# Coxeter notation

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 expresses Coxeter groups as a list of branch orders of a Coxeter diagram, like the polyhedral groups, = [p,q]. dihedral groups, , can be expressed a product [ ]×[n] or in a single symbol with an explicit order 2 branch, [2,n].

In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between with fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, with modifiers to indicate certain subgroups. The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson.

## Reflectional groupsEdit

For Coxeter groups, defined by pure reflections, there is a direct correspondence between the bracket notation and Coxeter-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 the An group is represented by [3n-1], to imply n nodes connected by n-1 order-3 branches. Example A2 = [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,r], starting with [31,1,1] =       as D4. Coxeter allowed for zeros as special cases to fit the An family, like A3 = [3,3,3,3] = [34,0,0] = [33,1,0] = [32,2,0], like           =         =      .

Coxeter groups formed by cyclic diagrams are represented by parenthesese inside of brackets, like [(p,q,r)] =   for the triangle 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. The paracompact 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 the rhombic 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 its regular tetrahedron coxeter diagram.

The Coxeter diagram usually leaves order-2 branches undrawn, but the bracket notation includes an explicit 2 to connect the subgraphs. So the Coxeter diagram         = A2×A2 = 2A2 can be represented by [3]×[3] = [3]2 = [3,2,3].

Finite groups
Rank Group
symbol
Bracket
notation
Coxeter
diagram
2 A2 [3]
2 B2 [4]
2 H2 [5]
2 G2 [6]
2 I2(p) [p]
3 Ih, H3 [5,3]
3 Td, A3 [3,3]
3 Oh, B3 [4,3]
4 A4 [3,3,3]
4 B4 [4,3,3]
4 D4 [31,1,1]
4 F4 [3,4,3]
4 H4 [5,3,3]
n An [3n-1]     ..
n Bn [4,3n-2]     ...
n Dn [3n-3,1,1]     ...
6 E6 [32,2,1]
7 E7 [33,2,1]
8 E8 [34,2,1]
Affine groups
Group
symbol
Bracket
notation
Coxeter diagram
${\displaystyle {\tilde {I}}_{1},{\tilde {A}}_{1}}$  [∞]
${\displaystyle {\tilde {A}}_{2}}$  [3[3]]
${\displaystyle {\tilde {C}}_{2}}$  [4,4]
${\displaystyle {\tilde {G}}_{2}}$  [6,3]
${\displaystyle {\tilde {A}}_{3}}$  [3[4]]
${\displaystyle {\tilde {B}}_{3}}$  [4,31,1]
${\displaystyle {\tilde {C}}_{3}}$  [4,3,4]
${\displaystyle {\tilde {A}}_{4}}$  [3[5]]
${\displaystyle {\tilde {B}}_{4}}$  [4,3,31,1]
${\displaystyle {\tilde {C}}_{4}}$  [4,3,3,4]
${\displaystyle {\tilde {D}}_{4}}$  [ 31,1,1,1]
${\displaystyle {\tilde {F}}_{4}}$  [3,4,3,3]
${\displaystyle {\tilde {A}}_{n}}$  [3[n+1]]     ...
or
...
${\displaystyle {\tilde {B}}_{n}}$  [4,3n-3,31,1]       ...
${\displaystyle {\tilde {C}}_{n}}$  [4,3n-2,4]       ...
${\displaystyle {\tilde {D}}_{n}}$  [ 31,1,3n-4,31,1]       ...
${\displaystyle {\tilde {E}}_{6}}$  [32,2,2]
${\displaystyle {\tilde {E}}_{7}}$  [33,3,1]
${\displaystyle {\tilde {E}}_{8},E_{9}}$  [35,2,1]
Hyperbolic groups
Group
symbol
Bracket
notation
Coxeter
diagram
[p,q]
with 2(p+q)<pq

[(p,q,r)]
with ${\displaystyle {\frac {1}{p}}+{\frac {1}{q}}+{\frac {1}{r}}<1}$

${\displaystyle {\overline {BH}}_{3}}$  [4,3,5]
${\displaystyle {\overline {K}}_{3}}$  [5,3,5]
${\displaystyle {\overline {J}}_{3},{\tilde {H}}_{3}}$  [3,5,3]
${\displaystyle {\overline {DH}}_{3}}$  [5,31,1]
${\displaystyle {\widehat {AB}}_{3}}$  [(3,3,3,4)]
${\displaystyle {\widehat {AH}}_{3}}$  [(3,3,3,5)]
${\displaystyle {\widehat {BB}}_{3}}$  [(3,4,3,4)]
${\displaystyle {\widehat {BH}}_{3}}$  [(3,4,3,5)]
${\displaystyle {\widehat {HH}}_{3}}$  [(3,5,3,5)]
${\displaystyle {\overline {H}}_{4},{\tilde {H}}_{4},H_{5}}$  [3,3,3,5]
${\displaystyle {\overline {BH}}_{4}}$  [4,3,3,5]
${\displaystyle {\overline {K}}_{4}}$  [5,3,3,5]
${\displaystyle {\overline {DH}}_{4}}$  [5,3,31,1]
${\displaystyle {\widehat {AF}}_{4}}$  [(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.

## SubgroupsEdit

Coxeter's notation represents rotational/translational symmetry by adding a + superscript operator outside the brackets which cuts the order of the group in half (called index 2 subgroup). This is called a direct subgroup because what remains are only direct isometries without reflective symmetry.

+ operators can also be applied inside of the brackets, and creates "semidirect" subgroups that include both reflective and nonreflective generators. Semidirect subgroups can only apply to Coxeter group subgroups that have even order branches next 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)+] (       ). The subgroup index is 2n for n + operators.

Groups without neighboring + elements can be seen in ringed nodes Coxeter-Dynkin diagram for uniform polytopes and honeycomb are related to hole nodes around the + elements, empty circles with the alternated nodes removed. So the snub cube,       has symmetry [4,3]+ (     ), and the snub tetrahedron,       has symmetry [4,3+] (     ), and a demicube, 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.

### Halving subgroups and extended groupsEdit

 [ 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. The 1 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, like tetrahedral symmetry is a half group of octahedral 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 symmetry Octahedral symmetry

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

Oh, [4,3] = [[3,3]]

(Order 48)

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

Johnson also added an asterisk 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 an extended 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], index 2 has generators {010,1,2}; while radical subgroup [4,3*], index 6, has generators {01210, 2, (012)3}; and finally [1+,4,3*], index 12 has generators {0(12)20, (012)201}.

### Trionic subgroupsEdit

[3,3] ≅ [2+,4] as one of 3 sets of 2 orthogonal mirrors in stereographic 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 specific trionic[4] subgroups 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)] (     or      ) as a reminder of its generators {02,1}. This symmetry reduction is the relationship between the regular tetrahedron and the tetragonal 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}.

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

### Central inversionEdit

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

A central inversion, order 2, is operationally differently by dimension. The group [ ]n = [2n-1] represents n orthogonal mirrors in n-dimensional space, or an n-flat subspace of a higher dimensional space. The mirrors of the group [2n-1] are numbered 0..n-1. The order of the mirrors doesn't matter in the case of an inversion.

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.

A Coxeter-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 single rotoreflection generator {012}.

Dimension Coxeter notation Order Coxeter diagram Operation Generator
2 [2]+ 2     180° rotation, C2 {01}
3 [2+,2+] 2       rotoreflection, Ci or S2 {012}
4 [2+,2+,2+] 2         double rotation {0123}
5 [2+,2+,2+,2+] 2           double rotary reflection {01234}
6 [2+,2+,2+,2+,2+] 2             triple rotation {012345}
7 [2+,2+,2+,2+,2+,2+] 2               triple rotary reflection {0123456}

### Rotations and rotary reflectionsEdit

Rotations and rotary reflections are constructed by a single single-generator product of all the reflections of a prismatic group, [2p]×[2q]×... When gcd(p,q,..)=1, they are isomorphic to the abstract cyclic group Zn, of order n=2pq.

The 4-dimensional double rotations, [2p+,2+,2q+], which include a central group, and are expressed by Conway as ±[Cp×Cq], order 2pq/gcd(p,q).[5]

Dimension Coxeter notation Order Coxeter diagram Operation Generator Direct subgroup
2 [p]+ p     Rotation {01} [p]+
3 [2p+,2+] 2p        rotary reflection {012}
4 [2p+,2+,2+]          double rotation {0123}
5 [2p+,2+,2+,2+]            double rotary reflection {01234}
6 [2p+,2+,2+,2+,2+]              triple rotation {012345}
7 [2p+,2+,2+,2+,2+,2+]                triple rotary reflection {0123456}
4 [2p+,2+,2q+] 2pq           double rotation {0123} [p+,2,q+]
5 [2p+,2+,2q+,2+]             double rotary reflection {01234}
6 [2p+,2+,2q+,2+,2+]               triple rotation {012345}
7 [2p+,2+,2q+,2+,2+,2+]                 triple rotary reflection {0123456}
6 [2p+,2+,2q+,2+,2r+] 2pqr                triple rotation {012345} [p+,2,q+,2,r+]
7 [2p+,2+,2q+,2+,2r+,2+]                  triple rotary reflection {0123456}

### Commutator subgroupsEdit

Simple groups with only odd-order branch elements have only a single rotational/translational subgroup of order 2, which is also the commutator 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 the 4s 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 subgroupsEdit

#### Rank 2 example subgroupsEdit

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.

#### Rank 3 Euclidean example subgroupsEdit

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 a transreflection, like {012} or {120}.

#### Hyperbolic example subgroupsEdit

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

## Extended symmetryEdit

Wallpaper
group
Triangle
symmetry
Extended
symmetry
Extended
diagram
Extended
group
Honeycombs
p3m1 (*333) a1   [3[3]]     ${\displaystyle {\tilde {A}}_{2}}$  (none)
p6m (*632) i2   [[3[3]]] ↔ [6,3]          ${\displaystyle {\tilde {A}}_{2}}$ ×2 ↔ ${\displaystyle {\tilde {G}}_{2}}$      1,     2
p31m (3*3) g3   [3+[3[3]]] ↔ [6,3+] ${\displaystyle {\tilde {A}}_{2}}$ ×3 ↔ ½ ${\displaystyle {\tilde {G}}_{2}}$  (none)
p6 (632) r6   [3[3[3]]]+ ↔ [6,3]+          ½${\displaystyle {\tilde {A}}_{2}}$ ×6 ↔ ½ ${\displaystyle {\tilde {G}}_{2}}$      (1)
p6m (*632) [3[3[3]]] ↔ [6,3] ${\displaystyle {\tilde {A}}_{2}}$ ×6 ↔ ${\displaystyle {\tilde {G}}_{2}}$      3
In the Euclidean plane, the ${\displaystyle {\tilde {A}}_{2}}$ , [3[3]] Coxeter group can be extended in two ways into the ${\displaystyle {\tilde {G}}_{2}}$ , [6,3] Coxeter group and relates uniform tilings as ringed diagrams.

Coxeter's notation includes double square bracket notation, [[X]] to express automorphic symmetry within a Coxeter diagram. Johnson added alternative of angled-bracket <[X]> or 〈[X]〉 option as equivalent to square brackets for doubling to distinguish diagram symmetry through the nodes versus through the branches. 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 equivalent rectangle and rhombic geometry diagrams of ${\displaystyle {\tilde {A}}_{3}}$ :      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 the square. But by considering the tetragonal 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 cyclic ${\displaystyle {\tilde {A}}_{n}}$  and branching ${\displaystyle D_{3}}$ , ${\displaystyle {\tilde {E}}_{6}}$ , and ${\displaystyle {\tilde {D}}_{4}}$  diagrams. ${\displaystyle {\tilde {A}}_{n}}$  has order 2n symmetry of a regular n-gon, {n}, and is represented by [n[3[n]]]. ${\displaystyle D_{3}}$  and ${\displaystyle {\tilde {E}}_{6}}$  are represented by [3[31,1,1]] = [3,4,3] and [3[32,2,2]] respectively while ${\displaystyle {\tilde {D}}_{4}}$  by [(3,3)[31,1,1,1]] = [3,3,4,3], with the diagram containing the order 24 symmetry of the regular tetrahedron, {3,3}. The paracompact hyperbolic group ${\displaystyle {\bar {L}}_{5}}$  = [31,1,1,1,1],       , contains the symmetry of a 5-cell, {3,3,3}, and thus is represented by [(3,3,3)[31,1,1,1,1]] = [3,4,3,3,3].

An asterisk * superscript is effectively an inverse operation, creating radical subgroups removing connected of odd-ordered mirrors.[7]

Examples:

Example Extended groups and radical subgroups
Extended groups Radical subgroups Coxeter diagrams Index
[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] [1+,4,3,3,4,3+] = [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 groups Radical subgroups Coxeter diagrams Index
[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 3D space 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).

## Computation with reflection matrices as symmetry generatorsEdit

A Coxeter group, represented by Coxeter 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). The generators 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.

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

If both p and q are even, [p+,q+],         , 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 are rotary 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. A translation generator τ0,1 (and matrix T0,1) is constructed as the product of two (or an even number of) reflections, including the affine reflection. A transreflection (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 the inversion, mapping a point to its inverse. For [4,3] and [5,3], ζ = (ρ0ρ1ρ2)h/2, where h is 6 and 10 respectively, the Coxeter 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 its Coxeter-Dynkin diagram. The structure of the groups are also given with their abstract group types: In this article, the abstract dihedral groups are represented as Dihn, and cyclic groups are represented by Zn, with Dih1=Z2.

### Rank 2Edit

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],
Reflections Rotation
Name R0

R1

S0,1=R0×R1

Order 2 2 p
Matrix

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

${\displaystyle \left[{\begin{smallmatrix}\cos 2\pi /p&\sin 2\pi /p\\\sin 2\pi /p&-\cos 2\pi /p\\\end{smallmatrix}}\right]}$

${\displaystyle \left[{\begin{smallmatrix}\cos 2\pi /p&\sin 2\pi /p\\-\sin 2\pi /p&\cos 2\pi /p\\\end{smallmatrix}}\right]}$

[2],
Reflections Rotation
Name R0

R1

S0,1=R0×R1

Order 2 2 2
Matrix

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

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

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

[3],
Reflections Rotation
Name R0

R1

S0,1=R0×R1

Order 2 2 3
Matrix

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

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

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

[4],
Reflections Rotation
Name R0

R1

S0,1=R0×R1

Order 2 2 4
Matrix

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

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

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

### Rank 3Edit

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

To reflect a point through a plane ${\displaystyle ax+by+cz=0}$  (which goes through the origin), one can use ${\displaystyle \mathbf {A} =\mathbf {I} -2\mathbf {NN} ^{T}}$ , where ${\displaystyle \mathbf {I} }$  is the 3x3 identity matrix and ${\displaystyle \mathbf {N} }$  is the three-dimensional unit vector for the vector normal of the plane. If the L2 norm of ${\displaystyle a,b,}$  and ${\displaystyle c}$  is unity, the transformation matrix can be expressed as:

${\displaystyle \mathbf {A} =\left[{\begin{smallmatrix}1-2a^{2}&-2ab&-2ac\\-2ab&1-2b^{2}&-2bc\\-2ac&-2bc&1-2c^{2}\end{smallmatrix}}\right]}$

#### Dihedral symmetryEdit

The reducible 3-dimensional finite reflective group is dihedral 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 order p rotoreflection is generated by V0,1,2, the product of all 3 reflections.

[p,2],
Reflections Rotation Rotoreflection
Name R0 R1 R2 S0,1 S1,2 S0,2 V0,1,2
Group
Order 2 2 2 p 2 2p
Matrix

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

${\displaystyle \left[{\begin{smallmatrix}\cos 2\pi /p&\sin 2\pi /p&0\\\sin 2\pi /p&-\cos 2\pi /p&0\\0&0&1\\\end{smallmatrix}}\right]}$

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

${\displaystyle \left[{\begin{smallmatrix}\cos 2\pi /p&\sin 2\pi /p&0\\-\sin 2\pi /p&\cos 2\pi /p&0\\0&0&1\\\end{smallmatrix}}\right]}$

${\displaystyle \left[{\begin{smallmatrix}\cos 2\pi /p&\sin 2\pi /p&0\\-\sin 2\pi /p&\cos 2\pi /p&0\\0&0&-1\\\end{smallmatrix}}\right]}$

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

${\displaystyle \left[{\begin{smallmatrix}\cos 2\pi /p&-\sin 2\pi /p&0\\-\sin 2\pi /p&-\cos 2\pi /p&0\\0&0&1\\\end{smallmatrix}}\right]}$

#### Tetrahedral symmetryEdit

reflection lines for [3,3] =

The simplest irreducible 3-dimensional finite reflective group is tetrahedral 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. A trionic subgroup, isomorphic to [2+,4], order 8, is generated by S0,2 and R1. An order 4 rotoreflection is generated by V0,1,2, the product of all 3 reflections.

[3,3],
Reflections Rotations Rotoreflection
Name R0 R1 R2 S0,1 S1,2 S0,2 V0,1,2
Name
Order 2 2 2 3 2 4
Matrix

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

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

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

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

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

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

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

(0,1,-1)n (1,-1,0)n (0,1,1)n (1,1,1)axis (1,1,-1)axis (1,0,0)axis

#### Octahedral symmetryEdit

Reflection lines for [4,3] =

Another irreducible 3-dimensional finite reflective group is octahedral 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-fold rotoreflection is generated by V0,1,2, the product of all 3 reflections.

[4,3],
Reflections Rotations Rotoreflection
Name R0 R1 R2 S0,1 S1,2 S0,2 V0,1,2
Group
Order 2 2 2 4 3 2 6
Matrix

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

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

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

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

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