# Coxeter notation

Last updated
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 as 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 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 groups

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] 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 the An family, like A3 = [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 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]. 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].

Finite groups
RankGroup
symbol
Bracket
notation
Coxeter
diagram
2A2[3]
2B2[4]
2H2[5]
2G2[6]
2I2(p)[p]
3 Ih, H3[5,3]
3 Td, A3[3,3]
3 Oh, B3[4,3]
4A4[3,3,3]
4B4[4,3,3]
4D4[31,1,1]
4 F4 [3,4,3]
4H4[5,3,3]
nAn[3n−1]..
nBn[4,3n−2]...
nDn[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.

## Subgroups

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 a direct 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 a pentagram, {5/2}, and [5,3+] relates to Schwarz triangle [5/2,3], density 2.

Examples on Rank 2 groups
[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 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. And chiral tetrahedral symmetry can be written as or , [1+,4,3+] = [3,3]+, with generators {12,0120}.

### Halving subgroups and extended groups

 [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)

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] ≅ [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

A trionic subgroup is an index 3 subgroup. Johnson defines a trionic 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 the regular tetrahedron and tetragonal 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 the density increases. For example, the icosahedral symmetry, [5,3], of the regular polyhedra icosahedron becomes [5/2,5], the symmetry of 2 regular star polyhedra. It also relates the hyperbolic tilings {p,3}, and star 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

Johnson identified two specific trionic 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 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.

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

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 ${\displaystyle 0\dots n-1}$. The order of the mirrors doesn't matter in the case of an inversion. The matrix of a central inversion is ${\displaystyle -I}$, 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.

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}.

DimensionCoxeter notationOrderCoxeter diagramOperationGenerator
2[2]+2180° 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+]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

Rotations and rotary reflections are constructed by a single single-generator product of all the reflections of a prismatic group, [2p]×[2q]×... where 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+] (with gcd(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], order pq, with one generator, like {0123}.

If there is a common factor f, the double rotation can be written as 1f[2pf+,2+,2qf+] (with gcd(p,q)=1), generators {0123,0132}, order 2pqf. For example, p=q=1, f=2, 12[4+,2+,4+] is order 4. And 1f[2pf+,2+,2qf+]+, generator {0123}, is order pqf. For example, 12[4+,2+,4+]+ is order 2, a central inversion.

In general a n-rotation group, [2p1+,2,2p2+,2,...,pn+] may require up to n 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]+2p Rotation {01}[2p]+2 = [p]+Simple rotation:
[2p]+2 = [p]+
order p
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+]+
order pq
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+]+
order pqr
7[2p+,2+,2q+,2+,2r+,2+]triple rotary reflection{0123456,
0123465,
0124356,
0213456}
[2p+,2+,2q+,2+,2r+,2+]+

### Commutator subgroups

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 subgroups

#### Rank 2 example subgroups

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 subgroups

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 subgroups

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

## Extended symmetry

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]${\displaystyle {\tilde {A}}_{2}\times 2\leftrightarrow {\tilde {G}}_{2}}$ 1 , 2
p31m (3*3)g3 [3+[3[3]]] ↔ [6,3+]${\displaystyle {\tilde {A}}_{2}\times 3\leftrightarrow {\tfrac {1}{2}}{\tilde {G}}_{2}}$(none)
p6 (632)r6 [3[3[3]]]+ ↔ [6,3]+${\displaystyle {\tfrac {1}{2}}{\tilde {A}}_{2}\times 6\leftrightarrow {\tfrac {1}{2}}{\tilde {G}}_{2}}$ (1)
p6m (*632)[3[3[3]]] ↔ [6,3]${\displaystyle {\tilde {A}}_{2}\times 6\leftrightarrow {\tilde {G}}_{2}}$ 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 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 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 groupsRadical 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][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
[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).

## Rank one groups

In one dimension, the bilateral group [ ] represents a single mirror symmetry, abstract Dih1 or Z2, symmetry order 2. It is represented as a Coxeter–Dynkin diagram with a single node, . The identity 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 notation Coxeter diagram OrderDescription
C1[ ]+1Identity
D2[ ]2Reflection group

## Rank two groups

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

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

The full p-gonal group [p], abstract dihedral group D2p, (nonabelian for p>2), of order 2p, is generated by two mirrors at angle π/p, represented by Coxeter diagram . The p-gonal subgroup [p]+, cyclic group Zp, of order p, generated by a rotation angle of π/p.

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

In the limit, going down to one dimensions, the full apeirogonal group is obtained when the angle goes to zero, so [∞], abstractly the infinite dihedral group D, represents two parallel mirrors and has a Coxeter diagram . The apeirogonal group [∞]+, , abstractly the infinite cyclic group Z, isomorphic to the additive group of the integers, is generated by a single nonzero translation.

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