,2] 768 tesseract prism F_{4}×A_{1} [3,4,3,2] 2304 24cell prism H_{4}×A_{1} [5,3,3,2] 28800 600cell or 120cell prism D_{4}×A_{1} [3^{1,1,1},2] 384 Demitesseract prism A_{3}×A_{2} [3,3,2,3] 144 Duoprism A_{3}×BC_{2} [3,3,2,4] 192 A_{3}×H_{2} [3,3,2,5] 240 A_{3}×G_{2} [3,3,2,6] 288 A_{3}×I_{2}(p) [3,3,2,p] 48p BC_{3}×A_{2} [4,3,2,3] 288 BC_{3}×BC_{2} [4,3,2,4] 384 BC_{3}×H_{2} [4,3,2,5] 480 BC_{3}×G_{2} [4,3,2,6] 576 BC_{3}×I_{2}(p) [4,3,2,p] 96p H_{3}×A_{2} [5,3,2,3] 720 H_{3}×BC_{2} [5,3,2,4] 960 H_{3}×H_{2} [5,3,2,5] 1200 H_{3}×G_{2} [5,3,2,6] 1440 H_{3}×I_{2}(p) [5,3,2,p] 240p A_{3}×A_{1}^{2} [3,3,2,2] 96 BC_{3}×A_{1}^{2} [4,3,2,2] 192 H_{3}×A_{1}^{2} [5,3,2,2] 480 A_{2}^{2}×A_{1} [3,2,3,2] 72 duoprism prism A_{2}×BC_{2}×A_{1} [3,2,4,2] 96 A_{2}×H_{2}×A_{1} [3,2,5,2] 120 A_{2}×G_{2}×A_{1} [3,2,6,2] 144 BC_{2}^{2}×A_{1} [4,2,4,2] 128 BC_{2}×H_{2}×A_{1} [4,2,5,2] 160 BC_{2}×G_{2}×A_{1} [4,2,6,2] 192 H_{2}^{2}×A_{1} [5,2,5,2] 200 H_{2}×G_{2}×A_{1} [5,2,6,2] 240 G_{2}^{2}×A_{1} [6,2,6,2] 288 I_{2}(p)×I_{2}(q)×A_{1} [p,2,q,2] 8pq A_{2}×A_{1}^{3} [3,2,2,2] 48 BC_{2}×A_{1}^{3} [4,2,2,2] 64 H_{2}×A_{1}^{3} [5,2,2,2] 80 G_{2}×A_{1}^{3} [6,2,2,2] 96 I_{2}(p)×A_{1}^{3} [p,2,2,2] 16p A_{1}^{5} [2,2,2,2] 32 5orthotope Six dimensions
The sixdimensional point groups, limiting to purely reflectional groups, can be listed by their Coxeter group. Related pure rotational 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]^{+} has five 3fold gyration points and symmetry order 2520.
Coxeter group Coxeter
diagramOrder Related regular/prismatic polytopes A_{6} [3,3,3,3,3] 5040 (7!) 6simplex A_{6}×2 [[3,3,3,3,3]] 10080 (2×7!) 6simplex dual compound BC_{6} [4,3,3,3,3] 46080 (2^{6}×6!) 6cube, 6orthoplex D_{6} [3,3,3,3^{1,1}] 23040 (2^{5}×6!) 6demicube E_{6} [3,3^{2,2}] 51840 (72×6!) 1_{22}, 2_{21} A_{5}×A_{1} [3,3,3,3,2] 1440 (2×6!) 5simplex prism BC_{5}×A_{1} [4,3,3,3,2] 7680 (2^{6}×5!) 5cube prism D_{5}×A_{1} [3,3,3^{1,1},2] 3840 (2^{5}×5!) 5demicube prism A_{4}×I_{2}(p) [3,3,3,2,p] 240p Duoprism BC_{4}×I_{2}(p) [4,3,3,2,p] 768p F_{4}×I_{2}(p) [3,4,3,2,p] 2304p H_{4}×I_{2}(p) [5,3,3,2,p] 28800p D_{4}×I_{2}(p) [3,3^{1,1},2,p] 384p A_{4}×A_{1}^{2} [3,3,3,2,2] 480 BC_{4}×A_{1}^{2} [4,3,3,2,2] 1536 F_{4}×A_{1}^{2} [3,4,3,2,2] 4608 H_{4}×A_{1}^{2} [5,3,3,2,2] 57600 D_{4}×A_{1}^{2} [3,3^{1,1},2,2] 768 A_{3}^{2} [3,3,2,3,3] 576 A_{3}×BC_{3} [3,3,2,4,3] 1152 A_{3}×H_{3} [3,3,2,5,3] 2880 BC_{3}^{2} [4,3,2,4,3] 2304 BC_{3}×H_{3} [4,3,2,5,3] 5760 H_{3}^{2} [5,3,2,5,3] 14400 A_{3}×I_{2}(p)×A_{1} [3,3,2,p,2] 96p Duoprism prism BC_{3}×I_{2}(p)×A_{1} [4,3,2,p,2] 192p H_{3}×I_{2}(p)×A_{1} [5,3,2,p,2] 480p A_{3}×A_{1}^{3} [3,3,2,2,2] 192 BC_{3}×A_{1}^{3} [4,3,2,2,2] 384 H_{3}×A_{1}^{3} [5,3,2,2,2] 960 I_{2}(p)×I_{2}(q)×I_{2}(r) [p,2,q,2,r] 8pqr Triaprism I_{2}(p)×I_{2}(q)×A_{1}^{2} [p,2,q,2,2] 16pq I_{2}(p)×A_{1}^{4} [p,2,2,2,2] 32p A_{1}^{6} [2,2,2,2,2] 64 6orthotope Seven dimensions
The sevendimensional point groups, limiting to purely reflectional groups, can be listed by their Coxeter group. Related pure rotational 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 3fold gyration points and symmetry order 20160.
Coxeter group Coxeter diagram Order Related polytopes A_{7} [3,3,3,3,3,3] 40320 (8!) 7simplex A_{7}×2 [[3,3,3,3,3,3]] 80640 (2×8!) 7simplex dual compound BC_{7} [4,3,3,3,3,3] 645120 (2^{7}×7!) 7cube, 7orthoplex D_{7} [3,3,3,3,3^{1,1}] 322560 (2^{6}×7!) 7demicube E_{7} [3,3,3,3^{2,1}] 2903040 (8×9!) 3_{21}, 2_{31}, 1_{32} A_{6}×A_{1} [3,3,3,3,3,2] 10080 (2×7!) BC_{6}×A_{1} [4,3,3,3,3,2] 92160 (2^{7}×6!) D_{6}×A_{1} [3,3,3,3^{1,1},2] 46080 (2^{6}×6!) E_{6}×A_{1} [3,3,3^{2,1},2] 103680 (144×6!) A_{5}×I_{2}(p) [3,3,3,3,2,p] 1440p BC_{5}×I_{2}(p) [4,3,3,3,2,p] 7680p D_{5}×I_{2}(p) [3,3,3^{1,1},2,p] 3840p A_{5}×A_{1}^{2} [3,3,3,3,2,2] 2880 BC_{5}×A_{1}^{2} [4,3,3,3,2,2] 15360 D_{5}×A_{1}^{2} [3,3,3^{1,1},2,2] 7680 A_{4}×A_{3} [3,3,3,2,3,3] 2880 A_{4}×BC_{3} [3,3,3,2,4,3] 5760 A_{4}×H_{3} [3,3,3,2,5,3] 14400 BC_{4}×A_{3} [4,3,3,2,3,3] 9216 BC_{4}×BC_{3} [4,3,3,2,4,3] 18432 BC_{4}×H_{3} [4,3,3,2,5,3] 46080 H_{4}×A_{3} [5,3,3,2,3,3] 345600 H_{4}×BC_{3} [5,3,3,2,4,3] 691200 H_{4}×H_{3} [5,3,3,2,5,3] 1728000 F_{4}×A_{3} [3,4,3,2,3,3] 27648 F_{4}×BC_{3} [3,4,3,2,4,3] 55296 F_{4}×H_{3} [3,4,3,2,5,3] 138240 D_{4}×A_{3} [3^{1,1,1},2,3,3] 4608 D_{4}×BC_{3} [3,3^{1,1},2,4,3] 9216 D_{4}×H_{3} [3,3^{1,1},2,5,3] 23040 A_{4}×I_{2}(p)×A_{1} [3,3,3,2,p,2] 480p BC_{4}×I_{2}(p)×A_{1} [4,3,3,2,p,2] 1536p D_{4}×I_{2}(p)×A_{1} [3,3^{1,1},2,p,2] 768p F_{4}×I_{2}(p)×A_{1} [3,4,3,2,p,2] 4608p H_{4}×I_{2}(p)×A_{1} [5,3,3,2,p,2] 57600p A_{4}×A_{1}^{3} [3,3,3,2,2,2] 960 BC_{4}×A_{1}^{3} [4,3,3,2,2,2] 3072 F_{4}×A_{1}^{3} [3,4,3,2,2,2] 9216 H_{4}×A_{1}^{3} [5,3,3,2,2,2] 115200 D_{4}×A_{1}^{3} [3,3^{1,1},2,2,2] 1536 A_{3}^{2}×A_{1} [3,3,2,3,3,2] 1152 A_{3}×BC_{3}×A_{1} [3,3,2,4,3,2] 2304 A_{3}×H_{3}×A_{1} [3,3,2,5,3,2] 5760 BC_{3}^{2}×A_{1} [4,3,2,4,3,2] 4608 BC_{3}×H_{3}×A_{1} [4,3,2,5,3,2] 11520 H_{3}^{2}×A_{1} [5,3,2,5,3,2] 28800 A_{3}×I_{2}(p)×I_{2}(q) [3,3,2,p,2,q] 96pq BC_{3}×I_{2}(p)×I_{2}(q) [4,3,2,p,2,q] 192pq H_{3}×I_{2}(p)×I_{2}(q) [5,3,2,p,2,q] 480pq A_{3}×I_{2}(p)×A_{1}^{2} [3,3,2,p,2,2] 192p BC_{3}×I_{2}(p)×A_{1}^{2} [4,3,2,p,2,2] 384p H_{3}×I_{2}(p)×A_{1}^{2} [5,3,2,p,2,2] 960p A_{3}×A_{1}^{4} [3,3,2,2,2,2] 384 BC_{3}×A_{1}^{4} [4,3,2,2,2,2] 768 H_{3}×A_{1}^{4} [5,3,2,2,2,2] 1920 I_{2}(p)×I_{2}(q)×I_{2}(r)×A_{1} [p,2,q,2,r,2] 16pqr I_{2}(p)×I_{2}(q)×A_{1}^{3} [p,2,q,2,2,2] 32pq I_{2}(p)×A_{1}^{5} [p,2,2,2,2,2] 64p A_{1}^{7} [2,2,2,2,2,2] 128 Eight dimensions
The eightdimensional point groups, limiting to purely reflectional groups, can be listed by their Coxeter group. Related pure rotational 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 3fold gyration points and symmetry order 181440.
Coxeter group Coxeter diagram Order Related polytopes A_{8} [3,3,3,3,3,3,3] 362880 (9!) 8simplex A_{8}×2 [[3,3,3,3,3,3,3]] 725760 (2x9!) 8simplex dual compound BC_{8} [4,3,3,3,3,3,3] 10321920 (2^{8}8!) 8cube,8orthoplex D_{8} [3,3,3,3,3,3^{1,1}] 5160960 (2^{7}8!) 8demicube E_{8} [3,3,3,3,3^{2,1}] 696729600 4_{21}, 2_{41}, 1_{42} A_{7}×A_{1} [3,3,3,3,3,3,2] 80640 7simplex prism BC_{7}×A_{1} [4,3,3,3,3,3,2] 645120 7cube prism D_{7}×A_{1} [3,3,3,3,3^{1,1},2] 322560 7demicube prism E_{7}×A_{1} [3,3,3,3^{2,1},2] 5806080 3_{21} prism, 2_{31} prism, 1_{42} prism A_{6}×I_{2}(p) [3,3,3,3,3,2,p] 10080p duoprism BC_{6}×I_{2}(p) [4,3,3,3,3,2,p] 92160p D_{6}×I_{2}(p) [3,3,3,3^{1,1},2,p] 46080p E_{6}×I_{2}(p) [3,3,3^{2,1},2,p] 103680p A_{6}×A_{1}^{2} [3,3,3,3,3,2,2] 20160 BC_{6}×A_{1}^{2} [4,3,3,3,3,2,2] 184320 D_{6}×A_{1}^{2} [3^{3,1,1},2,2] 92160 E_{6}×A_{1}^{2} [3,3,3^{2,1},2,2] 207360 A_{5}×A_{3} [3,3,3,3,2,3,3] 17280 BC_{5}×A_{3} [4,3,3,3,2,3,3] 92160 D_{5}×A_{3} [3^{2,1,1},2,3,3] 46080 A_{5}×BC_{3} [3,3,3,3,2,4,3] 34560 BC_{5}×BC_{3} [4,3,3,3,2,4,3] 184320 D_{5}×BC_{3} [3^{2,1,1},2,4,3] 92160 A_{5}×H_{3} [3,3,3,3,2,5,3] BC_{5}×H_{3} [4,3,3,3,2,5,3] D_{5}×H_{3} [3^{2,1,1},2,5,3] A_{5}×I_{2}(p)×A_{1} [3,3,3,3,2,p,2] BC_{5}×I_{2}(p)×A_{1} [4,3,3,3,2,p,2] D_{5}×I_{2}(p)×A_{1} [3^{2,1,1},2,p,2] A_{5}×A_{1}^{3} [3,3,3,3,2,2,2] BC_{5}×A_{1}^{3} [4,3,3,3,2,2,2] D_{5}×A_{1}^{3} [3^{2,1,1},2,2,2] A_{4}×A_{4} [3,3,3,2,3,3,3] BC_{4}×A_{4} [4,3,3,2,3,3,3] D_{4}×A_{4} [3^{1,1,1},2,3,3,3] F_{4}×A_{4} [3,4,3,2,3,3,3] H_{4}×A_{4} [5,3,3,2,3,3,3] BC_{4}×BC_{4} [4,3,3,2,4,3,3] D_{4}×BC_{4} [3^{1,1,1},2,4,3,3] F_{4}×BC_{4} [3,4,3,2,4,3,3] H_{4}×BC_{4} [5,3,3,2,4,3,3] D_{4}×D_{4} [3^{1,1,1},2,3^{1,1,1}] F_{4}×D_{4} [3,4,3,2,3^{1,1,1}] H_{4}×D_{4} [5,3,3,2,3^{1,1,1}] F_{4}×F_{4} [3,4,3,2,3,4,3] H_{4}×F_{4} [5,3,3,2,3,4,3] H_{4}×H_{4} [5,3,3,2,5,3,3] A_{4}×A_{3}×A_{1} [3,3,3,2,3,3,2] duoprism prisms A_{4}×BC_{3}×A_{1} [3,3,3,2,4,3,2] A_{4}×H_{3}×A_{1} [3,3,3,2,5,3,2] BC_{4}×A_{3}×A_{1} [4,3,3,2,3,3,2] BC_{4}×BC_{3}×A_{1} [4,3,3,2,4,3,2] BC_{4}×H_{3}×A_{1} [4,3,3,2,5,3,2] H_{4}×A_{3}×A_{1} [5,3,3,2,3,3,2] H_{4}×BC_{3}×A_{1} [5,3,3,2,4,3,2] H_{4}×H_{3}×A_{1} [5,3,3,2,5,3,2] F_{4}×A_{3}×A_{1} [3,4,3,2,3,3,2] F_{4}×BC_{3}×A_{1} [3,4,3,2,4,3,2] F_{4}×H_{3}×A_{1} [3,4,2,3,5,3,2] D_{4}×A_{3}×A_{1} [3^{1,1,1},2,3,3,2] D_{4}×BC_{3}×A_{1} [3^{1,1,1},2,4,3,2] D_{4}×H_{3}×A_{1} [3^{1,1,1},2,5,3,2] A_{4}×I_{2}(p)×I_{2}(q) [3,3,3,2,p,2,q] triaprism BC_{4}×I_{2}(p)×I_{2}(q) [4,3,3,2,p,2,q] F_{4}×I_{2}(p)×I_{2}(q) [3,4,3,2,p,2,q] H_{4}×I_{2}(p)×I_{2}(q) [5,3,3,2,p,2,q] D_{4}×I_{2}(p)×I_{2}(q) [3^{1,1,1},2,p,2,q] A_{4}×I_{2}(p)×A_{1}^{2} [3,3,3,2,p,2,2] BC_{4}×I_{2}(p)×A_{1}^{2} [4,3,3,2,p,2,2] F_{4}×I_{2}(p)×A_{1}^{2} [3,4,3,2,p,2,2] H_{4}×I_{2}(p)×A_{1}^{2} [5,3,3,2,p,2,2] D_{4}×I_{2}(p)×A_{1}^{2} [3^{1,1,1},2,p,2,2] A_{4}×A_{1}^{4} [3,3,3,2,2,2,2] BC_{4}×A_{1}^{4} [4,3,3,2,2,2,2] F_{4}×A_{1}^{4} [3,4,3,2,2,2,2] H_{4}×A_{1}^{4} [5,3,3,2,2,2,2] D_{4}×A_{1}^{4} [3^{1,1,1},2,2,2,2] A_{3}×A_{3}×I_{2}(p) [3,3,2,3,3,2,p] BC_{3}×A_{3}×I_{2}(p) [4,3,2,3,3,2,p] H_{3}×A_{3}×I_{2}(p) [5,3,2,3,3,2,p] BC_{3}×BC_{3}×I_{2}(p) [4,3,2,4,3,2,p] H_{3}×BC_{3}×I_{2}(p) [5,3,2,4,3,2,p] H_{3}×H_{3}×I_{2}(p) [5,3,2,5,3,2,p] A_{3}×A_{3}×A_{1}^{2} [3,3,2,3,3,2,2] BC_{3}×A_{3}×A_{1}^{2} [4,3,2,3,3,2,2] H_{3}×A_{3}×A_{1}^{2} [5,3,2,3,3,2,2] BC_{3}×BC_{3}×A_{1}^{2} [4,3,2,4,3,2,2] H_{3}×BC_{3}×A_{1}^{2} [5,3,2,4,3,2,2] H_{3}×H_{3}×A_{1}^{2} [5,3,2,5,3,2,2] A_{3}×I_{2}(p)×I_{2}(q)×A_{1} [3,3,2,p,2,q,2] BC_{3}×I_{2}(p)×I_{2}(q)×A_{1} [4,3,2,p,2,q,2] H_{3}×I_{2}(p)×I_{2}(q)×A_{1} [5,3,2,p,2,q,2] A_{3}×I_{2}(p)×A_{1}^{3} [3,3,2,p,2,2,2] BC_{3}×I_{2}(p)×A_{1}^{3} [4,3,2,p,2,2,2] H_{3}×I_{2}(p)×A_{1}^{3} [5,3,2,p,2,2,2] A_{3}×A_{1}^{5} [3,3,2,2,2,2,2] BC_{3}×A_{1}^{5} [4,3,2,2,2,2,2] H_{3}×A_{1}^{5} [5,3,2,2,2,2,2] I_{2}(p)×I_{2}(q)×I_{2}(r)×I_{2}(s) [p,2,q,2,r,2,s] 16pqrs I_{2}(p)×I_{2}(q)×I_{2}(r)×A_{1}^{2} [p,2,q,2,r,2,2] 32pqr I_{2}(p)×I_{2}(q)×A_{1}^{4} [p,2,q,2,2,2,2] 64pq I_{2}(p)×A_{1}^{6} [p,2,2,2,2,2,2] 128p A_{1}^{8} [2,2,2,2,2,2,2] 256 See also
 Point groups in two dimensions
 Point groups in three dimensions
 Crystallography
 Crystallographic point group
 Molecular symmetry
 Space group
 Xray diffraction
 Bravais lattice
Notes
References
 H.S.M. Coxeter: Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, WileyInterscience Publication, 1995, ISBN 9780471010036 [2]
 (Paper 23) H.S.M. Coxeter, Regular and SemiRegular Polytopes II, [Math. Zeit. 188 (1985) 559591]
 H.S.M. Coxeter and W. O. J. Moser. Generators and Relations for Discrete Groups 4th ed, SpringerVerlag. New York. 1980
 N.W. Johnson: Geometries and Transformations, Manuscript, (2011) Chapter 11: Finite symmetry groups
External links
Categories: Crystallography
 Euclidean symmetries
 Group theory
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