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. Point-group elements can either be rotations (determinant of M = 1) or else reflections, improper rotations, rotation-reflections, 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.
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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 |
Point groups in two dimensions, sometimes called rosette groups.
They come in two infinite families:
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 | nn | [n]+ | n | Cyclic: n-fold rotations. Abstract group Zn, the group of integers under addition modulo n. |
Dn | nm | *nn | [n] | 2n | Dihedral: cyclic with reflections. Abstract group Dihn, 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 | |
---|---|---|---|---|---|
D3 | A2 | [3] | 6 | Equilateral triangle | |
D4 | BC2 | [4] | 8 | Square | |
D5 | H2 | [5] | 10 | Regular pentagon | |
D6 | G2 | [6] | 12 | Regular hexagon | |
Dn | I2(n) | [n] | 2n | Regular polygon | |
D2n | I2(2n) | [[n]]=[2n] | 4n | Regular polygon | |
D2 | A12 | [2] | 4 | Rectangle | |
D1 | A1 | [ ] | 2 | Digon |
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,*
Applying the crystallographic restriction theorem to these groups yields 32 Crystallographic point groups.
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(*) 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 | |
---|---|---|---|---|---|
Td | A3 | [3,3] | 24 | Tetrahedron | |
Oh | BC3 | [4,3] =[[3,3]] |
48 | Cube, octahedron Stellated octahedron |
|
Ih | H3 | [5,3] | 120 | Icosahedron, dodecahedron | |
D3h | A2×A1 | [3,2] | 12 | Triangular prism | |
D4h | BC2×A1 | [4,2] | 16 | Square 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 | |
D2h | A13 | [2,2] | 8 | Cuboid | |
C3v | A2×A1 | [3] | 6 | Hosohedron | |
C4v | BC2×A1 | [4] | 8 | ||
C5v | H2×A1 | [5] | 10 | ||
C6v | G2×A1 | [6] | 12 | ||
Cnv | I2(n)×A1 | [n] | 2n | ||
C2v | A12 | [2] | 4 | ||
Cs | A1 | [ ] | 2 |
The four-dimensional 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 4-polytopes. 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 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 regular/prismatic 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 | |
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 | |
BC3×A1 | [4,3,2] | 96 | Octahedral prism | |
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 | ||
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 | ||
[[p,2,p]] | 8p2 | |||
A2×A12 | [3,2,2] | 24 | ||
BC2×A12 | [4,2,2] | 32 | ||
H2×A12 | [5,2,2] | 40 | ||
G2×A12 | [6,2,2] | 48 | ||
I2(p)×A12 | [p,2,2] | 8p | ||
A14 | [2,2,2] | 16 | 4-orthotope |
The five-dimensional 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 3-fold gyration points and symmetry order 360.
Coxeter group | Coxeter diagram |
Order | Related regular/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 | |
A4×A1 | [3,3,3,2] | 240 | 5-cell prism | |
BC4×A1 | [4,3,3,2] | 768 | tesseract prism | |
F4×A1 | [3,4,3,2] | 2304 | 24-cell prism | |
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×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 |
The six-dimensional 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 3-fold gyration points and symmetry order 2520.
Coxeter group | Coxeter diagram |
Order | Related regular/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 |
The seven-dimensional 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 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 |
The eight-dimensional 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 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 (2x9!) | 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 | 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 |