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Point group

Group of geometric symmetries with at least one fixed point

Summary

Group of geometric symmetries with at least one fixed point

[[Image:Flag of Hong Kong.svg	240px]]
The Bauhinia blakeana flower on the Hong Kong region flag has C5 symmetry; the star on each petal has D5 symmetry.	[[File:Yin and Yang.svg	160px]]
The Yin and Yang symbol has C2 symmetry of geometry with inverted colors

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.

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

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. 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	[ ]+		1	identity
	D1	[ ]		2	reflection group

Two dimensions

Point groups in two dimensions, sometimes called rosette groups.

They come in two infinite families:

Cyclic groups Cn of n-fold rotation groups
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

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
D1	A1	[ ]			2	C1	[]+	1	digon
D2	A12	[2]			4	C2	[2]+	2	rectangle
D3	A2	[3]			6	C3	[3]+	3	equilateral triangle
D4	BC2	[4]			8	C4	[4]+	4	square
D5	H2	[5]			10	C5	[5]+	5	regular pentagon
D6	G2	[6]			12	C6	[6]+	6	regular hexagon
Dn	I2(n)	[n]			2n	Cn	[n]+	n	regular polygon
D2×2	A12×2	= [4]		=	8
D3×2	A2×2	= [6]		=	12
D4×2	BC2×2	= [8]		=	16
D5×2	H2×2	= [10]		=	20
D6×2	G2×2	= [12]		=	24
Dn×2	I2(n)×2	= [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 Schoenflies 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.

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
[[Image:Spherical digonal hosohedron2.png	80px]]	[[Image:Spherical square hosohedron2.png	80px]]	[[Image:Spherical hexagonal hosohedron2.png	80px]]	[[Image:Spherical octagonal hosohedron2.png	80px]]	[[Image:Spherical decagonal hosohedron2.png	80px]]	[[Image:Spherical dodecagonal hosohedron2.png	80px]]
[[Image:Spherical digonal bipyramid2.svg	80px]]	[[Image:Spherical square bipyramid2.svg	80px]]	[[Image:Spherical hexagonal bipyramid2.svg	80px]]	[[Image:Spherical octagonal bipyramid2.svg	80px]]	[[Image:Spherical decagonal bipyramid2.svg	80px]]	[[Image:Spherical dodecagonal bipyramid2.svg	80px]]
[[Image:Tetrahedral reflection domains.png	80px]]	[[Image:Octahedral reflection domains.png	80px]]	[[Image:Icosahedral reflection domains.png	80px]]

Intl*	Geo
Orbifold	Schoenflies	Coxeter	Order
1		1	C1	[ ]+	1
		×1	Ci = S2	[2+,2+]	2
= m	1	*1	Cs = C1v = C1h	[ ]	2
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
4/m
6/m
n/m
2
2
2
2
2
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


2×
3×
4×
5×
6×
n×	S4
S6
S8
S10
S12
S2n	[2+,4+]
[2+,6+]
[2+,8+]
[2+,10+]
[2+,12+]
[2+,2n+]	4
6
8
10
12
2n

Intl	Geo	Orbifold	Schoenflies	Coxeter	Order
222
32
422
52
622
n22
n2
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
m2
4/mmm
m2
6/mmm
n/mmm
m2	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
2m
m
2m
m
2m
2m
m	4
6
8
10
12
n
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		332	T	[3,3]+	12
m	4	3*2	Th	[3+,4]	24
3m	3 3	*332	Td	[3,3]	24
432		432	O	[3,4]+	24
mm	4 3	*432	Oh	[3,4]	48
532		532	I	[3,5]+	60
m	5 3	*532	Ih	[3,5]	120

|- |}

Reflection groups

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 , mapping the first and last mirrors onto each other, doubling the symmetry to 48, and isomorphic to the [4,3] group.

Schoenflies	Coxeter group	Coxeter diagram	Order
prismatic polyhedra
	Td	A3	[3,3]
	Td×Dih1 = Oh	A3×2 = BC3	= [4,3]	node	4	node_c1	3	node_c2}}
	Oh	BC3	[4,3]
	Ih	H3	[5,3]
	D3h	A2×A1	[3,2]
	D3h×Dih1 = D6h	A2×A1×2	[[3],2]	node_c1	6	node	2	node_c2}}
	D4h	BC2×A1	[4,2]
	D4h×Dih1 = D8h	BC2×A1×2	[[4],2] = [8,2]	node_c1	8	node	2	node_c2}}
	D5h	H2×A1	[5,2]
	D6h	G2×A1	[6,2]
	Dnh	I2(n)×A1	[n,2]
	Dnh×Dih1 = D2nh	I2(n)×A1×2	[[n],2]	node_c1	2x	n	node	2	node_c2}}
	D2h	A13	[2,2]
	D2h×Dih1	A13×2	[[2],2] = [4,2]	node_c1	4	node	2	node_c2}}
	D2h×Dih3 = Oh	A13×6	[3[2,2]] = [4,3]	node_c1	4	node	3	node}}
	C3v	A2	[1,3]
	C4v	BC2	[1,4]
	C5v	H2	[1,5]
	C6v	G2	[1,6]
	Cnv	I2(n)	[1,n]
	Cnv×Dih1 = C2nv	I2(n)×2	[1,[n]] = [1,2n]	node_c1	2x	n	node}}
	C2v	A12	[1,2]
	C2v×Dih1	A12×2	[1,[2]]	node_c1	4	node}}
	Cs	A1	[1,1]

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, Section 4, Tables 4.1–4.3.

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 with its order doubled to 240.

Coxeter group/notation	Coxeter diagram	Related polytopes
A4	[3,3,3]
A4×2
BC4	[4,3,3]
D4	[31,1,1]
D4×2 = BC4	1,1] = [4,3,3]	node	4	node_c1	3	node_c2	3	node_c3}}
D4×6 = F4	[3[31,1,1]] = [3,4,3]	node_c2	3	node_c1	4	node	3	node}}
F4	[3,4,3]
F4×2
H4	[5,3,3]
A3×A1	[3,3,2]
A3×A1×2	[[3,3],2] = [4,3,2]	node	4	node_c1	3	node_c2	2	node_c3}}
BC3×A1	[4,3,2]
H3×A1	[5,3,2]
A2×A2	[3,2,3]
A2×BC2	[3,2,4]
A2×H2	[3,2,5]
A2×G2	[3,2,6]
BC2×BC2	[4,2,4]
BC22×2
BC2×H2	[4,2,5]
BC2×G2	[4,2,6]
H2×H2	[5,2,5]
H2×G2	[5,2,6]
G2×G2	[6,2,6]
I2(p)×I2(q)	[p,2,q]
I2(2p)×I2(q)	[[p],2,q] = [2p,2,q]	node_c1	2x	p	node	2	node_c2	q	node_c3}}
I2(2p)×I2(2q)	,2, = [2p,2,2q]	node_c1	2x	p	node	2	node_c2	2x	q	node}}
I2(p)2×2
I2(2p)2×2	2p,2,2p}}	node_c1	2x	p	node	2	node_c1	2x	p	node}}
A2×A1×A1	[3,2,2]
BC2×A1×A1	[4,2,2]
H2×A1×A1	[5,2,2]
G2×A1×A1	[6,2,2]
I2(p)×A1×A1	[p,2,2]
I2(2p)×A1×A1×2	[[p],2,2] = [2p,2,2]	node_c1	2x	p	node	2	node_c2	2	node_c3}}
I2(p)×A12×2	[p,2,[2]] = [p,2,4]	node_c1	p	node_c2	2	node_c3	4	node}}
I2(2p)×A12×4	,2, = [2p,2,4]	node_c1	2x	p	node	2	node_c2	4	node}}
A1×A1×A1×A1	[2,2,2]
A12×A1×A1×2	[[2],2,2] = [4,2,2]	node_c1	4	node	2	node_c2	2	node_c3}}
A12×A12×4	,2, = [4,2,4]	node_c1	4	node	2	node_c2	4	node}}
A13×A1×6	[3[2,2],2] = [4,3,2]	node_c1	4	node	3	node	2	node_c2}}
A14×24	[3,3[2,2,2]] = [4,3,3]	node_c1	4	node	3	node	3	node}}

Five dimensions

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]
	A5×2
	BC5	[4,3,3,3]
	D5	[32,1,1]
	D5×2	1,1]		=
A4×A1	[3,3,3,2]
A4×A1×2	[[3,3,3],2]
BC4×A1	[4,3,3,2]
F4×A1	[3,4,3,2]
F4×A1×2	[[3,4,3],2]
H4×A1	[5,3,3,2]
D4×A1	[31,1,1,2]
A3×A2	[3,3,2,3]
A3×A2×2	[[3,3],2,3]
A3×BC2	[3,3,2,4]
A3×H2	[3,3,2,5]
A3×G2	[3,3,2,6]
A3×I2(p)	[3,3,2,p]
BC3×A2	[4,3,2,3]
BC3×BC2	[4,3,2,4]
BC3×H2	[4,3,2,5]
BC3×G2	[4,3,2,6]
BC3×I2(p)	[4,3,2,p]
H3×A2	[5,3,2,3]
H3×BC2	[5,3,2,4]
H3×H2	[5,3,2,5]
H3×G2	[5,3,2,6]
H3×I2(p)	[5,3,2,p]
A3×A12	[3,3,2,2]
BC3×A12	[4,3,2,2]
H3×A12	[5,3,2,2]
A22×A1	[3,2,3,2]
A2×BC2×A1	[3,2,4,2]
A2×H2×A1	[3,2,5,2]
A2×G2×A1	[3,2,6,2]
BC22×A1	[4,2,4,2]
BC2×H2×A1	[4,2,5,2]
BC2×G2×A1	[4,2,6,2]
H22×A1	[5,2,5,2]
H2×G2×A1	[5,2,6,2]
G22×A1	[6,2,6,2]
I2(p)×I2(q)×A1	[p,2,q,2]
A2×A13	[3,2,2,2]
BC2×A13	[4,2,2,2]
H2×A13	[5,2,2,2]
G2×A13	[6,2,2,2]
I2(p)×A13	[p,2,2,2]
A15	[2,2,2,2]
A15×(2!)	[[2],2,2,2]		=
A15×(2!×2!)	,2,[2],2]		=
A15×(3!)	[3[2,2],2,2]		=
A15×(3!×2!)	[3[2,2],2,		=
A15×(4!)	[3,3[2,2,2],2]]		=
A15×(5!)	[3,3,3[2,2,2,2]]		=

Six dimensions

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!)
	A6×2		10080 (2×7!)
	BC6	[4,3,3,3,3]	46080 (26×6!)
	D6	[3,3,3,31,1]	23040 (25×6!)
	E6	[3,32,2]	51840 (72×6!)
	A5×A1	[3,3,3,3,2]	1440 (2×6!)
	BC5×A1	[4,3,3,3,2]	7680 (26×5!)
	D5×A1	[3,3,31,1,2]	3840 (25×5!)
	A4×I2(p)	[3,3,3,2,p]	240p
	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
	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
	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

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 diagram	Order
A7	[3,3,3,3,3,3]	40320 (8!)
A7×2		80640 (2×8!)
BC7	[4,3,3,3,3,3]	645120 (27×7!)
D7	[3,3,3,3,31,1]	322560 (26×7!)
E7	[3,3,3,32,1]	2903040 (8×9!)
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 diagram	Order
A8	[3,3,3,3,3,3,3]	362880 (9!)
A8×2		725760 (2×9!)
BC8	[4,3,3,3,3,3,3]	10321920 (288!)
D8	[3,3,3,3,3,31,1]	5160960 (278!)
E8	[3,3,3,3,32,1]	696729600 (192×10!)
A7×A1	[3,3,3,3,3,3,2]	80640
BC7×A1	[4,3,3,3,3,3,2]	645120
D7×A1	[3,3,3,3,31,1,2]	322560
E7×A1	[3,3,3,32,1,2]	5806080
A6×I2(p)	[3,3,3,3,3,2,p]	10080p
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]
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]
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

References

Wikipedia Source

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