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Coxeter notation

From Wikipedia, the free encyclopedia
Classification system for symmetry groups in geometry
Fundamental domains of reflective 3D point groups
,[ ] = [1]
C1v		,[2]
C2v		,[3]
C3v		,[4]
C4v		,[5]
C5v		,[6]
C6v

Order 2
Order 4
Order 6
Order 8
Order 10
Order 12

[2] = [2,1]
D1h
[2,2]
D2h
[2,3]
D3h
[2,4]
D4h
[2,5]
D5h
[2,6]
D6h

Order 4
Order 8
Order 12
Order 16
Order 20
Order 24
,[3,3],Td,[4,3],Oh,[5,3],Ih

Order 24
Order 48
Order 120
Coxeter notation expressesCoxeter groups as a list of branch orders of aCoxeter diagram, like thepolyhedral groups, = [p,q].Dihedral groups,, can be expressed as a product[ ]×[n] or in a single symbol with an explicit order 2 branch,[2,n].

Ingeometry,Coxeter notation (alsoCoxeter symbol) is a system of classifyingsymmetry groups, describing the angles between fundamental reflections of aCoxeter group in a bracketed notation expressing the structure of aCoxeter-Dynkin diagram, with modifiers to indicate certain subgroups. The notation is named afterH. S. M. Coxeter, and has been more comprehensively defined byNorman Johnson.

Reflectional groups

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Further information:Point group

ForCoxeter groups, defined by pure reflections, there is a direct correspondence between the bracket notation andCoxeter-Dynkin diagram. The numbers in the bracket notation represent the mirror reflection orders in the branches of the Coxeter diagram. It uses the same simplification, suppressing 2s between orthogonal mirrors.

The Coxeter notation is simplified with exponents to represent the number of branches in a row for linear diagram. So theAn group is represented by [3n−1], to implyn nodes connected byn−1 order-3 branches. ExampleA2 = [3,3] = [32] or [31,1] represents diagrams or.

Coxeter initially represented bifurcating diagrams with vertical positioning of numbers, but later abbreviated with an exponent notation, like [...,3p,q] or [3p,q,r], starting with [31,1,1] or [3,31,1] = or as D4. Coxeter allowed for zeros as special cases to fit theAn family, likeA3 = [3,3,3,3] = [34,0,0] = [34,0] = [33,1] = [32,2], like = =.

Coxeter groups formed by cyclic diagrams are represented by parentheseses inside of brackets, like [(p,q,r)] = for thetriangle group (p q r). If the branch orders are equal, they can be grouped as an exponent as the length the cycle in brackets, like [(3,3,3,3)] = [3[4]], representing Coxeter diagram or. can be represented as [3,(3,3,3)] or [3,3[3]].

More complicated looping diagrams can also be expressed with care. Theparacompact Coxeter group can be represented by Coxeter notation [(3,3,(3),3,3)], with nested/overlapping parentheses showing two adjacent [(3,3,3)] loops, and is also represented more compactly as [3[ ]×[ ]], representing therhombic symmetry of the Coxeter diagram. The paracompact complete graph diagram or, is represented as [3[3,3]] with the superscript [3,3] as the symmetry of itsregular tetrahedron coxeter diagram.

Finite groups
RankGroup
symbol		Bracket
notation		Coxeter
diagram
2A2[3]
2B2[4]
2H2[5]
2G2[6]
2I2(p)[p]
3Ih,H3[5,3]
3Td,A3[3,3]
3Oh,B3[4,3]
4A4[3,3,3]
4B4[4,3,3]
4D4[31,1,1]
4F4[3,4,3]
4H4[5,3,3]
nAn[3n−1]..
nBn[4,3n−2]...
nDn[3n−3,1,1]...
6E6[32,2,1]
7E7[33,2,1]
8E8[34,2,1]
Affine groups
Group
symbol		Bracket
notation		Coxeter diagram
I~1,A~1{\displaystyle {\tilde {I}}_{1},{\tilde {A}}_{1}}[∞]
A~2{\displaystyle {\tilde {A}}_{2}}[3[3]]
C~2{\displaystyle {\tilde {C}}_{2}}[4,4]
G~2{\displaystyle {\tilde {G}}_{2}}[6,3]
A~3{\displaystyle {\tilde {A}}_{3}}[3[4]]
B~3{\displaystyle {\tilde {B}}_{3}}[4,31,1]
C~3{\displaystyle {\tilde {C}}_{3}}[4,3,4]
A~4{\displaystyle {\tilde {A}}_{4}}[3[5]]
B~4{\displaystyle {\tilde {B}}_{4}}[4,3,31,1]
C~4{\displaystyle {\tilde {C}}_{4}}[4,3,3,4]
D~4{\displaystyle {\tilde {D}}_{4}}[ 31,1,1,1]
F~4{\displaystyle {\tilde {F}}_{4}}[3,4,3,3]
A~n{\displaystyle {\tilde {A}}_{n}}[3[n+1]]...
or
...
B~n{\displaystyle {\tilde {B}}_{n}}[4,3n−3,31,1]...
C~n{\displaystyle {\tilde {C}}_{n}}[4,3n−2,4]...
D~n{\displaystyle {\tilde {D}}_{n}}[ 31,1,3n−4,31,1]...
E~6{\displaystyle {\tilde {E}}_{6}}[32,2,2]
E~7{\displaystyle {\tilde {E}}_{7}}[33,3,1]
E~8=E9{\displaystyle {\tilde {E}}_{8}=E_{9}}[35,2,1]
Hyperbolic groups
Group
symbol		Bracket
notation		Coxeter
diagram
[p,q]
with2(p + q) < pq
[(p,q,r)]
with1p+1q+1r<1{\displaystyle {\frac {1}{p}}+{\frac {1}{q}}+{\frac {1}{r}}<1}
BH¯3{\displaystyle {\overline {BH}}_{3}}[4,3,5]
K¯3{\displaystyle {\overline {K}}_{3}}[5,3,5]
J¯3,H~3{\displaystyle {\overline {J}}_{3},{\tilde {H}}_{3}}[3,5,3]
DH¯3{\displaystyle {\overline {DH}}_{3}}[5,31,1]
AB^3{\displaystyle {\widehat {AB}}_{3}}[(3,3,3,4)] 
AH^3{\displaystyle {\widehat {AH}}_{3}}[(3,3,3,5)] 
BB^3{\displaystyle {\widehat {BB}}_{3}}[(3,4,3,4)]
BH^3{\displaystyle {\widehat {BH}}_{3}}[(3,4,3,5)]
HH^3{\displaystyle {\widehat {HH}}_{3}}[(3,5,3,5)]
H¯4,H~4,H5{\displaystyle {\overline {H}}_{4},{\tilde {H}}_{4},H_{5}}[3,3,3,5]
BH¯4{\displaystyle {\overline {BH}}_{4}}[4,3,3,5]
K¯4{\displaystyle {\overline {K}}_{4}}[5,3,3,5]
DH¯4{\displaystyle {\overline {DH}}_{4}}[5,3,31,1]
AF^4{\displaystyle {\widehat {AF}}_{4}}[(3,3,3,3,4)]

For the affine and hyperbolic groups, the subscript is one less than the number of nodes in each case, since each of these groups was obtained by adding a node to a finite group's diagram.

Unconnected groups

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The Coxeter diagram usually leaves order-2 branches undrawn, but the bracket notation includes an explicit2 to connect the subgraphs. So the Coxeter diagram =A2×A2 = 2A2 can be represented by [3]×[3] = [3]2 = [3,2,3]. Sometimes explicit 2-branches may be included either with a 2 label, or with a line with a gap: or, as an identical presentation as [3,2,3].

Rank and dimension

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Coxeter point group rank is equal to the number of nodes which is also equal to the dimension. A single mirror exists in 1-dimension, [ ],, while in 2-dimensions [1], or [ ]×[ ]+. The1 is a place-holder, not an actual branch order, but a marker for an orthogonal inactive mirror. The notation [n,1], represents a rank 3 group, as [n]×[ ]+ or. Similarly, [1,1] as [ ]×[ ]+×[ ]+ or order 2 and [1,1]+ as [ ]+×[ ]+×[ ]+ or, order 1!

Subgroups

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Coxeter's notation represents rotational/translational symmetry by adding a+ superscript operator outside the brackets, [X]+ which cuts the order of the group [X] in half, thus an index 2 subgroup. This operator implies an even number of operators must be applied, replacing reflections with rotations (or translations). When applied to a Coxeter group, this is called adirect subgroup because what remains are only direct isometries without reflective symmetry.

The+ operators can also be applied inside of the brackets, like [X,Y+] or [X,(Y,Z)+], and creates"semidirect" subgroups that may include both reflective and nonreflective generators. Semidirect subgroups can only apply to Coxeter group subgroups that have even order branches adjacent to it. Elements by parentheses inside of a Coxeter group can be give a+ superscript operator, having the effect of dividing adjacent ordered branches into half order, thus is usually only applied with even numbers. For example, [4,3+] and [4,(3,3)+] ().

If applied with adjacent odd branch, it doesn't create a subgroup of index 2, but instead creates overlapping fundamental domains, like [5,1+] = [5/2], which can define doubly wrapped polygons like apentagram, {5/2}, and [5,3+] relates toSchwarz triangle [5/2,3],density 2.

Examples on Rank 2 groups
GroupOrderGeneratorsSubgroupOrderGeneratorsNotes
[p]2p{0,1}[p]+p{01}Direct subgroup
[2p+] = [2p]+2p{01}[2p+]+ = [2p]+2 = [p]+p{0101}
[2p]4p{0,1}[1+,2p] = [p] = =2p{101,1}Half subgroups
[2p,1+] = [p] = ={0,010}
[1+,2p,1+] = [2p]+2 = [p]+ = =p{0101}Quarter group

Groups without neighboring+ elements can be seen in ringed nodes Coxeter-Dynkin diagram foruniform polytopes and honeycomb are related tohole nodes around the+ elements, empty circles with the alternated nodes removed. So thesnub cube, has symmetry [4,3]+ (), and thesnub tetrahedron, has symmetry [4,3+] (), and ademicube, h{4,3} = {3,3} ( or =) has symmetry [1+,4,3] = [3,3] ( or = =).

Note:Pyritohedral symmetry can be written as, separating the graph with gaps for clarity, with the generators {0,1,2} from the Coxeter group, producing pyritohedral generators {0,12}, a reflection and 3-fold rotation. And chiral tetrahedral symmetry can be written as or, [1+,4,3+] = [3,3]+, with generators {12,0120}.

Halving subgroups and extended groups

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Further information:§ Extended_symmetry
Example halving operations

[1,4,1] = [4]		 = =
[1+,4,1]=[2]=[ ]×[ ]
= =
[1,4,1+]=[2]=[ ]×[ ]		 = = =
[1+,4,1+] = [2]+

Johnson extends the+ operator to work with a placeholder 1+ nodes, which removes mirrors, doubling the size of the fundamental domain and cuts the group order in half.[1] In general this operation only applies to individual mirrors bounded by even-order branches. The1 represents a mirror so [2p] can be seen as [2p,1], [1,2p], or [1,2p,1], like diagram or, with 2 mirrors related by an order-2p dihedral angle. The effect of a mirror removal is to duplicate connecting nodes, which can be seen in the Coxeter diagrams: =, or in bracket notation:[1+,2p,1] = [1,p,1] = [p].

Each of these mirrors can be removed so h[2p] = [1+,2p,1] = [1,2p,1+] = [p], a reflective subgroup index 2. This can be shown in a Coxeter diagram by adding a+ symbol above the node: = =.

If both mirrors are removed, a quarter subgroup is generated, with the branch order becoming a gyration point of half the order:

q[2p] = [1+,2p,1+] = [p]+, a rotational subgroup of index 4. = = = =.

For example, (with p=2): [4,1+] = [1+,4] = [2] = [ ]×[ ], order 4. [1+,4,1+] = [2]+, order 2.

The opposite to halving is doubling[2] which adds a mirror, bisecting a fundamental domain, and doubling the group order.

[[p]] = [2p]

Halving operations apply for higher rank groups, liketetrahedral symmetry is a half group ofoctahedral group: h[4,3] = [1+,4,3] = [3,3], removing half the mirrors at the 4-branch. The effect of a mirror removal is to duplicate all connecting nodes, which can be seen in the Coxeter diagrams: =, h[2p,3] = [1+,2p,3] = [(p,3,3)].

If nodes are indexed, half subgroups can be labeled with new mirrors as composites. Like, generators {0,1} has subgroup =, generators {1,010}, where mirror 0 is removed, and replaced by a copy of mirror 1 reflected across mirror 0. Also given, generators {0,1,2}, it has half group =, generators {1,2,010}.

Doubling by adding a mirror also applies in reversing the halving operation: [[3,3]] = [4,3], or more generally [[(q,q,p)]] = [2p,q].

Tetrahedral symmetryOctahedral symmetry

Td, [3,3] = [1+,4,3]
= =
(Order 24)
Oh, [4,3] = [[3,3]]

(Order 48)

Radical subgroups

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A radical subgroup is similar to an alternation, but removes the rotational generators.

Johnson also added anasterisk or star* operator for "radical" subgroups,[3] that acts similar to the+ operator, but removes rotational symmetry. The index of the radical subgroup is the order of the removed element. For example, [4,3*] ≅ [2,2]. The removed [3] subgroup is order 6 so [2,2] is an index 6 subgroup of [4,3].

The radical subgroups represent the inverse operation to anextended symmetry operation. For example, [4,3*] ≅ [2,2], and in reverse [2,2] can be extended as [3[2,2]] ≅ [4,3]. The subgroups can be expressed as a Coxeter diagram: or ≅. The removed node (mirror) causes adjacent mirror virtual mirrors to become real mirrors.

If [4,3] has generators {0,1,2}, [4,3+], index 2, has generators {0,12}; [1+,4,3] ≅ [3,3], index 2 has generators {010,1,2}; while radical subgroup [4,3*] ≅ [2,2], index 6, has generators {01210, 2, (012)3}; and finally [1+,4,3*], index 12 has generators {0(12)20, (012)201}.

Trionic subgroups

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Rank 2 example, [6] trionic subgroups with 3 colors of mirror lines
Example on octahedral symmetry: [4,3⅄] = [2,4].
Example trionic subgroup on hexagonal symmetry [6,3] maps onto a larger [6,3] symmetry.
Rank 3
Example trionic subgroups on octagonal symmetry [8,3] maps onto larger [4,8] symmetries.
Rank 4

Atrionic subgroup is an index 3 subgroup. Johnson defines atrionic subgroup with operator ⅄, index 3. For rank 2 Coxeter groups, [3], the trionic subgroup, [3⅄] is [ ], a single mirror. And for [3p], the trionic subgroup is [3p]⅄ ≅ [p]. Given, with generators {0,1}, has 3 trionic subgroups. They can be differentiated by putting the ⅄ symbol next to the mirror generator to be removed, or on a branch for both: [3p,1⅄] = =, =, and [3p⅄] = = with generators {0,10101}, {01010,1}, or {101,010}.

Trionic subgroups of tetrahedral symmetry: [3,3]⅄ ≅ [2+,4], relating the symmetry of theregular tetrahedron andtetragonal disphenoid.

For rank 3 Coxeter groups, [p,3], there is a trionic subgroup [p,3⅄] ≅ [p/2,p], or =. For example, the finite group [4,3⅄] ≅ [2,4], and Euclidean group [6,3⅄] ≅ [3,6], and hyperbolic group [8,3⅄] ≅ [4,8].

An odd-order adjacent branch,p, will not lower the group order, but create overlapping fundamental domains. The group order stays the same, while thedensity increases. For example, theicosahedral symmetry, [5,3], of the regular polyhedraicosahedron becomes [5/2,5], the symmetry of 2 regular star polyhedra. It also relates the hyperbolic tilings {p,3}, andstar hyperbolic tilings {p/2,p}

For rank 4, [q,2p,3⅄] = [2p,((p,q,q))], =.

For example, [3,4,3⅄] = [4,3,3], or =, generators {0,1,2,3} in [3,4,3] with the trionic subgroup [4,3,3] generators {0,1,2,32123}. For hyperbolic groups, [3,6,3⅄] = [6,3[3]], and [4,4,3⅄] = [4,4,4].

Trionic subgroups of tetrahedral symmetry

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[3,3]⅄ ≅ [2+,4] as one of 3 sets of 2 orthogonal mirrors instereographic projection. The red, green, and blue represent 3 sets of mirrors, and the gray lines are removed mirrors, leaving 2-fold gyrations (purple diamonds).
Trionic relations of [3,3]

Johnson identified two specifictrionic subgroups[4] of [3,3], first an index 3 subgroup [3,3]⅄ ≅ [2+,4], with [3,3] ( = =) generators {0,1,2}. It can also be written as [(3,3,2⅄)] () as a reminder of its generators {02,1}. This symmetry reduction is the relationship between the regulartetrahedron and thetetragonal disphenoid, represent a stretching of a tetrahedron perpendicular to two opposite edges.

Secondly he identifies a related index 6 subgroup [3,3]Δ or [(3,3,2⅄)]+ (), index 3 from [3,3]+ ≅ [2,2]+, with generators {02,1021}, from [3,3] and its generators {0,1,2}.

These subgroups also apply within larger Coxeter groups with [3,3] subgroup with neighboring branches all even order.

Trionic subgroup relations of [3,3,4]

For example, [(3,3)+,4], [(3,3)⅄,4], and [(3,3)Δ,4] are subgroups of [3,3,4], index 2, 3 and 6 respectively. The generators of [(3,3)⅄,4] ≅ [[4,2,4]] ≅ [8,2+,8], order 128, are {02,1,3} from [3,3,4] generators {0,1,2,3}. And [(3,3)Δ,4] ≅ [[4,2+,4]], order 64, has generators {02,1021,3}. As well, [3⅄,4,3⅄] ≅ [(3,3)⅄,4].

Also related [31,1,1] = [3,3,4,1+] has trionic subgroups: [31,1,1]⅄ = [(3,3)⅄,4,1+], order 64, and 1=[31,1,1]Δ = [(3,3)Δ,4,1+] ≅ [[4,2+,4]]+, order 32.

Central inversion

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A 2D central inversion is a 180 degree rotation, [2]+

Acentral inversion, order 2, is operationally differently by dimension. The group [ ]n = [2n−1] representsn orthogonal mirrors in n-dimensional space, or ann-flat subspace of a higher dimensional space. The mirrors of the group [2n−1] are numbered⁠0…n−1{\displaystyle 0\dots n-1}⁠. The order of the mirrors doesn't matter in the case of an inversion. The matrix of a central inversion is⁠−I{\displaystyle -I}⁠, the Identity matrix with negative one on the diagonal.

From that basis, the central inversion has a generator as the product of all the orthogonal mirrors. In Coxeter notation this inversion group is expressed by adding an alternation+ to each 2 branch. The alternation symmetry is marked on Coxeter diagram nodes as open nodes.

ACoxeter-Dynkin diagram can be marked up with explicit 2 branches defining a linear sequence of mirrors, open-nodes, and shared double-open nodes to show the chaining of the reflection generators.

For example, [2+,2] and [2,2+] are subgroups index 2 of [2,2],, and are represented as (or) and (or) with generators {01,2} and {0,12} respectively. Their common subgroup index 4 is [2+,2+], and is represented by (or), with the double-open marking a shared node in the two alternations, and a singlerotoreflection generator {012}.

DimensionCoxeter notationOrderCoxeter diagramOperationGenerator
2[2]+2180°rotation, C2{01}
3[2+,2+]2rotoreflection, Ci or S2{012}
4[2+,2+,2+]2double rotation{0123}
5[2+,2+,2+,2+]2double rotary reflection{01234}
6[2+,2+,2+,2+,2+]2triple rotation{012345}
7[2+,2+,2+,2+,2+,2+]2triple rotary reflection{0123456}

Rotations and rotary reflections

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Rotations androtary reflections are constructed by a single single-generator product of all the reflections of a prismatic group, [2p]×[2q]×... wheregcd(p,q,...)=1, they are isomorphic to the abstractcyclic group Zn, of ordern=2pq.

The 4-dimensional double rotations, [2p+,2+,2q+] (withgcd(p,q)=1), which include a central group, and are expressed by Conway as ±[Cp×Cq],[5] order 2pq. From Coxeter diagram, generators {0,1,2,3}, requires two generator for [2p+,2+,2q+], as {0123,0132}. Half groups, [2p+,2+,2q+]+, or cyclic graph, [(2p+,2+,2q+,2+)], expressed by Conway is [Cp×Cq], orderpq, with one generator, like {0123}.

If there is a common factorf, the double rotation can be written as1⁄f[2pf+,2+,2qf+] (withgcd(p,q)=1), generators {0123,0132}, order 2pqf. For example,p=q=1,f=2,1⁄2[4+,2+,4+] is order 4. And1⁄f[2pf+,2+,2qf+]+, generator {0123}, is orderpqf. For example,1⁄2[4+,2+,4+]+ is order 2, acentral inversion.

In general an-rotation group, [2p1+,2,2p2+,2,...,pn+] may require up ton generators if gcd(p1,..,pn)>1, as a product of all mirrors, and then swapping sequential pairs. The half group, [2p1+,2,2p2+,2,...,pn+]+ has generators squared.n-rotary reflections are similar.

Examples
DimensionCoxeter notationOrderCoxeter diagramOperationGeneratorsDirect subgroup
2[2p]+2pRotation{01}[2p]+2 = [p]+Simple rotation:
[2p]+2 = [p]+
orderp
3[2p+,2+]rotary reflection{012}[2p+,2+]+ = [p]+
4[2p+,2+,2+]double rotation{0123}[2p+,2+,2+]+ = [p]+
5[2p+,2+,2+,2+]double rotary reflection{01234}[2p+,2+,2+,2+]+ = [p]+
6[2p+,2+,2+,2+,2+]triple rotation{012345}[2p+,2+,2+,2+,2+]+ = [p]+
7[2p+,2+,2+,2+,2+,2+]triple rotary reflection{0123456}[2p+,2+,2+,2+,2+,2+]+ = [p]+
4[2p+,2+,2q+]2pqdouble rotation{0123,
0132}		[2p+,2+,2q+]+Double rotation:
[2p+,2+,2q+]+
orderpq
5[2p+,2+,2q+,2+]double rotary reflection{01234,
01243}		[2p+,2+,2q+,2+]+
6[2p+,2+,2q+,2+,2+]triple rotation{012345,
012354,
013245}		[2p+,2+,2q+,2+,2+]+
7[2p+,2+,2q+,2+,2+,2+]triple rotary reflection{0123456,
0123465,
0124356,
0124356}		[2p+,2+,2q+,2+,2+,2+]+
6[2p+,2+,2q+,2+,2r+]2pqrtriple rotation{012345,
012354,
013245}		[2p+,2+,2q+,2+,2r+]+Triple rotation:
[2p+,2+,2q+,2+,2r+]+
orderpqr
7[2p+,2+,2q+,2+,2r+,2+]triple rotary reflection{0123456,
0123465,
0124356,
0213456}		[2p+,2+,2q+,2+,2r+,2+]+

Commutator subgroups

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Hasse diagram subgroups of [4,4], down to its commutator subgroup, index 8

Simple groups with only odd-order branch elements have only a single rotational/translational subgroup of order 2, which is also thecommutator subgroup, examples [3,3]+, [3,5]+, [3,3,3]+, [3,3,5]+. For other Coxeter groups with even-order branches, the commutator subgroup has index 2c, where c is the number of disconnected subgraphs when all the even-order branches are removed.[6]

For example, [4,4] has three independent nodes in the Coxeter diagram when the4s are removed, so its commutator subgroup is index 23, and can have different representations, all with three+ operators: [4+,4+]+, [1+,4,1+,4,1+], [1+,4,4,1+]+, or [(4+,4+,2+)]. A general notation can be used with +c as a group exponent, like [4,4]+3.

Example subgroups

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Rank 2 example subgroups

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Dihedral symmetry groups with even-orders have a number of subgroups. This example shows two generator mirrors of [4] in red and green, and looks at all subgroups by halfing, rank-reduction, and their direct subgroups. The group [4], has two mirror generators 0, and 1. Each generate two virtual mirrors 101 and 010 by reflection across the other.

Subgroups of [4]
Index12 (half)4 (Rank-reduction)
Diagram
Coxeter

[1,4,1] = [4]		 = =
[1+,4,1] = [1+,4] = [2]		 = =
[1,4,1+] = [4,1+] = [2]
[2,1+] = [1] = [ ]
[1+,2] = [1] = [ ]
Generators{0,1}{101,1}{0,010}{0}{1}
Direct subgroups
Index248
Diagram
Coxeter
[4]+		 = = =
[4]+2 = [1+,4,1+] = [2]+
[ ]+
Generators{01}{(01)2}{02} = {12} = {(01)4} = { }

Rank 3 Euclidean example subgroups

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The [4,4] group has 15 small index subgroups. This table shows them all, with a yellow fundamental domain for pure reflective groups, and alternating white and blue domains which are paired up to make rotational domains. Cyan, red, and green mirror lines correspond to the same colored nodes in the Coxeter diagram. Subgroup generators can be expressed as products of the original 3 mirrors of the fundamental domain, {0,1,2}, corresponding to the 3 nodes of the Coxeter diagram,. A product of two intersecting reflection lines makes a rotation, like {012}, {12}, or {02}. Removing a mirror causes two copies of neighboring mirrors, across the removed mirror, like {010}, and {212}. Two rotations in series cut the rotation order in half, like {0101} or {(01)2}, {1212} or {(02)2}. A product of all three mirrors creates atransreflection, like {012} or {120}.

Small index subgroups of [4,4]
Index124
Diagram
Coxeter
[1,4,1,4,1] = [4,4]
[1+,4,4]
=		[4,4,1+]
=		[4,1+,4]
=		[1+,4,4,1+]
=		[4+,4+]
Generators{0,1,2}{010,1,2}{0,1,212}{0,101,121,2}{010,1,212,20102}{(01)2,(12)2,012,120}
Orbifold*442*222222×
Semidirect subgroups
Index24
Diagram
Coxeter[4,4+]
[4+,4]
[(4,4,2+)]
=		[4,1+,4,1+]
= =		[1+,4,1+,4]
= =
Generators{0,12}{01,2}{02,1,212}{0,101,(12)2}{(01)2,121,2}
Orbifold4*22*22
Direct subgroups
Index248
Diagram
Coxeter[4,4]+
=		[4,4+]+
=		[4+,4]+
=		[(4,4,2+)]+
=		[4,4]+3 = [(4+,4+,2+)] = [1+,4,1+,4,1+] = [4+,4+]+
= = = =
Generators{01,12}{(01)2,12}{01,(12)2}{02,(01)2,(12)2}{(01)2,(12)2,2(01)22}
Orbifold4422222
Radical subgroups
Index816
Diagram
Coxeter[4,4*]
=		[4*,4]
=		[4,4*]+
=		[4*,4]+
=
Orbifold*22222222

Hyperbolic example subgroups

[edit]

The same set of 15 small subgroups exists on all triangle groups with even order elements, like [6,4] in the hyperbolic plane:

Small index subgroups of [6,4]
Index124
Diagram
Coxeter
[1,6,1,4,1] = [6,4]
[1+,6,4]
=		[6,4,1+]
=		[6,1+,4]
=		[1+,6,4,1+]
=		[6+,4+]
Generators{0,1,2}{010,1,2}{0,1,212}{0,101,121,2}{010,1,212,20102}{(01)2,(12)2,012}
Orbifold*642*443*662*3222*323232×
Semidirect subgroups
Diagram
Coxeter[6,4+]
[6+,4]
[(6,4,2+)]
[6,1+,4,1+]
= =
= =		[1+,6,1+,4]
= =
= =
Generators{0,12}{01,2}{02,1,212}{0,101,(12)2}{(01)2,121,2}
Orbifold4*36*22*322*333*22
Direct subgroups
Index248
Diagram
Coxeter[6,4]+
=		[6,4+]+
=		[6+,4]+
=		[(6,4,2+)]+
=		[6+,4+]+ = [1+,6,1+,4,1+]
=
= =
Generators{01,12}{(01)2,12}{01,(12)2}{02,(01)2,(12)2}{(01)2,(12)2,201012}
Orbifold64244366232223232
Radical subgroups
Index8121624
Diagram
Generators{0,101,21012,1210121}{2,121,101020101,0102010,
010101020101010,
10101010201010101}
Coxeter
(orbifold)		[6,4*]
=
(*3333)		[6*,4]

(*222222)		[6,4*]+
=
(3333)		[6*,4]+

(222222)

Parabolic subgroups

[edit]

Aparabolic subgroup of a Coxeter group can be identified by removing one or more generator mirrors represented with a Coxeter diagram. For example the octahedral group has parabolic subgroups,,,,,. In bracket notation [4,3] has parabolic subgroups [4],[2],[3], and a single mirror []. The order of the subgroup is known, and always an integer divisor group order, or index. Parabolic subgroups can also be written with x nodes, like=[4,3] subgroup by removing second mirror: or = = [4,1×,3] = [2].

Petrie subgroup

[edit]

Apetrie subgroup of an irreducible coxeter group can be created by the product of all of the generators. It can be seen in the skew regularpetrie polygon of aregular polytope. The order of the new group is called theCoxeter number of the original Coxeter group. TheCoxeter number of a Coxeter group is 2m/n, wheren is the rank, andm is the number of reflections. A petrie subgroup can be written with aπ superscript. For example, [3,3]π is the petrie subgroup of a tetrahedral group, cyclic group order 4, generated by arotoreflection. A rank 4 Coxeter group will have adouble rotation generator, like [4,3,3]π is order 8.

Extended symmetry

[edit]
Wallpaper
group		Triangle
symmetry		Extended
symmetry		Extended
diagram		Extended
group		Honeycombs
p3m1 (*333)a1[3[3]]A~2{\displaystyle {\tilde {A}}_{2}}(none)
p6m (*632)i2[[3[3]]] ↔ [6,3] ↔A~2×2↔G~2{\displaystyle {\tilde {A}}_{2}\times 2\leftrightarrow {\tilde {G}}_{2}}1,2
p31m (3*3)g3[3+[3[3]]] ↔ [6,3+] ↔A~2×3↔12G~2{\displaystyle {\tilde {A}}_{2}\times 3\leftrightarrow {\tfrac {1}{2}}{\tilde {G}}_{2}}(none)
p6 (632)r6[3[3[3]]]+ ↔ [6,3]+ ↔12A~2×6↔12G~2{\displaystyle {\tfrac {1}{2}}{\tilde {A}}_{2}\times 6\leftrightarrow {\tfrac {1}{2}}{\tilde {G}}_{2}}(1)
p6m (*632)[3[3[3]]] ↔ [6,3]A~2×6↔G~2{\displaystyle {\tilde {A}}_{2}\times 6\leftrightarrow {\tilde {G}}_{2}}3
In the Euclidean plane, theA~2{\displaystyle {\tilde {A}}_{2}}, [3[3]] Coxeter group can be extended in two ways into theG~2{\displaystyle {\tilde {G}}_{2}}, [6,3] Coxeter group and relates uniform tilings as ringed diagrams.
See also:Goursat tetrahedron § Extended symmetry

Coxeter's notation includes double square bracket notation, [[X]] to expressautomorphic symmetry within a Coxeter diagram. Johnson added alternative doubling by angled-bracket <[X]>. Johnson also added a prefix symmetry modifier [Y[X]], where Y can either represent symmetry of the Coxeter diagram of [X], or symmetry of the fundamental domain of [X].

For example, in 3D these equivalentrectangle andrhombic geometry diagrams ofA~3{\displaystyle {\tilde {A}}_{3}}: and, the first doubled with square brackets, [[3[4]]] or twice doubled as [2[3[4]]], with [2], order 4 higher symmetry. To differentiate the second, angled brackets are used for doubling, <[3[4]]> and twice doubled as <2[3[4]]>, also with a different [2], order 4 symmetry. Finally a full symmetry where all 4 nodes are equivalent can be represented by [4[3[4]]], with the order 8, [4] symmetry of thesquare. But by considering thetetragonal disphenoid fundamental domain the [4] extended symmetry of the square graph can be marked more explicitly as [(2+,4)[3[4]]] or [2+,4[3[4]]].

Further symmetry exists in the cyclicA~n{\displaystyle {\tilde {A}}_{n}} and branchingD3{\displaystyle D_{3}},E~6{\displaystyle {\tilde {E}}_{6}}, andD~4{\displaystyle {\tilde {D}}_{4}} diagrams.A~n{\displaystyle {\tilde {A}}_{n}} has order 2n symmetry of a regularn-gon, {n}, and is represented by [n[3[n]]].D3{\displaystyle D_{3}} andE~6{\displaystyle {\tilde {E}}_{6}} are represented by [3[31,1,1]] = [3,4,3] and [3[32,2,2]] respectively whileD~4{\displaystyle {\tilde {D}}_{4}} by [(3,3)[31,1,1,1]] = [3,3,4,3], with the diagram containing the order 24 symmetry of the regulartetrahedron, {3,3}. The paracompact hyperbolic groupL¯5{\displaystyle {\bar {L}}_{5}} = [31,1,1,1,1],, contains the symmetry of a5-cell, {3,3,3}, and thus is represented by [(3,3,3)[31,1,1,1,1]] = [3,4,3,3,3].

Anasterisk * superscript is effectively an inverse operation, creatingradical subgroups removing connected of odd-ordered mirrors.[7]

Examples:

Example Extended groups and radical subgroups
Extended groupsRadical subgroupsCoxeter diagramsIndex
[3[2,2]] = [4,3][4,3*] = [2,2] =6
[(3,3)[2,2,2]] = [4,3,3][4,(3,3)*] = [2,2,2] =24
[1[31,1]] = [[3,3]] = [3,4][3,4,1+] = [3,3] =2
[3[31,1,1]] = [3,4,3][3*,4,3] = [31,1,1] =6
[2[31,1,1,1]] = [4,3,3,4][1+,4,3,3,4,1+] = [31,1,1,1] =4
[3[3,31,1,1]] = [3,3,4,3][3*,4,3,3] = [31,1,1,1] =6
[(3,3)[31,1,1,1]] = [3,4,3,3][3,4,(3,3)*] = [31,1,1,1] =24
[2[3,31,1,1,1]] = [3,(3,4)1,1][3,(3,4,1+)1,1] = [3,31,1,1,1] =4
[(2,3)[1,131,1,1]] = [4,3,3,4,3][3*,4,3,3,4,1+] = [31,1,1,1,1] =12
[(3,3)[3,31,1,1,1]] = [3,3,4,3,3][3,3,4,(3,3)*] = [31,1,1,1,1] =24
[(3,3,3)[31,1,1,1,1]] = [3,4,3,3,3][3,4,(3,3,3)*] = [31,1,1,1,1] =120
Extended groupsRadical subgroupsCoxeter diagramsIndex
[1[3[3]]] = [3,6][3,6,1+] = [3[3]] =2
[3[3[3]]] = [6,3][6,3*] = [3[3]] =6
[1[3,3[3]]] = [3,3,6][3,3,6,1+] = [3,3[3]] =2
[(3,3)[3[3,3]]] = [6,3,3][6,(3,3)*] = [3[3,3]] =24
[1[∞]2] = [4,4][4,1+,4] = [∞]2 = [∞,2,∞] =2
[2[∞]2] = [4,4][1+,4,4,1+] = [(4,4,2*)] = [∞]2 =4
[4[∞]2] = [4,4][4,4*] = [∞]2 =8
[2[3[4]]] = [4,3,4][1+,4,3,4,1+] = [(4,3,4,2*)] = [3[4]] = =4
[3[∞]3] = [4,3,4][4,3*,4] = [∞]3 = [∞,2,∞,2,∞] =6
[(3,3)[∞]3] = [4,31,1][4,(31,1)*] = [∞]3 =24
[(4,3)[∞]3] = [4,3,4][4,(3,4)*] = [∞]3 =48
[(3,3)[∞]4] = [4,3,3,4][4,(3,3)*,4] = [∞]4 =24
[(4,3,3)[∞]4] = [4,3,3,4][4,(3,3,4)*] = [∞]4 =384

Looking at generators, the double symmetry is seen as adding a new operator that maps symmetric positions in the Coxeter diagram, making some original generators redundant. For 3Dspace groups, and 4D point groups, Coxeter defines an index two subgroup of [[X]], [[X]+], which he defines as the product of the original generators of [X] by the doubling generator. This looks similar to [[X]]+, which is the chiral subgroup of [[X]]. So for example the 3D space groups [[4,3,4]]+ (I432, 211) and [[4,3,4]+] (Pm3n, 223) are distinct subgroups of [[4,3,4]] (Im3m, 229).

Rank one groups

[edit]
Further information:One-dimensional symmetry group

In one dimension, thebilateral group [ ] represents a single mirror symmetry, abstractDih1 orZ2, symmetryorder 2. It is represented as aCoxeter–Dynkin diagram with a single node,. Theidentity group is the direct subgroup [ ]+, Z1, symmetry order 1. The+ superscript simply implies that alternate mirror reflections are ignored, leaving the identity group in this simplest case. Coxeter used a single open node to represent an alternation,.

GroupCoxeter notationCoxeter diagramOrderDescription
C1[ ]+1Identity
D2[ ]2Reflection group

Rank two groups

[edit]
Further information:Point groups in two dimensions
A regularhexagon, with markings on edges and vertices has 8 symmetries: [6], [3], [2], [1], [6]+, [3]+, [2]+, [1]+, with [3] and [1] existing in two forms, depending whether the mirrors are on the edges or vertices.

In two dimensions, therectangular group [2], abstract D22 or D4, also can be represented as adirect product [ ]×[ ], being the product of two bilateral groups, represents two orthogonal mirrors, with Coxeter diagram,, withorder 4. The2 in [2] comes from linearization of the orthogonal subgraphs in the Coxeter diagram, as with explicit branch order 2. Therhombic group, [2]+ ( or), half of the rectangular group, thepoint reflection symmetry, Z2, order 2.

Coxeter notation to allow a1 place-holder for lower rank groups, so [1] is the same as [ ], and [1+] or [1]+ is the same as [ ]+ and Coxeter diagram.

Thefull p-gonal group [p], abstractdihedral group D2p, (nonabelian for p>2), oforder 2p, is generated by two mirrors at angleπ/p, represented by Coxeter diagram. Thep-gonal subgroup [p]+,cyclic groupZp, of orderp, generated by a rotation angle of π/p.

Coxeter notation uses double-bracking to represent anautomorphicdoubling of symmetry by adding a bisecting mirror to thefundamental domain. For example, [[p]] adds a bisecting mirror to [p], and is isomorphic to [2p].

In the limit, going down to one dimensions, thefullapeirogonal group is obtained when the angle goes to zero, so [∞], abstractly theinfinite dihedral group D∞, represents two parallel mirrors and has a Coxeter diagram. Theapeirogonal group [∞]+,, abstractly the infinitecyclic group Z∞,isomorphic to theadditive group of theintegers, is generated by a single nonzero translation.

In the hyperbolic plane, there is afullpseudogonal group [iπ/λ], andpseudogonal subgroup [iπ/λ]+,. These groups exist in regular infinite-sided polygons, with edge length λ. The mirrors are all orthogonal to a single line.

Example rank 2 finite and hyperbolic symmetries
TypeFiniteAffineHyperbolic
Geometry...
Coxeter
[ ]		 =
[2]=[ ]×[ ]
[3]
[4]
[p]
[∞]
[∞]
[iπ/λ]
Order24682p∞
Mirror lines are colored to correspond to Coxeter diagram nodes.
Fundamental domains are alternately colored.
Even
images
(direct)		...
Odd
images
(inverted)
Coxeter
[ ]+
[2]+
[3]+
[4]+
[p]+
[∞]+
[∞]+
[iπ/λ]+
Order1234p∞
Cyclic subgroups represent alternate reflections, all even (direct) images.
GroupIntlOrbifoldCoxeterCoxeter diagramOrderDescription
Finite
Znnn•[n]+nCyclic:n-fold rotations. Abstract group Zn, the group of integers under addition modulon.
D2nnm*n•[n]2nDihedral: cyclic with reflections. Abstract group Dihn, thedihedral group.
Affine
Z∞∞∞•[∞]+∞Cyclic:apeirogonal group. Abstract group Z∞, the group of integers under addition.
Dih∞∞m*∞•[∞]∞Dihedral: parallel reflections. Abstractinfinite dihedral group Dih∞.
Hyperbolic
Z∞[πi/λ]+∞pseudogonal group
Dih∞[πi/λ]∞full pseudogonal group

Rank three groups

[edit]
Further information:Point groups in three dimensions andList of spherical symmetry groups

Point groups in 3 dimensions can be expressed in bracket notation related to the rank 3 Coxeter groups:

Finite groups of isometries in 3-space[2]
Rotation groupsExtended groups
NameBracketOrbSchAbstractOrderNameBracketOrbSchAbstractOrder
Identity[ ]+11C1Z11Bilateral[1,1] = [ ]*D2Z22
Central[2+,2+]×Ci2×Z12
Acrorhombic[1,2]+ = [2]+22C2Z22Acrorectangular[1,2] = [2]*22C2vD44
Gyrorhombic[2+,4+]2×S4Z44
Orthorhombic[2,2+]2*D1dZ2×Z24
Pararhombic[2,2]+222D2D44Gyrorectangular[2+,4]2*2D2dD88
Orthorectangular[2,2]*222D2hZ2×D48
Acro-p-gonal[1,p]+ = [p]+ppCpZppFull acro-p-gonal[1,p] = [p]*ppCpvD2p2p
Gyro-p-gonal[2+,2p+]p×S2pZ2p2p
Ortho-p-gonal[2,p+]p*CphZ2×Zp2p
Para-p-gonal[2,p]+p22D2pD2p2pFull gyro-p-gonal[2+,2p]2*pDpdD4p4p
Full ortho-p-gonal[2,p]*p22DphZ2×D2p4p
Tetrahedral[3,3]+332TA412Full tetrahedral[3,3]*332TdS424
Pyritohedral[3+,4]3*2Th2×A424
Octahedral[3,4]+432OS424Full octahedral[3,4]*432Oh2×S448
Icosahedral[3,5]+532IA560Full icosahedral[3,5]*532Ih2×A5120

In three dimensions, thefull orthorhombic group ororthorectangular [2,2], abstractly Z23,order 8, represents three orthogonal mirrors, (also represented by Coxeter diagram as three separate dots). It can also can be represented as adirect product [ ]×[ ]×[ ], but the [2,2] expression allows subgroups to be defined:

First there is a "semidirect" subgroup, theorthorhombic group, [2,2+] ( or), abstractly Z2×Z2, of order 4. When the+ superscript is given inside of the brackets, it means reflections generated only from the adjacent mirrors (as defined by the Coxeter diagram,) are alternated. In general, the branch orders neighboring the+ node must be even. In this case [2,2+] and [2+,2] represent two isomorphic subgroups that are geometrically distinct. The other subgroups are thepararhombic group [2,2]+ ( or), also order 4, and finally thecentral group [2+,2+] ( or) of order 2.

Next there is thefull ortho-p-gonal group, [2,p] (), abstractly Z2×D2p, of order 4p, representing two mirrors at adihedral angle π/p, and both are orthogonal to a third mirror. It is also represented by Coxeter diagram as.

The direct subgroup is called the para-p-gonal group, [2,p]+ ( or), abstractly D2p, of order 2p, and another subgroup is [2,p+] () abstractly Z2×Zp, also of order 2p.

Thefull gyro-p-gonal group, [2+,2p] ( or), abstractly D4p, of order 4p. The gyro-p-gonal group, [2+,2p+] ( or), abstractlyZ2p, of order 2p is a subgroup of both [2+,2p] and [2,2p+].

Thepolyhedral groups are based on the symmetry ofplatonic solids: thetetrahedron,octahedron,cube,icosahedron, anddodecahedron, withSchläfli symbols {3,3}, {3,4}, {4,3}, {3,5}, and {5,3} respectively. The Coxeter groups for these are: [3,3] (), [3,4] (), [3,5] () called fulltetrahedral symmetry,octahedral symmetry, andicosahedral symmetry, with orders of 24, 48, and 120.

Pyritohedral symmetry, [3+,4] is an index 5 subgroup oficosahedral symmetry, [5,3].

In all these symmetries, alternate reflections can be removed producing the rotational tetrahedral [3,3]+(), octahedral [3,4]+ (), and icosahedral [3,5]+ () groups of order 12, 24, and 60. The octahedral group also has a unique index 2 subgroup called thepyritohedral symmetry group, [3+,4] ( or), of order 12, with a mixture of rotational and reflectional symmetry. Pyritohedral symmetry is also an index 5 subgroup of icosahedral symmetry: -->, with virtual mirror1 across0, {010}, and 3-fold rotation {12}.

The tetrahedral group, [3,3] (), has a doubling [[3,3]] (which can be represented by colored nodes), mapping the first and last mirrors onto each other, and this produces the [3,4] ( or) group. The subgroup [3,4,1+] ( or) is the same as [3,3], and [3+,4,1+] ( or) is the same as [3,3]+.

Example rank 3 finite Coxeter groups subgroup trees
Tetrahedral symmetryOctahedral symmetry
Icosahedral symmetry
Finite (point groups in three dimensions)
Intl*Geo
[8]		Orb.Schön.Struct.CoxeterOrd.
111C1Z1[] = [ ]+ =1
2 = m1*CsZ2[ ]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		Z2
Z3
Z4
Z5
Z6
Zn		[1,2]+
[1,3]+
[1,4]+
[1,5]+
[1,6]+
[1,n]+





2
3
4
5
6
n
2mm
3m
4mm
5m
6mm
nmm
nm		2
3
4
5
6
n		*22
*33
*44
*55
*66
*nn		C2v
C3v
C4v
C5v
C6v
Cnv		D2
D3
D4
D5
D6
Dn		[1,2]
[1,3]
[1,4]
[1,5]
[1,6]
[1,n]




4
6
8
10
12
2n
2/m
3/m
4/m
5/m
6/m
n/m		2 2
3 2
4 2
5 2
6 2
n 2		2*
3*
4*
5*
6*
n*		C2h
C3h
C4h
C5h
C6h
Cnh		Z2×Z2
Z2×Z3
Z2×Z4
Z2×Z5
Z2×Z6
Z2×Zn		[2,2+]
[2,3+]
[2,4+]
[2,5+]
[2,6+]
[2,n+]




4
6
8
10
12
2n
1
4
3
8
5
12
2n
n		2 2
4 2
6 2
8 2
10 2
12 2
2n 2		1×
2×
3×
4×
5×
6×
n×		Ci = S2
S4
S6
S8
S10
S12
S2n		Z2
Z4
Z6=Z2×Z3
Z8
Z10=Z2×Z5
Z12
Z2n		[2+,2+]
[2+,4+]
[2+,6+]
[2+,8+]
[2+,10+]
[2+,12+]
[2+,2n+]





2
4
6
8
10
12
2n
IntlGeoOrb.Schön.Struct.CoxeterOrd.
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		D4
D6
D8
D10
D12
D2n		[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		Z2×D4
Z2×D6
Z2×D8
Z2×D10
Z2×D12
Z2×D2n		[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		42
62
82
102
122
n2
2*2
2*3
2*4
2*5
2*6
2*n		D2d
D3d
D4d
D5d
D6d
Dnd		D4
D6
D8
D10
D12
D2n		[2+,4]
[2+,6]
[2+,8]
[2+,10]
[2+,12]
[2+,2n]




8
12
16
20
24
4n
2333332TA4[3,3]+12
m3433*2ThA4×S2[3+,4]24
43m3 3*332TdS4[3,3]24
43243432OS4[3,4]+24
m3m4 3*432OhS4×S2[3,4]48
53253532IA5[3,5]+60
53m5 3*532IhA4×S2[3,5]120

Affine

[edit]
Further information:List of planar symmetry groups

In the Euclidean plane there's 3 fundamental reflective groups generated by 3 mirrors, represented by Coxeter diagrams,, and, and are given Coxeter notation as [4,4], [6,3], and [(3,3,3)]. The parentheses of the last group imply the diagram cycle, and also has a shorthand notation [3[3]].

[[4,4]] as a doubling of the [4,4] group produced the same symmetry rotated π/4 from the original set of mirrors.

Direct subgroups of rotational symmetry are: [4,4]+, [6,3]+, and [(3,3,3)]+. [4+,4] and [6,3+] are semidirect subgroups.

Semiaffine (frieze groups)
IUCOrb.GeoSch.Coxeter
p1∞∞p1C∞[∞] = [∞,1]+ = [∞+,2,1+] =
p1m1*∞∞p1C∞v[∞] = [∞,1] = [∞,2,1+] =
p11g∞×p.g1S2∞[∞+,2+]
p11m∞*p. 1C∞h[∞+,2]
p222∞p2D∞[∞,2]+
p2mg2*∞p2gD∞d[∞,2+]
p2mm*22∞p2D∞h[∞,2]
Affine (Wallpaper groups)
IUCOrb.Geo.Coxeter
p22222p2[4,1+,4]+
p2gg22×pg2g[4+,4+]
p2mm*2222p2[4,1+,4]
c2mm2*22c2[[4+,4+]]
p4442p4[4,4]+
p4gm4*2pg4[4+,4]
p4mm*442p4[4,4]
p3333p3[1+,6,3+] = [3[3]]+ =
p3m1*333p3[1+,6,3] = [3[3]] =
p31m3*3h3[6,3+] = [3[3[3]]+]
p6632p6[6,3]+ = [3[3[3]]]+
p6mm*632p6[6,3] = [3[3[3]]]

Given in Coxeter notation (orbifold notation), some low index affine subgroups are:

Reflective
group		Reflective
subgroup		Mixed
subgroup		Rotation
subgroup		Improper rotation/
translation		Commutator
subgroup
[4,4], (*442)[1+,4,4], (*442)
[4,1+,4], (*2222)
[1+,4,4,1+], (*2222)		[4+,4], (4*2)
[(4,4,2+)], (2*22)
[1+,4,1+,4], (2*22)		[4,4]+, (442)
[1+,4,4+], (442)
[1+,4,1+4,1+], (2222)		[4+,4+], (22×)[4+,4+]+, (2222)
[6,3], (*632)[1+,6,3] = [3[3]], (*333)[3+,6], (3*3)[6,3]+, (632)
[1+,6,3+], (333)		[1+,6,3+], (333)

Rank four groups

[edit]

Hasse diagram subgroup relations (partial!)

Point groups

[edit]
Further information:Point groups in four dimensions

Rank four groups defined the 4-dimensionalpoint groups:

Finite groups
[ ]:
SymbolOrder
[1]+1.1
[1] = [ ]2.1
[2]:
SymbolOrder
[1+,2]+1.1
[2]+2.1
[2]4.1
[2,2]:
SymbolOrder
[2+,2+]+
= [(2+,2+,2+)]		1.1
[2+,2+]2.1
[2,2]+4.1
[2+,2]4.1
[2,2]8.1
[2,2,2]:
SymbolOrder
[(2+,2+,2+,2+)]
= [2+,2+,2+]+		1.1
[2+,2+,2+]2.1
[2+,2,2+]4.1
[(2,2)+,2+]4
[[2+,2+,2+]]4
[2,2,2]+8
[2+,2,2]8.1
[(2,2)+,2]8
[[2+,2,2+]]8.1
[2,2,2]16.1
[[2,2,2]]+16
[[2,2+,2]]16
[[2,2,2]]32
[p]:
SymbolOrder
[p]+p
[p]2p
[p,2]:
SymbolOrder
[p,2]+2p
[p,2]4p
[2p,2+]:
SymbolOrder
[2p,2+]4p
[2p+,2+]2p
[p,2,2]:
SymbolOrder
[p+,2,2+]2p
[(p,2)+,2+]2p
[p,2,2]+4p
[p,2,2+]4p
[p+,2,2]4p
[(p,2)+,2]4p
[p,2,2]8p
[2p,2+,2]:
SymbolOrder
[2p+,2+,2+]+p
[2p+,2+,2+]2p
[2p+,2+,2]4p
[2p+,(2,2)+]4p
[2p,(2,2)+]8p
[2p,2+,2]8p
[p,2,q]:
SymbolOrder
[p+,2,q+]pq
[p,2,q]+2pq
[p+,2,q]2pq
[p,2,q]4pq
[(p,2)+,2q]:
SymbolOrder
[(p,2)+,2q+]2pq
[(p,2)+,2q]4pq
[2p,2,2q]:
SymbolOrder
[2p+,2+,2q+]+=
[(2p+,2+,2q+,2+)]		pq
[2p+,2+,2q+]2pq
[2p,2+,2q+]4pq
[((2p,2)+,(2q,2)+)]4pq
[2p,2+,2q]8pq
[[p,2,p]]:
SymbolOrder
[[p+,2,p+]]2p2
[[p,2,p]]+4p2
[[p,2,p]+]4p2
[[p,2,p]]8p2
[[2p,2,2p]]:
SymbolOrder
[[(2p+,2+,2p+,2+)]]2p2
[[2p+,2+,2p+]]4p2
[[((2p,2)+,(2p,2)+)]]8p2
[[2p,2+,2p]]16p2
[3,3,2]:
SymbolOrder
[(3,3)Δ,2,1+]
≅ [2,2]+		4
[(3,3)Δ,2]
≅ [2,(2,2)+]		8
[(3,3)⅄,2,1+]
≅ [4,2+]		8
[(3,3)+,2,1+]
= [3,3]+		12.5
[(3,3)⅄,2]
≅ [2,4,2+]		16
[3,3,2,1+]
= [3,3]		24
[(3,3)+,2]24.10
[3,3,2]+24.10
[3,3,2]48.36
[4,3,2]:
SymbolOrder
[1+,4,3+,2,1+]
= [3,3]+		12
[3+,4,2+]24
[(3,4)+,2+]24
[1+,4,3+,2]
= [(3,3)+,2]		24.10
[3+,4,2,1+]
= [3+,4]		24.10
[(4,3)+,2,1+]
= [4,3]+		24.15
[1+,4,3,2,1+]
= [3,3]		24
[1+,4,(3,2)+]
= [3,3,2]+		24
[3,4,2+]48
[4,3+,2]48.22
[4,(3,2)+]48
[(4,3)+,2]48.36
[1+,4,3,2]
= [3,3,2]		48.36
[4,3,2,1+]
= [4,3]		48.36
[4,3,2]+48.36
[4,3,2]96.5
[5,3,2]:
SymbolOrder
[(5,3)+,2,1+]
= [5,3]+		60.13
[5,3,2,1+]
= [5,3]		120.2
[(5,3)+,2]120.2
[5,3,2]+120.2
[5,3,2]240 (nc)
[31,1,1]:
SymbolOrder
[31,1,1]Δ
≅[[4,2+,4]]+		32
[31,1,1]⅄64
[31,1,1]+96.1
[31,1,1]192.2
<[3,31,1]>
= [4,3,3]		384.1
[3[31,1,1]]
= [3,4,3]		1152.1
[3,3,3]:
SymbolOrder
[3,3,3]+60.13
[3,3,3]120.1
[[3,3,3]]+120.2
[[3,3,3]+]120.1
[[3,3,3]]240.1
[4,3,3]:
SymbolOrder
[1+,4,(3,3)Δ]
= [31,1,1]Δ
≅[[4,2+,4]]+		32
[4,(3,3)Δ]
= [2+,4[2,2,2]+]
≅[[4,2+,4]]		64
[1+,4,(3,3)⅄]
= [31,1,1]⅄		64
[1+,4,(3,3)+]
= [31,1,1]+		96.1
[4,(3,3)⅄]
≅ [[4,2,4]]		128
[1+,4,3,3]
= [31,1,1]		192.2
[4,(3,3)+]192.1
[4,3,3]+192.3
[4,3,3]384.1
[3,4,3]:
SymbolOrder
[3+,4,3+]288.1
[3,4,3⅄]
= [4,3,3]		384.1
[3,4,3]+576.2
[3+,4,3]576.1
[[3+,4,3+]]576 (nc)
[3,4,3]1152.1
[[3,4,3]]+1152 (nc)
[[3,4,3]+]1152 (nc)
[[3,4,3]]2304 (nc)
[5,3,3]:
SymbolOrder
[5,3,3]+7200 (nc)
[5,3,3]14400 (nc)

Subgroups

[edit]
1D-4D reflective point groups and subgroups
OrderReflectionSemidirect
subgroups		Direct
subgroups		Commutator
subgroup
2[ ][ ]+[ ]+1[ ]+
4[2][2]+[2]+2
8[2,2][2+,2][2+,2+][2,2]+[2,2]+3
16[2,2,2][2+,2,2]
[(2,2)+,2]
[2+,2+,2]
[(2,2)+,2+]
[2+,2+,2+]

[2,2,2]+
[2+,2,2+]
[2,2,2]+4
[21,1,1][(2+)1,1,1]
2n[n][n]+[n]+1[n]+
4n[2n][2n]+[2n]+2
4n[2,n][2,n+][2,n]+[2,n]+2
8n[2,2n][2+,2n][2+,2n+][2,2n]+[2,2n]+3
8n[2,2,n][2+,2,n]
[2,2,n+]
[2+,(2,n)+][2,2,n]+
[2+,2,n+]
[2,2,n]+3
16n[2,2,2n][2,2+,2n][2+,2+,2n]
[2,2+,2n+]
[(2,2)+,2n+]
[2+,2+,2n+]


[2,2,2n]+
[2+,2n,2+]
[2,2,2n]+4
[2,2n,2][2+,2n+,2+]
[2n,21,1][2n+,(2+)1,1]
24[3,3][3,3]+[3,3]+1[3,3]+
48[3,3,2][(3,3)+,2][3,3,2]+[3,3,2]+2
48[4,3][4,3+][4,3]+[4,3]+2
96[4,3,2][(4,3)+,2]
[4,(3,2)+]
[4,3,2]+[4,3,2]+3
[3,4,2][3,4,2+]
[3+,4,2]
[(3,4)+,2+][3+,4,2+]
120[5,3][5,3]+[5,3]+1[5,3]+
240[5,3,2][(5,3)+,2][5,3,2]+[5,3,2]+2
4pq[p,2,q][p+,2,q][p,2,q]+
[p+,2,q+]
[p,2,q]+2[p+,2,q+]
8pq[2p,2,q][2p,(2,q)+][2p+,(2,q)+][2p,2,q]+[2p,2,q]+3
16pq[2p,2,2q][2p,2+,2q][2p+,2+,2q]
[2p+,2+,2q+]
[(2p,(2,2q)+,2+)]

-		[2p,2,2q]+[2p,2,2q]+4
120[3,3,3][3,3,3]+[3,3,3]+1[3,3,3]+
192[31,1,1][31,1,1]+[31,1,1]+1[31,1,1]+
384[4,3,3][4,(3,3)+][4,3,3]+[4,3,3]+2
1152[3,4,3][3+,4,3][3,4,3]+
[3+,4,3+]
[3,4,3]+2[3+,4,3+]
14400[5,3,3][5,3,3]+[5,3,3]+1[5,3,3]+

Space groups

[edit]
Space groups

Affine isomorphism and correspondences
8 cubic space groups as extended symmetry from [3[4]], with square Coxeter diagrams and reflective fundamental domains
35 cubic space groups in International,Fibrifold notation, and Coxeter notation
Rank four groups as 3-dimensionalspace groups
Triclinic (1-2)
CoxeterSpace group
[∞+,2,∞+,2,∞+](1) P1
Monoclinic (3-15)
CoxeterSpace group
[(∞,2,∞)+,2,∞+](3) P2
[∞+,2,∞+,2,∞](6) Pm
[(∞,2,∞)+,2,∞](10) P2/m
Orthorhombic (16-74)
CoxeterSpace group
[∞,2,∞,2,∞]+(16) P222
[[∞,2,∞,2,∞]]+(23) I222
[∞+,2,∞,2,∞](25) Pmm2
[∞,2,∞,2,∞](47) Pmmm
[[∞,2,∞,2,∞]](71) Immm
[∞+,2,∞+,2,∞+]
[∞,2,∞,2+,∞]
[∞,2+,∞,2+,∞]
Tetragonal (75-142)
CoxeterSpace group
[(4,4)+,2,∞+](75) P4
[2+[(4,4)+,2,∞+]](79) I4
[(4,4)+,2,∞](83) P4/m
[2+[(4,4)+,2,∞]](87) I4/m
[4,4,2,∞]+(89) P422
[2+[4,4,2,∞]]+(97) I422
[4,4,2,∞+](99) P4mm
[4,4,2,∞](123) P4/mmm
[2+[4,4,2,∞]](139) I4/mmm
[4,(4,2)+,∞](140) I4/mcm
[4,4,2+,∞]
[(4,4)+,2+,∞]
[4,4,2+,∞+]
[(4,4)+,2+,∞+]
[4+,4+,2+,∞]
[4,4+,2,∞]
[4,4+,2+,∞]
[((4,2+,4)),2,∞]
[4,4+,2,∞+]
[4,4+,2+,∞+]
[((4,2+,4)),2,∞+]
Trigonal (143-167), rhombohedral
CoxeterSpace group
Hexagonal (168-194)
[(6,3)+,2,∞+](168) P6
[(6,3)+,2,∞](175) P6/m
[6,3,2,∞]+(177) P622
[6,3,2,∞+](183) P6mm
[6,3,2,∞](191) P6/mmm
[(3[3])+,2,∞+]
[3[3],2,∞]
[6,3+,2,∞]
[6,3+,2,∞+]
[3[3],2,∞]+
[3[3],2,∞+]
[(3[3])+,2,∞]
Cubic (195-230)
GroupCoxeterSpace groupIndex
[[4,3,4]][[4,3,4]](229) Im3m1
[[4,3,4]]+(211) I4322
[[4,3,4]+](223) Pm3n2
[[4,3+,4]](204) I32
[[(4,3,4,2+)]](217) I43m2
[[4,3+,4]]+(197) I234
[[4,3,4]+]+(208) P42324
[[4,3+,4)]+](201) Pn434
[[(4,3,4,2+)]+](218) P43n4
[4,3,4]
[4,3,4](221) Pm3m2
[4,3,4]+(207) P4324
[4,3+,4](200) Pm34
[4,(3,4)+](226) Fm3c4
[(4,3,4,2+)](215) P43m4
[[{4,(3}+,4)+]](228) Fd3c4
[4,3+,4]+(195) P238
[{4,(3}+,4)+](219) F43c8
[4,31,1]

[4,31,1](225) Fm3m4
[4,(31,1)+](202) Fm38
[4,31,1]+(209) F4328
[[3[4]]]
[(4+,2+)[3[4]]](222) Pn3n2
[[3[4]]](227) Fd3m4
[[3[4]]]+(203) Fd38
[[3[4]]+](210) F41328
[3[4]]

[3[4]](216) F43m8
[3[4]]+(196) F2316

Line groups

[edit]

Rank four groups also defined the 3-dimensionalline groups:

Semiaffine (3D) groups
Point groupLine group
Hermann-MauguinSchönfliesHermann-MauguinOffset typeWallpaperCoxeter
[∞h,2,pv]
EvennOddnEvennOddnIUCOrbifoldDiagram
nCnPnqHelical:qp1o[∞+,2,n+]
2nnS2nP2nPnNonep11g, pg(h)××[(∞,2)+,2n+]
n/m2nCnhPn/mP2nNonep11m, pm(h)**[∞+,2,n]
2n/mC2nhP2nn/mZigzagc11m, cm(h)*×[∞+,2+,2n]
nmmnmCnvPnmmPnmNonep1m1, pm(v)**[∞,2,n+]
PnccPncPlanar reflectionp1g1, pg(v)××[∞+,(2,n)+]
2nmmC2nvP2nnmcZigzagc1m1, cm(v)*×[∞,2+,2n+]
n22n2DnPnq22Pnq2Helical:qp22222[∞,2,n]+
2n2mnmDndP2n2mPnmNonep2mg, pmg(h)22*[(∞,2)+,2n]
P2n2cPncPlanar reflectionp2gg, pgg22×[+(∞,(2),2n)+]
n/mmm2n2mDnhPn/mmmP2n2mNonep2mm, pmm*2222[∞,2,n]
Pn/mccP2n2cPlanar reflectionp2mg, pmg(v)22*[∞,(2,n)+]
2n/mmmD2nhP2nn/mcmZigzagc2mm, cmm2*22[∞,2+,2n]

Duoprismatic group

[edit]
Extended duoprismatic symmetry
Extended duoprismatic groups, [p]×[p] or [p,2,p] or, expressed in relation to itstetragonal disphenoid fundamental domain symmetry.

Rank four groups defined the 4-dimensional duoprismatic groups. In the limit as p and q go to infinity, they degenerate into 2 dimensions and the wallpaper groups.

Duoprismatic groups (4D)
WallpaperCoxeter
[p,2,q]
Coxeter
[[p,2,p]]
Wallpaper
IUCOrbifoldDiagramIUCOrbifoldDiagram
p1o[p+,2,q+][[p+,2,p+]]p1o
pg××[(p,2)+,2q+]-
pm**[p+,2,q]-
cm*×[2p+,2+,2q]-
p22222[p,2,q]+[[p,2,p]]+p4442
pmg22*[(p,2)+,2q]-
pgg22×[+(2p,(2),2q)+][[+(2p,(2),2p)+]]cmm2*22
pmm*2222[p,2,q][[p,2,p]]p4m*442
cmm2*22[2p,2+,2q][[2p,2+,2p]]p4g4*2

Wallpaper groups

[edit]

Rank four groups also defined some of the 2-dimensionalwallpaper groups, as limiting cases of the four-dimensional duoprism groups:

Affine (2D plane)
IUCOrb.GeoCoxeterDiagram
p1op1[∞+,2,∞+]
[∞+,2+,∞+]
[(∞+,2+,∞+,2+)]
p22222p2[∞,2,∞]+
[(∞,2+,∞,2+)]
p11g××pg1h: [∞+,(2,∞)+]
p1g1v: [(∞,2)+,∞+]
p2gm22*pg2h: [(∞,2)+,∞]
p2mgv: [∞,(2,∞)+]
IUCOrb.GeoCoxeterDiagram
p11m**p1h: [∞+,2,∞]
p1m1v: [∞,2,∞+]
p2mm*2222p2[∞,2,∞]
c11m*×c1h: [∞+,2+,∞]
c1m1v: [∞,2+,∞+]
p2gg22×pg2g[+(∞,(2),∞)+]
[((∞,2)+)[2]]
c2mm2*22c2[∞,2+,∞]

Subgroups of [∞,2,∞], (*2222) can be expressed down to its index 16 commutator subgroup:

Subgroups of [∞,2,∞]
Reflective
group		Reflective
subgroup		Mixed
subgroup		Rotation
subgroup		Improper rotation/
translation		Commutator
subgroup
[∞,2,∞], (*2222)[1+,∞,2,∞], (*2222)[∞+,2,∞], (**)[∞,2,∞]+, (2222)[∞,2+,∞]+, (°)
[∞+,2+,∞+], (°)
[∞+,2,∞+], (°)
[∞+,2+,∞], (*×)
[(∞,2)+,∞+], (××)
[((∞,2)+,(∞,2)+)], (22×)		[(∞+,2+,∞+,2+)], (°)
[∞,2+,∞], (2*22)
[(∞,2)+,∞], (22*)

Complex reflections

[edit]
Hasse diagram with all subgroup relations on rank 2 Shephard groups.

Coxeter notation has been extended toComplex space, Cn where nodes areunitary reflections of period 2 or greater. Nodes are labeled by an index, assumed to be 2 for ordinary real reflection if suppressed.Complex reflection groups are calledShephard groups rather thanCoxeter groups, and can be used to constructcomplex polytopes.

InC1{\displaystyle \mathbb {C} ^{1}}, a rank 1 Shephard group, orderp, is represented asp[ ], [ ]p or ]p[. It has a single generator, representing a 2π/p radian rotation in theComplex plane:e2πi/p{\displaystyle e^{2\pi i/p}}.

Coxeter writes the rank 2 complex group,p[q]r representsCoxeter diagram. Thep andr should only be suppressed if both are 2, which is the real case [q]. The order of a rank 2 groupp[q]r isg=8/q(1/p+2/q+1/r−1)−2{\displaystyle g=8/q(1/p+2/q+1/r-1)^{-2}}.[9]

The rank 2 solutions that generate complex polygons are:p[4]2 (p is 2,3,4,...),3[3]3,3[6]2,3[4]3,4[3]4,3[8]2,4[6]2,4[4]3,3[5]3,5[3]5,3[10]2,5[6]2, and5[4]3 with Coxeter diagrams,,,,,,,,,,,,.

Some subgroup relations among infinite Shephard groups

Infinite groups are3[12]2,4[8]2,6[6]2,3[6]3,6[4]3,4[4]4, and6[3]6 or,,,,,,.

Index 2 subgroups exists by removing a real reflection:p[2q]2 →p[q]p. Also indexr subgroups exist for 4 branches:p[4]r →p[r]p.

For the infinite familyp[4]2, for anyp = 2, 3, 4,..., there are two subgroups:p[4]2 → [p], indexp, while andp[4]2 →p[ ]×p[ ], index 2.

Computation with reflection matrices as symmetry generators

[edit]

A Coxeter group, represented byCoxeter diagram, is given Coxeter notation [p,q] for the branch orders. Each node in the Coxeter diagram represents a mirror, by convention called ρi (and matrix Ri). Thegenerators of this group [p,q] are reflections: ρ0, ρ1, and ρ2. Rotational subsymmetry is given as products of reflections: By convention, σ0,1 (and matrix S0,1) = ρ0ρ1 represents a rotation of angle π/p, and σ1,2 = ρ1ρ2 is a rotation of angle π/q, and σ0,2 = ρ0ρ2 represents a rotation of angle π/2.

[p,q]+,, is an index 2 subgroup represented by two rotation generators, each a products of two reflections: σ0,1, σ1,2, and representing rotations of π/p, and π/q angles respectively.

With one even branch, [p+,2q], or, is another subgroup of index 2, represented by rotation generator σ0,1, and reflectional ρ2.

With even branches, [2p+,2q+],, is a subgroup of index 4 with two generators, constructed as a product of all three reflection matrices: By convention as: ψ0,1,2 and ψ1,2,0, which arerotary reflections, representing a reflection and rotation or reflection.

In the case of affine Coxeter groups like, or, one mirror, usually the last, is translated off the origin. Atranslation generator τ0,1 (and matrix T0,1) is constructed as the product of two (or an even number of) reflections, including the affine reflection. Atransreflection (reflection plus a translation) can be the product of an odd number of reflections φ0,1,2 (and matrix V0,1,2), like the index 4 subgroup: [4+,4+] =.

Another composite generator, by convention as ζ (and matrix Z), represents theinversion, mapping a point to its inverse. For [4,3] and [5,3], ζ = (ρ0ρ1ρ2)h/2, whereh is 6 and 10 respectively, theCoxeter number for each family. For 3D Coxeter group [p,q] (), this subgroup is a rotary reflection [2+,h+].

Coxeter groups are categorized by their rank, being the number of nodes in itsCoxeter-Dynkin diagram. The structure of the groups are also given with their abstract group types: In this article, the abstractdihedral groups are represented asDihn, andcyclic groups are represented byZn, withDih1=Z2.

Rank 2

[edit]
Dihedral groupsCyclic groups

[2]
[2]+

[3]
[3]+

[4]
[4]+

[6]
[6]+

Example, in 2D, the Coxeter group [p] () is represented by two reflection matrices R0 and R1, The cyclic symmetry [p]+ () is represented by rotation generator of matrix S0,1.

[p],
ReflectionsRotation
NameR0
R1
S0,1=R0×R1
Order22p
Matrix

[100−1]{\displaystyle \left[{\begin{smallmatrix}1&0\\0&-1\\\end{smallmatrix}}\right]}

[cos⁡2π/psin⁡2π/psin⁡2π/p−cos⁡2π/p]{\displaystyle \left[{\begin{smallmatrix}\cos 2\pi /p&\sin 2\pi /p\\\sin 2\pi /p&-\cos 2\pi /p\\\end{smallmatrix}}\right]}

[cos⁡2π/psin⁡2π/p−sin⁡2π/pcos⁡2π/p]{\displaystyle \left[{\begin{smallmatrix}\cos 2\pi /p&\sin 2\pi /p\\-\sin 2\pi /p&\cos 2\pi /p\\\end{smallmatrix}}\right]}

[2],
ReflectionsRotation
NameR0
R1
S0,1=R0×R1
Order222
Matrix

[100−1]{\displaystyle \left[{\begin{smallmatrix}1&0\\0&-1\\\end{smallmatrix}}\right]}

[−1001]{\displaystyle \left[{\begin{smallmatrix}-1&0\\0&1\\\end{smallmatrix}}\right]}

[−100−1]{\displaystyle \left[{\begin{smallmatrix}-1&0\\0&-1\\\end{smallmatrix}}\right]}

[3],
ReflectionsRotation
NameR0
R1
S0,1=R0×R1
Order223
Matrix

[100−1]{\displaystyle \left[{\begin{smallmatrix}1&0\\0&-1\\\end{smallmatrix}}\right]}

[1/23/23/2−1/2]{\displaystyle \left[{\begin{smallmatrix}1/2&{\sqrt {3}}/2\\{\sqrt {3}}/2&-1/2\\\end{smallmatrix}}\right]}

[1/23/2−3/21/2]{\displaystyle \left[{\begin{smallmatrix}1/2&{\sqrt {3}}/2\\-{\sqrt {3}}/2&1/2\\\end{smallmatrix}}\right]}

[4],
ReflectionsRotation
NameR0
R1
S0,1=R0×R1
Order224
Matrix

[100−1]{\displaystyle \left[{\begin{smallmatrix}1&0\\0&-1\\\end{smallmatrix}}\right]}

[0110]{\displaystyle \left[{\begin{smallmatrix}0&1\\1&0\\\end{smallmatrix}}\right]}

[01−10]{\displaystyle \left[{\begin{smallmatrix}0&1\\-1&0\\\end{smallmatrix}}\right]}

[6],
ReflectionsRotation
NameR0
R1
S0,1=R0×R1
Order226
Matrix

[100−1]{\displaystyle \left[{\begin{smallmatrix}1&0\\0&-1\\\end{smallmatrix}}\right]}

[3/21/21/2−3/2]{\displaystyle \left[{\begin{smallmatrix}{\sqrt {3}}/2&1/2\\1/2&-{\sqrt {3}}/2\\\end{smallmatrix}}\right]}

[3/21/2−1/23/2]{\displaystyle \left[{\begin{smallmatrix}{\sqrt {3}}/2&1/2\\-1/2&{\sqrt {3}}/2\\\end{smallmatrix}}\right]}

[8],
ReflectionsRotation
NameR0
R1
S0,1=R0×R1
Order228
Matrix

[100−1]{\displaystyle \left[{\begin{smallmatrix}1&0\\0&-1\\\end{smallmatrix}}\right]}

[2/22/22/2−2/2]{\displaystyle \left[{\begin{smallmatrix}{\sqrt {2}}/2&{\sqrt {2}}/2\\{\sqrt {2}}/2&-{\sqrt {2}}/2\\\end{smallmatrix}}\right]}

[2/22/2−2/22/2]{\displaystyle \left[{\begin{smallmatrix}{\sqrt {2}}/2&{\sqrt {2}}/2\\-{\sqrt {2}}/2&{\sqrt {2}}/2\\\end{smallmatrix}}\right]}

Rank 3

[edit]

The finite rank 3 Coxeter groups are [1,p], [2,p], [3,3], [3,4], and [3,5].

To reflect a point through a planeax+by+cz=0{\displaystyle ax+by+cz=0} (which goes through the origin), one can useA=I−2NNT{\displaystyle \mathbf {A} =\mathbf {I} -2\mathbf {NN} ^{T}}, whereI{\displaystyle \mathbf {I} } is the 3×3 identity matrix andN{\displaystyle \mathbf {N} } is the three-dimensionalunit vector for the vector normal of the plane. If theL2 norm ofa,b,{\displaystyle a,b,} andc{\displaystyle c} is unity, the transformation matrix can be expressed as:

A=[1−2a2−2ab−2ac−2ab1−2b2−2bc−2ac−2bc1−2c2]{\displaystyle \mathbf {A} =\left[{\begin{smallmatrix}1-2a^{2}&-2ab&-2ac\\-2ab&1-2b^{2}&-2bc\\-2ac&-2bc&1-2c^{2}\end{smallmatrix}}\right]}

[p,2]

[edit]
Example fundamental domains, [5,2], as spherical triangles

The reducible 3-dimensional finite reflective group isdihedral symmetry, [p,2], order 4p,. The reflection generators are matrices R0, R1, R2. R02=R12=R22=(R0×R1)3=(R1×R2)3=(R0×R2)2=Identity. [p,2]+ () is generated by 2 of 3 rotations: S0,1, S1,2, and S0,2. An orderprotoreflection is generated by V0,1,2, the product of all 3 reflections.

[p,2],
ReflectionsRotationRotoreflection
NameR0R1R2S0,1S1,2S0,2V0,1,2
Group
Order222p22p
Matrix

[1000−10001]{\displaystyle \left[{\begin{smallmatrix}1&0&0\\0&-1&0\\0&0&1\\\end{smallmatrix}}\right]}

[cos⁡2π/psin⁡2π/p0sin⁡2π/p−cos⁡2π/p0001]{\displaystyle \left[{\begin{smallmatrix}\cos 2\pi /p&\sin 2\pi /p&0\\\sin 2\pi /p&-\cos 2\pi /p&0\\0&0&1\\\end{smallmatrix}}\right]}

[10001000−1]{\displaystyle \left[{\begin{smallmatrix}1&0&0\\0&1&0\\0&0&-1\\\end{smallmatrix}}\right]}

[cos⁡2π/psin⁡2π/p0−sin⁡2π/pcos⁡2π/p0001]{\displaystyle \left[{\begin{smallmatrix}\cos 2\pi /p&\sin 2\pi /p&0\\-\sin 2\pi /p&\cos 2\pi /p&0\\0&0&1\\\end{smallmatrix}}\right]}

[cos⁡2π/psin⁡2π/p0−sin⁡2π/pcos⁡2π/p000−1]{\displaystyle \left[{\begin{smallmatrix}\cos 2\pi /p&\sin 2\pi /p&0\\-\sin 2\pi /p&\cos 2\pi /p&0\\0&0&-1\\\end{smallmatrix}}\right]}

[1000−1000−1]{\displaystyle \left[{\begin{smallmatrix}1&0&0\\0&-1&0\\0&0&-1\\\end{smallmatrix}}\right]}

[cos⁡2π/p−sin⁡2π/p0−sin⁡2π/p−cos⁡2π/p0001]{\displaystyle \left[{\begin{smallmatrix}\cos 2\pi /p&-\sin 2\pi /p&0\\-\sin 2\pi /p&-\cos 2\pi /p&0\\0&0&1\\\end{smallmatrix}}\right]}

[3,3]

[edit]
reflection lines for [3,3] =

The simplest irreducible 3-dimensional finite reflective group istetrahedral symmetry, [3,3], order 24,. The reflection generators, from a D3=A3 construction, are matrices R0, R1, R2. R02=R12=R22=(R0×R1)3=(R1×R2)3=(R0×R2)2=Identity. [3,3]+ () is generated by 2 of 3 rotations: S0,1, S1,2, and S0,2. Atrionic subgroup, isomorphic to [2+,4], order 8, is generated by S0,2 and R1. An order 4rotoreflection is generated by V0,1,2, the product of all 3 reflections.

[3,3],
ReflectionsRotationsRotoreflection
NameR0R1R2S0,1S1,2S0,2V0,1,2
Name
Order222324
Matrix

[100001010]{\displaystyle \left[{\begin{smallmatrix}1&0&0\\0&0&1\\0&1&0\\\end{smallmatrix}}\right]}

[010100001]{\displaystyle \left[{\begin{smallmatrix}0&1&0\\1&0&0\\0&0&1\\\end{smallmatrix}}\right]}

[10000−10−10]{\displaystyle \left[{\begin{smallmatrix}1&0&0\\0&0&-1\\0&-1&0\\\end{smallmatrix}}\right]}

[010001100]{\displaystyle \left[{\begin{smallmatrix}0&1&0\\0&0&1\\1&0&0\\\end{smallmatrix}}\right]}

[00−11000−10]{\displaystyle \left[{\begin{smallmatrix}0&0&-1\\1&0&0\\0&-1&0\\\end{smallmatrix}}\right]}

[1000−1000−1]{\displaystyle \left[{\begin{smallmatrix}1&0&0\\0&-1&0\\0&0&-1\\\end{smallmatrix}}\right]}

[00−10−10100]{\displaystyle \left[{\begin{smallmatrix}0&0&-1\\0&-1&0\\1&0&0\\\end{smallmatrix}}\right]}

(0,1,−1)n(1,−1,0)n(0,1,1)n(1,1,1)axis(1,1,−1)axis(1,0,0)axis

[4,3]

[edit]
Reflection lines for [4,3] =

Another irreducible 3-dimensional finite reflective group isoctahedral symmetry, [4,3], order 48,. The reflection generators matrices are R0, R1, R2. R02=R12=R22=(R0×R1)4=(R1×R2)3=(R0×R2)2=Identity. Chiral octahedral symmetry, [4,3]+, () is generated by 2 of 3 rotations: S0,1, S1,2, and S0,2.Pyritohedral symmetry [4,3+], () is generated by reflection R0 and rotation S1,2. A 6-foldrotoreflection is generated by V0,1,2, the product of all 3 reflections.

[4,3],
ReflectionsRotationsRotoreflection
NameR0R1R2S0,1S1,2S0,2V0,1,2
Group
Order2224326
Matrix

[10001000−1]{\displaystyle \left[{\begin{smallmatrix}1&0&0\\0&1&0\\0&0&-1\\\end{smallmatrix}}\right]}

[100001010]{\displaystyle \left[{\begin{smallmatrix}1&0&0\\0&0&1\\0&1&0\\\end{smallmatrix}}\right]}

[010100001]{\displaystyle \left[{\begin{smallmatrix}0&1&0\\1&0&0\\0&0&1\\\end{smallmatrix}}\right]}

[1000010−10]{\displaystyle \left[{\begin{smallmatrix}1&0&0\\0&0&1\\0&-1&0\\\end{smallmatrix}}\right]}

[010001100]{\displaystyle \left[{\begin{smallmatrix}0&1&0\\0&0&1\\1&0&0\\\end{smallmatrix}}\right]}

[01010000−1]{\displaystyle \left[{\begin{smallmatrix}0&1&0\\1&0&0\\0&0&-1\\\end{smallmatrix}}\right]}

[010001−100]{\displaystyle \left[{\begin{smallmatrix}0&1&0\\0&0&1\\-1&0&0\\\end{smallmatrix}}\right]}

(0,0,1)n(0,1,−1)n(1,−1,0)n(1,0,0)axis(1,1,1)axis(1,−1,0)axis

[5,3]

[edit]
Reflection lines for [5,3] =

A final irreducible 3-dimensional finite reflective group isicosahedral symmetry, [5,3], order 120,. The reflection generators matrices are R0, R1, R2. R02=R12=R22=(R0×R1)5=(R1×R2)3=(R0×R2)2=Identity. [5,3]+ () is generated by 2 of 3 rotations: S0,1, S1,2, and S0,2. A 10-foldrotoreflection is generated by V0,1,2, the product of all 3 reflections.

[5,3],
ReflectionsRotationsRotoreflection
NameR0R1R2S0,1S1,2S0,2V0,1,2
Group
Order22253210
Matrix[−100010001]{\displaystyle \left[{\begin{smallmatrix}-1&0&0\\0&1&0\\0&0&1\end{smallmatrix}}\right]}[1−ϕ2−ϕ2−12−ϕ2121−ϕ2−121−ϕ2ϕ2]{\displaystyle \left[{\begin{smallmatrix}{\frac {1-\phi }{2}}&{\frac {-\phi }{2}}&{\frac {-1}{2}}\\{\frac {-\phi }{2}}&{\frac {1}{2}}&{\frac {1-\phi }{2}}\\{\frac {-1}{2}}&{\frac {1-\phi }{2}}&{\frac {\phi }{2}}\end{smallmatrix}}\right]}[1000−10001]{\displaystyle \left[{\begin{smallmatrix}1&0&0\\0&-1&0\\0&0&1\end{smallmatrix}}\right]}[ϕ−12ϕ212−ϕ2121−ϕ2−121−ϕ2ϕ2]{\displaystyle \left[{\begin{smallmatrix}{\frac {\phi -1}{2}}&{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {-\phi }{2}}&{\frac {1}{2}}&{\frac {1-\phi }{2}}\\{\frac {-1}{2}}&{\frac {1-\phi }{2}}&{\frac {\phi }{2}}\end{smallmatrix}}\right]}[1−ϕ2ϕ2−12−ϕ2−121−ϕ2−12ϕ−12ϕ2]{\displaystyle \left[{\begin{smallmatrix}{\frac {1-\phi }{2}}&{\frac {\phi }{2}}&{\frac {-1}{2}}\\{\frac {-\phi }{2}}&{\frac {-1}{2}}&{\frac {1-\phi }{2}}\\{\frac {-1}{2}}&{\frac {\phi -1}{2}}&{\frac {\phi }{2}}\end{smallmatrix}}\right]}[−1000−10001]{\displaystyle \left[{\begin{smallmatrix}-1&0&0\\0&-1&0\\0&0&1\end{smallmatrix}}\right]}[ϕ−12−ϕ212−ϕ2−121−ϕ2−12ϕ−12ϕ2]{\displaystyle \left[{\begin{smallmatrix}{\frac {\phi -1}{2}}&{\frac {-\phi }{2}}&{\frac {1}{2}}\\{\frac {-\phi }{2}}&{\frac {-1}{2}}&{\frac {1-\phi }{2}}\\{\frac {-1}{2}}&{\frac {\phi -1}{2}}&{\frac {\phi }{2}}\end{smallmatrix}}\right]}
(1,0,0)n(φ,1,φ−1)n(0,1,0)n(φ,1,0)axis(1,1,1)axis(1,0,0)axis

Rank 4

[edit]
Further information:Point groups in four dimensions

There are 4 irreducibleCoxeter groups in 4 dimensions: [3,3,3], [4,3,3], [31,1,1], [3,4,4], [5,3,3], as well as an infinite family of duoprismatic groups [p,2,q].

[p,2,q]

[edit]

The duprismatic group, [p,2,q], has order 4pq.

[p,2,q],
Reflections
NameR0R1R2R3
Group element
Order2222
Matrix

[10000−10000100001]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&-1&0&0\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[cos⁡2π/psin⁡2π/p00sin⁡2π/p−cos⁡2π/p0000100001]{\displaystyle \left[{\begin{smallmatrix}\cos 2\pi /p&\sin 2\pi /p&0&0\\\sin 2\pi /p&-\cos 2\pi /p&0&0\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[100001000010000−1]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&-1\\\end{smallmatrix}}\right]}

[1000010000cos⁡2π/qsin⁡2π/q00sin⁡2π/q−cos⁡2π/q]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&\cos 2\pi /q&\sin 2\pi /q\\0&0&\sin 2\pi /q&-\cos 2\pi /q\\\end{smallmatrix}}\right]}

[[p,2,p]]
[edit]

The duoprismatic group can double in order, to 8p2, with a 2-fold rotation between the two planes.

[[p,2,p]],
RotationReflections
NameTR0R1R2=TR1TR3=TR0T
Element
Order222
Matrix

[10000−10000100001]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&-1&0&0\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[cos⁡2π/psin⁡2π/p00sin⁡2π/p−cos⁡2π/p0000100001]{\displaystyle \left[{\begin{smallmatrix}\cos 2\pi /p&\sin 2\pi /p&0&0\\\sin 2\pi /p&-\cos 2\pi /p&0&0\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[000−100−100−100−1000]{\displaystyle \left[{\begin{smallmatrix}0&0&0&-1\\0&0&-1&0\\0&-1&0&0\\-1&0&0&0\\\end{smallmatrix}}\right]}

[100001000010000−1]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&-1\\\end{smallmatrix}}\right]}

[1000010000cos⁡2π/psin⁡2π/p00sin⁡2π/p−cos⁡2π/p]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&\cos 2\pi /p&\sin 2\pi /p\\0&0&\sin 2\pi /p&-\cos 2\pi /p\\\end{smallmatrix}}\right]}

[3,3,3]

[edit]

Hypertetrahedral symmetry, [3,3,3], order 120, is easiest to represent with 4 mirrors in 5-dimensions, as a subgroup of [4,3,3,3].

[3,3,3],
ReflectionsRotationsRotoreflectionsDouble rotation
NameR0R1R2R3S0,1S1,2S2,3S0,2S1,3S2,3V0,1,2V0,1,3W0,1,2,3
Element group
Order222232465
Matrix

[1000001000001000000100010]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0&0\\0&1&0&0&0\\0&0&1&0&0\\0&0&0&0&1\\0&0&0&1&0\\\end{smallmatrix}}\right]}

[1000001000000100010000001]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0&0\\0&1&0&0&0\\0&0&0&1&0\\0&0&1&0&0\\0&0&0&0&1\\\end{smallmatrix}}\right]}

[1000000100010000001000001]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0&0\\0&0&1&0&0\\0&1&0&0&0\\0&0&0&1&0\\0&0&0&0&1\\\end{smallmatrix}}\right]}

[0100010000001000001000001]{\displaystyle \left[{\begin{smallmatrix}0&1&0&0&0\\1&0&0&0&0\\0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1\\\end{smallmatrix}}\right]}

[1000001000000100000100100]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0&0\\0&1&0&0&0\\0&0&0&1&0\\0&0&0&0&1\\0&0&1&0&0\\\end{smallmatrix}}\right]}

[1000000100000100100000001]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0&0\\0&0&1&0&0\\0&0&0&1&0\\0&1&0&0&0\\0&0&0&0&1\\\end{smallmatrix}}\right]}

[1000000100000100100000001]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0&0\\0&0&1&0&0\\0&0&0&1&0\\0&1&0&0&0\\0&0&0&0&1\\\end{smallmatrix}}\right]}

[1000000100010000000100010]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0&0\\0&0&1&0&0\\0&1&0&0&0\\0&0&0&0&1\\0&0&0&1&0\\\end{smallmatrix}}\right]}

[0100010000000100010000001]{\displaystyle \left[{\begin{smallmatrix}0&1&0&0&0\\1&0&0&0&0\\0&0&0&1&0\\0&0&1&0&0\\0&0&0&0&1\\\end{smallmatrix}}\right]}

[0100010000001000000100010]{\displaystyle \left[{\begin{smallmatrix}0&1&0&0&0\\1&0&0&0&0\\0&0&1&0&0\\0&0&0&0&1\\0&0&0&1&0\\\end{smallmatrix}}\right]}

[1000000100000100000101000]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0&0\\0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1\\0&1&0&0&0\\\end{smallmatrix}}\right]}

[0100010000000100000100100]{\displaystyle \left[{\begin{smallmatrix}0&1&0&0&0\\1&0&0&0&0\\0&0&0&1&0\\0&0&0&0&1\\0&0&1&0&0\\\end{smallmatrix}}\right]}

[0100000100000100000110000]{\displaystyle \left[{\begin{smallmatrix}0&1&0&0&0\\0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1\\1&0&0&0&0\\\end{smallmatrix}}\right]}

(0,0,0,1,-1)n(0,0,1,−1,0)n(0,1,−1,0,0)n(1,−1,0,0,0)n
[[3,3,3]]
[edit]

The extended group [[3,3,3]], order 240, is doubled by a 2-fold rotation matrix T, here reversing coordinate order and sign: There are 3 generators {T, R0, R1}. Since T is self-reciprocal R3=TR0T, and R2=TR1T.

[[3,3,3]],
RotationReflections
NameTR0R1TR1T=R2TR0T=R3
Element group
Order22222
Matrix

[0000−1000−1000−1000−1000−10000]{\displaystyle \left[{\begin{smallmatrix}0&0&0&0&-1\\0&0&0&-1&0\\0&0&-1&0&0\\0&-1&0&0&0\\-1&0&0&0&0\\\end{smallmatrix}}\right]}

[1000001000001000000100010]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0&0\\0&1&0&0&0\\0&0&1&0&0\\0&0&0&0&1\\0&0&0&1&0\\\end{smallmatrix}}\right]}

[1000001000000100010000001]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0&0\\0&1&0&0&0\\0&0&0&1&0\\0&0&1&0&0\\0&0&0&0&1\\\end{smallmatrix}}\right]}

[1000000100010000001000001]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0&0\\0&0&1&0&0\\0&1&0&0&0\\0&0&0&1&0\\0&0&0&0&1\\\end{smallmatrix}}\right]}

[0100010000001000001000001]{\displaystyle \left[{\begin{smallmatrix}0&1&0&0&0\\1&0&0&0&0\\0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1\\\end{smallmatrix}}\right]}

(0,0,0,1,-1)n(0,0,1,−1,0)n(0,1,−1,0,0)n(1,−1,0,0,0)n

[4,3,3]

[edit]

A irreducible 4-dimensional finite reflective group ishyperoctahedral group (or hexadecachoric group (for16-cell), B4=[4,3,3], order 384,. The reflection generators matrices are R0, R1, R2, R3. R02=R12=R22=R32=(R0×R1)4=(R1×R2)3=(R2×R3)3=(R0×R2)2=(R1×R3)2=(R0×R3)2=Identity.

Chiral hyperoctahedral symmetry, [4,3,3]+, () is generated by 3 of 6 rotations: S0,1, S1,2, S2,3, S0,2, S1,3, and S0,3.Hyperpyritohedral symmetry [4,(3,3)+], () is generated by reflection R0 and rotations S1,2 and S2,3. An 8-folddouble rotation is generated by W0,1,2,3, the product of all 4 reflections.

[4,3,3],
ReflectionsRotationsRotoreflectionDouble rotation
NameR0R1R2R3S0,1S1,2S2,3S0,2S1,3S0,3V1,2,3V0,1,3V0,1,2V0,2,3W0,1,2,3
Group
Order2222432468
Matrix

[100001000010000−1]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&-1\\\end{smallmatrix}}\right]}

[1000010000010010]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\\\end{smallmatrix}}\right]}

[1000001001000001]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[0100100000100001]{\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[10000100000100−10]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&-1&0\\\end{smallmatrix}}\right]}

[1000001000010100]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&0&0&1\\0&1&0&0\\\end{smallmatrix}}\right]}

[0100001010000001]{\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\0&0&1&0\\1&0&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[100000100100000−1]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&-1\\\end{smallmatrix}}\right]}

[0100100000010010]{\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&0&1\\0&0&1&0\\\end{smallmatrix}}\right]}

[010010000010000−1]{\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&1&0\\0&0&0&-1\\\end{smallmatrix}}\right]}

[0100001000011000]{\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\0&0&1&0\\0&0&0&1\\1&0&0&0\\\end{smallmatrix}}\right]}

[01001000000100−10]{\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&0&1\\0&0&-1&0\\\end{smallmatrix}}\right]}

[10000010000100−10]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&0&0&1\\0&0&-1&0\\\end{smallmatrix}}\right]}

[010000101000000−1]{\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\0&0&1&0\\1&0&0&0\\0&0&0&-1\\\end{smallmatrix}}\right]}

[010000100001−1000]{\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\0&0&1&0\\0&0&0&1\\-1&0&0&0\\\end{smallmatrix}}\right]}

(0,0,0,1)n(0,0,1,−1)n(0,1,−1,0)n(1,−1,0,0)n
[3,31,1]
[edit]

A half group of [4,3,3] is [3,31,1],, order 192. It shares 3 generators with [4,3,3] group, but has two copies of an adjacent generator, one reflected across the removed mirror.

[3,31,1],
Reflections
NameR0R1R2R3
Group
Order2222
Matrix

[0100100000100001]{\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[1000001001000001]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[1000010000010010]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\\\end{smallmatrix}}\right]}

[10000100000−100−10]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&0&-1\\0&0&-1&0\\\end{smallmatrix}}\right]}

(1,−1,0,0)n(0,1,−1,0)n(0,0,1,−1)n(0,0,1,1)n

[3,4,3]

[edit]

A irreducible 4-dimensional finite reflective group isIcositetrachoric group (for24-cell), F4=[3,4,3], order 1152,. The reflection generators matrices are R0, R1, R2, R3. R02=R12=R22=R32=(R0×R1)3=(R1×R2)4=(R2×R3)3=(R0×R2)2=(R1×R3)2=(R0×R3)2=Identity.

Chiral icositetrachoric symmetry, [3,4,3]+, () is generated by 3 of 6 rotations: S0,1, S1,2, S2,3, S0,2, S1,3, and S0,3. Ionic diminished [3,4,3+] group, () is generated by reflection R0 and rotations S1,2 and S2,3. A 12-folddouble rotation is generated by W0,1,2,3, the product of all 4 reflections.

[3,4,3],
ReflectionsRotations
NameR0R1R2R3S0,1S1,2S2,3S0,2S1,3S0,3
Element group
Order22223432
Matrix

[0100100000100001]{\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[1000001001000001]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[1000010000−100001]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[1/2−1/2−1/2−1/2−1/21/2−1/2−1/2−1/2−1/21/2−1/2−1/2−1/2−1/21/2]{\displaystyle \left[{\begin{smallmatrix}1/2&-1/2&-1/2&-1/2\\-1/2&1/2&-1/2&-1/2\\-1/2&-1/2&1/2&-1/2\\-1/2&-1/2&-1/2&1/2\\\end{smallmatrix}}\right]}

[0100001010000001]{\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\0&0&1&0\\1&0&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[100000100−1000001]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&-1&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[1/2−1/21/2−1/2−1/21/21/2−1/2−1/2−1/2−1/2−1/2−1/2−1/21/21/2]{\displaystyle \left[{\begin{smallmatrix}1/2&-1/2&1/2&-1/2\\-1/2&1/2&1/2&-1/2\\-1/2&-1/2&-1/2&-1/2\\-1/2&-1/2&1/2&1/2\\\end{smallmatrix}}\right]}

[0100100000−100001]{\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&-1&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[1/2−1/2−1/2−1/2−1/2−1/21/2−1/2−1/21/2−1/2−1/2−1/2−1/2−1/21/2]{\displaystyle \left[{\begin{smallmatrix}1/2&-1/2&-1/2&-1/2\\-1/2&-1/2&1/2&-1/2\\-1/2&1/2&-1/2&-1/2\\-1/2&-1/2&-1/2&1/2\\\end{smallmatrix}}\right]}

[−1/21/2−1/2−1/21/2−1/2−1/2−1/2−1/2−1/21/2−1/2−1/21/2−1/21/2]{\displaystyle \left[{\begin{smallmatrix}-1/2&1/2&-1/2&-1/2\\1/2&-1/2&-1/2&-1/2\\-1/2&-1/2&1/2&-1/2\\-1/2&1/2&-1/2&1/2\\\end{smallmatrix}}\right]}

(1,−1,0,0)n(0,1,−1,0)n(0,0,1,0)n(−1,−1,−1,−1)n
[3,4,3],
RotoreflectionDouble rotation
NameV1,2,3V0,1,3V0,1,2V0,2,3W0,1,2,3
Element group
Order612
Matrix

[−1/21/2−1/2−1/2−1/2−1/21/2−1/21/2−1/2−1/2−1/2−1/2−1/2−1/21/2]{\displaystyle \left[{\begin{smallmatrix}-1/2&1/2&-1/2&-1/2\\-1/2&-1/2&1/2&-1/2\\1/2&-1/2&-1/2&-1/2\\-1/2&-1/2&-1/2&1/2\\\end{smallmatrix}}\right]}

[01000010−10000001]{\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\0&0&1&0\\-1&0&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[1/21/2−1/2−1/2−1/21/21/2−1/2−1/2−1/2−1/2−1/2−1/21/2−1/21/2]{\displaystyle \left[{\begin{smallmatrix}1/2&1/2&-1/2&-1/2\\-1/2&1/2&1/2&-1/2\\-1/2&-1/2&-1/2&-1/2\\-1/2&1/2&-1/2&1/2\\\end{smallmatrix}}\right]}

[−1/21/21/2−1/21/2−1/21/2−1/2−1/2−1/2−1/2−1/2−1/2−1/21/21/2]{\displaystyle \left[{\begin{smallmatrix}-1/2&1/2&1/2&-1/2\\1/2&-1/2&1/2&-1/2\\-1/2&-1/2&-1/2&-1/2\\-1/2&-1/2&1/2&1/2\\\end{smallmatrix}}\right]}

[1/21/2−1/2−1/21/2−1/21/2−1/2−1/2−1/2−1/2−1/21/2−1/2−1/21/2]{\displaystyle \left[{\begin{smallmatrix}1/2&1/2&-1/2&-1/2\\1/2&-1/2&1/2&-1/2\\-1/2&-1/2&-1/2&-1/2\\1/2&-1/2&-1/2&1/2\\\end{smallmatrix}}\right]}

[[3,4,3]]
[edit]

The group [[3,4,3]] extends [3,4,3] by a 2-fold rotation, T, doubling order to 2304.

[[3,4,3]],
RotationReflections
NameTR0R1R2 = TR1TR3 = TR0T
Element group
Order22222
Matrix

[0100100000100001]{\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[1000001001000001]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[1000010000−100001]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[1/2−1/2−1/2−1/2−1/21/2−1/2−1/2−1/2−1/21/2−1/2−1/2−1/2−1/21/2]{\displaystyle \left[{\begin{smallmatrix}1/2&-1/2&-1/2&-1/2\\-1/2&1/2&-1/2&-1/2\\-1/2&-1/2&1/2&-1/2\\-1/2&-1/2&-1/2&1/2\\\end{smallmatrix}}\right]}

(1,−1,0,0)n(0,1,−1,0)n(0,0,1,0)n(−1,−1,−1,−1)n

[5,3,3]

[edit]
Stereographic projections

[5,3,3]+ 72 order-5 gyrations
[5,3,3]+ 200 order-3 gyrations

[5,3,3]+ 450 order-2 gyrations
[5,3,3]+ all gyrations

The hyper-icosahedral symmetry, [5,3,3], order 14400,. The reflection generators matrices are R0, R1, R2, R3. R02=R12=R22=R32=(R0×R1)5=(R1×R2)3=(R2×R3)3=(R0×R2)2=(R0×R3)2=(R1×R3)2=Identity. [5,3,3]+ () is generated by 3 rotations: S0,1 = R0×R1, S1,2 = R1×R2, S2,3 = R2×R3, etc.

[5,3,3],
Reflections
NameR0R1R2R3
Element group
Order2222
Matrix[−1000010000100001]{\displaystyle \left[{\begin{smallmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{smallmatrix}}\right]}[1−ϕ2−ϕ2−120−ϕ2121−ϕ20−121−ϕ2ϕ200001]{\displaystyle \left[{\begin{smallmatrix}{\frac {1-\phi }{2}}&{\frac {-\phi }{2}}&{\frac {-1}{2}}&0\\{\frac {-\phi }{2}}&{\frac {1}{2}}&{\frac {1-\phi }{2}}&0\\{\frac {-1}{2}}&{\frac {1-\phi }{2}}&{\frac {\phi }{2}}&0\\0&0&0&1\end{smallmatrix}}\right]}[10000−10000100001]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&-1&0&0\\0&0&1&0\\0&0&0&1\end{smallmatrix}}\right]}[1000012ϕ21−ϕ20ϕ21−ϕ21201−ϕ212ϕ2]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&{\frac {1}{2}}&{\frac {\phi }{2}}&{\frac {1-\phi }{2}}\\0&{\frac {\phi }{2}}&{\frac {1-\phi }{2}}&{\frac {1}{2}}\\0&{\frac {1-\phi }{2}}&{\frac {1}{2}}&{\frac {\phi }{2}}\end{smallmatrix}}\right]}
(1,0,0,0)n(φ,1,φ−1,0)n(0,1,0,0)n(0,−1,φ,1−φ)n

Rank 8

[edit]

[34,2,1]

[edit]

TheE8 Coxeter group, [34,2,1],, has 8 mirror nodes, order 696729600 (192x10!). E7 and E6, [33,2,1],, and [32,2,1], can be constructed by ignoring the first mirror or the first two mirrors respectively.

E8=[34,2,1],
Reflections
NameR0R1R2R3R4R5R6R7
Element group
Order22222222
Matrix[0100000010000000001000000001000000001000000001000000001000000001]{\displaystyle \left[{\begin{smallmatrix}0&1&0&0&0&0&0&0\\1&0&0&0&0&0&0&0\\0&0&1&0&0&0&0&0\\0&0&0&1&0&0&0&0\\0&0&0&0&1&0&0&0\\0&0&0&0&0&1&0&0\\0&0&0&0&0&0&1&0\\0&0&0&0&0&0&0&1\end{smallmatrix}}\right]}[1000000000100000010000000001000000001000000001000000001000000001]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0&0&0&0&0\\0&0&1&0&0&0&0&0\\0&1&0&0&0&0&0&0\\0&0&0&1&0&0&0&0\\0&0&0&0&1&0&0&0\\0&0&0&0&0&1&0&0\\0&0&0&0&0&0&1&0\\0&0&0&0&0&0&0&1\end{smallmatrix}}\right]}[1000000001000000000100000010000000001000000001000000001000000001]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0&0&0&0&0\\0&1&0&0&0&0&0&0\\0&0&0&1&0&0&0&0\\0&0&1&0&0&0&0&0\\0&0&0&0&1&0&0&0\\0&0&0&0&0&1&0&0\\0&0&0&0&0&0&1&0\\0&0&0&0&0&0&0&1\end{smallmatrix}}\right]}[1000000001000000001000000000100000010000000001000000001000000001]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0&0&0&0&0\\0&1&0&0&0&0&0&0\\0&0&1&0&0&0&0&0\\0&0&0&0&1&0&0&0\\0&0&0&1&0&0&0&0\\0&0&0&0&0&1&0&0\\0&0&0&0&0&0&1&0\\0&0&0&0&0&0&0&1\end{smallmatrix}}\right]}[1000000001000000001000000001000000000100000010000000001000000001]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0&0&0&0&0\\0&1&0&0&0&0&0&0\\0&0&1&0&0&0&0&0\\0&0&0&1&0&0&0&0\\0&0&0&0&0&1&0&0\\0&0&0&0&1&0&0&0\\0&0&0&0&0&0&1&0\\0&0&0&0&0&0&0&1\end{smallmatrix}}\right]}[1000000001000000001000000001000000001000000000100000010000000001]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0&0&0&0&0\\0&1&0&0&0&0&0&0\\0&0&1&0&0&0&0&0\\0&0&0&1&0&0&0&0\\0&0&0&0&1&0&0&0\\0&0&0&0&0&0&1&0\\0&0&0&0&0&1&0&0\\0&0&0&0&0&0&0&1\end{smallmatrix}}\right]}[1000000001000000001000000001000000001000000000−1000000−10000000001]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0&0&0&0&0\\0&1&0&0&0&0&0&0\\0&0&1&0&0&0&0&0\\0&0&0&1&0&0&0&0\\0&0&0&0&1&0&0&0\\0&0&0&0&0&0&-1&0\\0&0&0&0&0&-1&0&0\\0&0&0&0&0&0&0&1\end{smallmatrix}}\right]}[3/4−1/4−1/4−1/4−1/4−1/4−1/4−1/4−1/43/4−1/4−1/4−1/4−1/4−1/4−1/4−1/4−1/43/4−1/4−1/4−1/4−1/4−1/4−1/4−1/4−1/43/4−1/4−1/4−1/4−1/4−1/4−1/4−1/4−1/43/4−1/4−1/4−1/4−1/4−1/4−1/4−1/4−1/43/4−1/4−1/4−1/4−1/4−1/4−1/4−1/4−1/43/4−1/4−1/4−1/4−1/4−1/4−1/4−1/4−1/43/4]{\displaystyle \left[{\begin{smallmatrix}3/4&-1/4&-1/4&-1/4&-1/4&-1/4&-1/4&-1/4\\-1/4&3/4&-1/4&-1/4&-1/4&-1/4&-1/4&-1/4\\-1/4&-1/4&3/4&-1/4&-1/4&-1/4&-1/4&-1/4\\-1/4&-1/4&-1/4&3/4&-1/4&-1/4&-1/4&-1/4\\-1/4&-1/4&-1/4&-1/4&3/4&-1/4&-1/4&-1/4\\-1/4&-1/4&-1/4&-1/4&-1/4&3/4&-1/4&-1/4\\-1/4&-1/4&-1/4&-1/4&-1/4&-1/4&3/4&-1/4\\-1/4&-1/4&-1/4&-1/4&-1/4&-1/4&-1/4&3/4\end{smallmatrix}}\right]}
(1,-1,0,0,0,0,0,0)n(0,1,-1,0,0,0,0,0)n(0,0,1,-1,0,0,0,0)n(0,0,0,1,-1,0,0,0)n(0,0,0,0,1,-1,0,0)n(0,0,0,0,0,1,-1,0)n(0,0,0,0,0,1,1,0)n(1,1,1,1,1,1,1,1)n

Affine rank 2

[edit]

Affine matrices are represented by adding an extra row and column, the last row being zero except last entry 1. The last column represents a translation vector.

[∞]

[edit]

The affine group [∞],, can be given by two reflection matrices, x=0 and x=1.

[∞],
ReflectionsTranslation
NameR0R1S0,1
Element group
Order22∞
Matrix

[−1001]{\displaystyle \left[{\begin{smallmatrix}-1&0\\0&1\\\end{smallmatrix}}\right]}

[−1101]{\displaystyle \left[{\begin{smallmatrix}-1&1\\0&1\\\end{smallmatrix}}\right]}

[1−101]{\displaystyle \left[{\begin{smallmatrix}1&-1\\0&1\\\end{smallmatrix}}\right]}

Hyperplanex=0x=1

Affine rank 3

[edit]

[4,4]

[edit]
Further information:Tetrakis_square_tiling § Symmetry

The affine group [4,4],, (p4m), can be given by three reflection matrices, reflections across the x axis (y=0), a diagonal (x=y), and the affine reflection across the line (x=1). [4,4]+ () (p4) is generated by S0,1 S1,2, and S0,2. [4+,4+] () (pgg) is generated by 2-fold rotation S0,2 andglide reflection (transreflection) V0,1,2. [4+,4] () (p4g) is generated by S0,1 and R3. The group [(4,4,2+)] () (cmm), is generated by 2-fold rotation S1,3 and reflection R2.

[4,4],
ReflectionsRotationsGlides
NameR0R1R2S0,1S1,2S0,2V0,1,2V0,2,1
Element group
Order22242∞ (2)
Matrix

[1000−10001]{\displaystyle \left[{\begin{smallmatrix}1&0&0\\0&-1&0\\0&0&1\\\end{smallmatrix}}\right]}

[010100001]{\displaystyle \left[{\begin{smallmatrix}0&1&0\\1&0&0\\0&0&1\\\end{smallmatrix}}\right]}

[−102010001]{\displaystyle \left[{\begin{smallmatrix}-1&0&2\\0&1&0\\0&0&1\\\end{smallmatrix}}\right]}

[010−100001]{\displaystyle \left[{\begin{smallmatrix}0&1&0\\-1&0&0\\0&0&1\\\end{smallmatrix}}\right]}

[010−102001]{\displaystyle \left[{\begin{smallmatrix}0&1&0\\-1&0&2\\0&0&1\\\end{smallmatrix}}\right]}

[−1020−10001]{\displaystyle \left[{\begin{smallmatrix}-1&0&2\\0&-1&0\\0&0&1\\\end{smallmatrix}}\right]}

[01010−2001]{\displaystyle \left[{\begin{smallmatrix}0&1&0\\1&0&-2\\0&0&1\\\end{smallmatrix}}\right]}

[0−12−100001]{\displaystyle \left[{\begin{smallmatrix}0&-1&2\\-1&0&0\\0&0&1\\\end{smallmatrix}}\right]}

Hyperplaney=0x=yx=1

[3,6]

[edit]

The affine group [3,6],, (p6m), can be given by three reflection matrices, reflections across the x axis (y=0), line y=(√3/2)x, and vertical line x=1.

[3,6],
ReflectionsRotationsGlides
NameR0R1R2S0,1S1,2S0,2V0,1,2V0,2,1
Element group
Order222362∞ (2)
Matrix

[1000−10001]{\displaystyle \left[{\begin{smallmatrix}1&0&0\\0&-1&0\\0&0&1\\\end{smallmatrix}}\right]}

[−1/23/203/21/20001]{\displaystyle \left[{\begin{smallmatrix}-1/2&{\sqrt {3}}/2&0\\{\sqrt {3}}/2&1/2&0\\0&0&1\\\end{smallmatrix}}\right]}

[−102010001]{\displaystyle \left[{\begin{smallmatrix}-1&0&2\\0&1&0\\0&0&1\\\end{smallmatrix}}\right]}

[−1/23/20−3/2−1/20001]{\displaystyle \left[{\begin{smallmatrix}-1/2&{\sqrt {3}}/2&0\\-{\sqrt {3}}/2&-1/2&0\\0&0&1\\\end{smallmatrix}}\right]}

[1/23/2−1−3/21/23001]{\displaystyle \left[{\begin{smallmatrix}1/2&{\sqrt {3}}/2&-1\\-{\sqrt {3}}/2&1/2&{\sqrt {3}}\\0&0&1\\\end{smallmatrix}}\right]}

[−1020−10001]{\displaystyle \left[{\begin{smallmatrix}-1&0&2\\0&-1&0\\0&0&1\\\end{smallmatrix}}\right]}

[1/23/2−13/2−1/2−3001]{\displaystyle \left[{\begin{smallmatrix}1/2&{\sqrt {3}}/2&-1\\{\sqrt {3}}/2&-1/2&-{\sqrt {3}}\\0&0&1\\\end{smallmatrix}}\right]}

[1/2−3/22−3/2−1/20001]{\displaystyle \left[{\begin{smallmatrix}1/2&-{\sqrt {3}}/2&2\\-{\sqrt {3}}/2&-1/2&0\\0&0&1\\\end{smallmatrix}}\right]}

Hyperplaney=0y=(√3/2)xx=1

[3[3]]

[edit]

The affine group [3[3]] can be constructed as a half group of. R2 is replaced by R'2 = R2×R1×R2, presented by the hyperplane: y+(√3/2)x=2. The fundamental domain is anequilateral triangle with edge length 2.

[3[3]],
ReflectionsRotationsGlides
NameR0R1R'2 = R2×R1×R2S0,1S1,2S0,2V0,1,2V0,2,1
Element group
Order2223∞ (2)
Matrix

[1000−10001]{\displaystyle \left[{\begin{smallmatrix}1&0&0\\0&-1&0\\0&0&1\\\end{smallmatrix}}\right]}

[−1/23/203/21/20001]{\displaystyle \left[{\begin{smallmatrix}-1/2&{\sqrt {3}}/2&0\\{\sqrt {3}}/2&1/2&0\\0&0&1\\\end{smallmatrix}}\right]}

[−1/2−3/23−3/21/23001]{\displaystyle \left[{\begin{smallmatrix}-1/2&-{\sqrt {3}}/2&3\\-{\sqrt {3}}/2&1/2&{\sqrt {3}}\\0&0&1\\\end{smallmatrix}}\right]}

[010−100001]{\displaystyle \left[{\begin{smallmatrix}0&1&0\\-1&0&0\\0&0&1\\\end{smallmatrix}}\right]}

[−1/23/20−3/2−1/223001]{\displaystyle \left[{\begin{smallmatrix}-1/2&{\sqrt {3}}/2&0\\-{\sqrt {3}}/2&-1/2&2{\sqrt {3}}\\0&0&1\\\end{smallmatrix}}\right]}

[−1/2−3/233/2−1/2−3001]{\displaystyle \left[{\begin{smallmatrix}-1/2&-{\sqrt {3}}/2&3\\{\sqrt {3}}/2&-1/2&-{\sqrt {3}}\\0&0&1\\\end{smallmatrix}}\right]}

[−1/23/203/21/2−23001]{\displaystyle \left[{\begin{smallmatrix}-1/2&{\sqrt {3}}/2&0\\{\sqrt {3}}/2&1/2&-2{\sqrt {3}}\\0&0&1\\\end{smallmatrix}}\right]}

[−1/2−3/23−3/21/2−3001]{\displaystyle \left[{\begin{smallmatrix}-1/2&-{\sqrt {3}}/2&3\\-{\sqrt {3}}/2&1/2&-{\sqrt {3}}\\0&0&1\\\end{smallmatrix}}\right]}

Hyperplaney=0y=(√3/2)xy+(√3/2)x=2

Affine rank 4

[edit]

[4,3,4]

[edit]
[4,3,4] fundamental domain

The affine group is [4,3,4] (), can be given by four reflection matrices. Mirror R0 can be put on z=0 plane. Mirror R1 can be put on plane y=z. Mirror R2 can be put on x=y plane. Mirror R3 can be put on x=1 plane. [4,3,4]+ () is generated by S0,1, S1,2, and S2,3.

[4,3,4],
ReflectionsRotationsTransflectionsScrew axis
NameR0R1R2R3S0,1S1,2S2,3S0,2S0,3S1,3T0,1,2T1,2,3U0,1,2,3
Element group
Order222243426∞ (3)
Matrix

[1000010000−100001]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[1000001001000001]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[0100100000100001]{\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[−1002010000100001]{\displaystyle \left[{\begin{smallmatrix}-1&0&0&2\\0&1&0&0\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[100000100−1000001]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&-1&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[0100001010000001]{\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\0&0&1&0\\1&0&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[0100−100200100001]{\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\-1&0&0&2\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[0100100000−100001]{\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&-1&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[−1002010000−100001]{\displaystyle \left[{\begin{smallmatrix}-1&0&0&2\\0&1&0&0\\0&0&-1&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[−1002001001000001]{\displaystyle \left[{\begin{smallmatrix}-1&0&0&2\\0&0&1&0\\0&1&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[01000010−10000001]{\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\0&0&1&0\\-1&0&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[01000010−10020001]{\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\0&0&1&0\\-1&0&0&2\\0&0&0&1\\\end{smallmatrix}}\right]}

[01000010100−20001]{\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\0&0&1&0\\1&0&0&-2\\0&0&0&1\\\end{smallmatrix}}\right]}

Hyperplanez=0y=zx=yx=1
[[4,3,4]]
[edit]

The extended group [[4,3,4]] doubles the group order, adding with a 2-fold rotation matrix T, with a fixed axis through points (1,1/2,0) and (1/2,1/2,1/2). The generators are {R0,R1,T}. R2 = T×R1×T and R3 = T×R0×T.

[[4,3,4]],
RotationReflections
NameTR0R1R2 = T×R1×TR3 = T×R0×T
Element group
Order22222
Matrix

[00−110−101−10010001]{\displaystyle \left[{\begin{smallmatrix}0&0&-1&1\\0&-1&0&1\\-1&0&0&1\\0&0&0&1\\\end{smallmatrix}}\right]}

[1000010000−100001]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[1000001001000001]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[0100100000100001]{\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[−1002010000100001]{\displaystyle \left[{\begin{smallmatrix}-1&0&0&2\\0&1&0&0\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}

HyperplanePoint (1/2,1/2,1/2)
Axis (-1,0,1)		z=0y=zx=yx=1

[4,31,1]

[edit]
[4,31,1] fundamental domain

The group [4,31,1] can be constructed from [4,3,4], by computing [4,3,4,1+],, as R'3=R3×R2×R3, with new R'3 as an image of R2 across R3.

[4,31,1],
ReflectionsRotations
NameR0R1R2R'3S0,1S1,2S1,3S0,2S0,3S2,3
Element group
Order22223332
Matrix

[1000010000−100001]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[1000001001000001]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[0100100000100001]{\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[0−102−100200100001]{\displaystyle \left[{\begin{smallmatrix}0&-1&0&2\\-1&0&0&2\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[100000100−1000001]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&-1&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[0100001010000001]{\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\0&0&1&0\\1&0&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[0−1020010−10020001]{\displaystyle \left[{\begin{smallmatrix}0&-1&0&2\\0&0&1&0\\-1&0&0&2\\0&0&0&1\\\end{smallmatrix}}\right]}

[0100100000−100001]{\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&-1&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[0−102−100200−100001]{\displaystyle \left[{\begin{smallmatrix}0&-1&0&2\\-1&0&0&2\\0&0&-1&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[−10020−10200100001]{\displaystyle \left[{\begin{smallmatrix}-1&0&0&2\\0&-1&0&2\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}

Hyperplanez=0y=zx=yx+y=2

[3[4]]

[edit]
[3[4]] fundamental domain

The group [3[4]] can be constructed from [4,3,4], by removing first and last mirrors, [1+,4,3,4,1+],, by R'1=R0×R1×R0 and R'3=R3×R2×R3.

[3[4]]
ReflectionsRotations
NameR'0R1R2R'3S0,1S1,2S1,3S0,2S0,3S2,3
Element group
Order22223332
Matrix

[100000−100−1000001]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&0&-1&0\\0&-1&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[1000001001000001]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[0100100000100001]{\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[0−102−100200100001]{\displaystyle \left[{\begin{smallmatrix}0&-1&0&2\\-1&0&0&2\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[10000−10000−100001]{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[0100001010000001]{\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\0&0&1&0\\1&0&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[0−1020010−10020001]{\displaystyle \left[{\begin{smallmatrix}0&-1&0&2\\0&0&1&0\\-1&0&0&2\\0&0&0&1\\\end{smallmatrix}}\right]}

[010000−10−10000001]{\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\0&0&-1&0\\-1&0&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}

[0−10200−10100−20001]{\displaystyle \left[{\begin{smallmatrix}0&-1&0&2\\0&0&-1&0\\1&0&0&-2\\0&0&0&1\\\end{smallmatrix}}\right]}

[−10020−10200100001]{\displaystyle \left[{\begin{smallmatrix}-1&0&0&2\\0&-1&0&2\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}

Hyperplaney=-zy=zx=yx+y=2

Notes

[edit]
  1. ^Johnson (2018), 11.6Subgroups and extensions, p 255, halving subgroups
  2. ^abJohnson (2018), pp.231-236, and p 245 Table 11.4Finite groups of isometries in 3-space
  3. ^Johnson (2018), 11.6Subgroups and extensions, p 259, radical subgroup
  4. ^Johnson (2018), 11.6Subgroups and extensions, p 258, trionic subgroups
  5. ^Conway, 2003, p.46, Table 4.2 Chiral groups II
  6. ^Coxeter and Moser, 1980, Sec 9.5 Commutator subgroup, p. 124–126
  7. ^Johnson, Norman W.; Weiss, Asia Ivić (1999)."Quaternionic modular groups".Linear Algebra and Its Applications.295 (1–3):159–189.doi:10.1016/S0024-3795(99)00107-X.
  8. ^The Crystallographic Space groups in Geometric algebra, D. Hestenes and J. Holt, Journal of Mathematical Physics. 48, 023514 (2007) (22 pages)PDF[1]
  9. ^Coxeter, Regular Complex Polytopes, 9.7 Two-generator subgroups reflections. pp. 178–179

References

[edit]
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