In geometry, Coxeter notation is a system of classifying symmetry groups, describing the angles between with fundamental reflections of a Coxeter group. It uses a bracketed notation, with modifiers to indicate certain subgroups. The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson.
Further information: Point group 
For Coxeter groups defined by pure reflections, there is a direct correspondence between the bracket notation and CoxeterDynkin diagram graphs. The numbers in the bracket notation represent the mirror reflection orders in the branches of the Coxeter graph.
The Coxeter notation is simplified with exponents to represent the number of branches in a row for linear graphs. So the A_{n} group is represented by [3^{n1}], to imply n nodes connected by n1 order3 branches.
Coxeter initially represented bifurcating diagrams with vertical positioning of numbers, but later abbreviated with an exponent notation, like [3^{p,q,r}], starting with [3^{1,1,1}] as D_{4}. Coxeter allowed for zeros as special cases to fit the rectified nsimplex polytopes into the same notation, and also allowed one 1 index for sequences that remove the common node to all the branches.
Coxeter groups formed by cyclic graphs are represented by parenthesese inside of brackets, like [(a,b,c)] for the triangle group (a b c). If they are equal, they can be grouped as an exponent as the length the cycle in brackets, like [(3,3,3,3)] = [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 graph.
The Coxeter graph usually leaves order2 branches undrawn, but the bracket notation includes an explicit 2 to connect the subgraphs. So the Coxeter graph, , H_{3}×A_{2} can be represented by [5,3]×[3] and [5,3,2,3].
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). Multiple + operators may exist if neighboring elements are all even order, and the subgroup index is 2^{n} for n operators.
For Coxeter groups with even order branches, 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. Example [4,3^{+}].
Groups without neighboring + elements can be seen in ringed CoxeterDynkin diagram for uniform polytopes and honeycomb are related to holes on the nodes around the + elements, empty circles with the alternated nodes removed. So , has symmetry in Coxeter notation of [4,3]^{+}, and has group notation [4,3^{+}].
Johnson extends the + operator to work with a placeholder 1 nodes, and a mirror connected to only evenorder branches can be removed as 1^{+}. For example [1^{+},4,3], removes a mirror between two order4 rotation points, reducing its reflection in half to 2, and dividing the 3 order mirrors into 2 copies as the [3,3] group. If all mirrors are removed, the identity symmetry is restored: [1^{+},2n,1^{+}] = [ ]^{+}, or [(1^{+},2,1^{+},2,1^{+})] = [ ]^{+}.
Simple groups with only oddorder 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 evenorder branches, the commutator subgroup has index 2^{c}, where c is the number of disconnected subgraphs when all the evenorder branches are removed.^{[1]} For example [4,4] has 3 independent nodes in the Coxeter graph when the 4s are removed, so its commutator subgroup is index 2^{3}, and can have different representions, all with three + operators: [4^{+},4^{+}]^{+}, or [1^{+},4,4,1^{+}]^{+}, or [(4^{+},4^{+},2^{+})]. A general notation can be used with +c as a group exponent, like [4,4]^{+3}.
Coxeter's notation includes double square bracket notation, [[X]] to express isomorphic symmetry within a Coxeter diagram. Johnson added alternative of angledbracket <[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 is the bracket notation symmetry of the diagram for [X].
For example in these equivalent rectangle and rhombic geometry diagrams of : 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.
Further symmetry exists in the cyclic and branching , , and diagrams. graph has order 2n symmetry of a regular ngon, {n}, and is represented by [n[3^{[n]}]]. and are represented by [3[3^{1,1,1}]] and [3[3^{2,2,2}]] respectively while by [3,3[3^{1,1,1,1}]], with the diagram containing the order 24 symmetry of the regular tetrahedron, {3,3}. The noncompact hyperbolic group graph = [3^{1,1,1,1,1}], , contains the symmetry of a 5cell, {3,3,3}, and thus is represented by [3,3,3[3^{1,1,1,1,1}]], but as suggested by the grouping 3 and 2 branches, it can also be given lower symmetry like [3[3^{1,1,1}],[[3^{1,1}]].
A Coxeter group, represented by Coxeter diagram , is given Coxeter notation [p,q] for the branch orders. Each node represents a mirror, which represented by a reflection matrix R_{i} or ρ_{i}. So the generators of this group are ρ_{1}, ρ_{2}, and ρ_{3}.
[p,q]^{+} is an index 2 subgroup represented by rotations, and generators as products of reflections: ρ_{1}.ρ_{2}, ρ_{2}.ρ_{3}. These products define rotation matrices and are sometimes labeled σ_{1,2}, and σ_{2,3} and represent rotations of π/p, and π/q angles respectively.
[2p^{+},2q^{+}] is a subgroup of index 4 has two +s with a single generator type as a product of all three reflection matrices: ψ_{1,2,3} = σ_{1,2}ρ_{3} = = ρ_{1}&sigma_{2,3} = ρ_{1}.ρ_{2}.ρ_{3}, which is an improper rotation or rotoreflection, representing a reflection and rotation. Or in the case of affine Coxeter groups with parallel mirrors , represents a translation τ_{1,2}, or transreflection for 3 reflections φ_{1,2,3}, like for the hyperbolic coxeter group , [&infin^{+},4^{+}].
Coxeter groups are categorized by their rank, being the number of nodes in its Coxeter graph. The structure of the groups are also given with their abstract group types: In this article, the abstract dihedral groups are represented as Dih_{n}, and cyclic groups are represented by Z_{n}, with Dih_{1}=Z_{2}.
In one dimension, the bilateral group [ ] represents a single mirror symmetry, abstract Dih_{1} or Z_{2}, symmetry order 2. It is represented as a Coxeter graph with a single node, . The identity group is the direct subgroup [ ]^{+}, Z_{1}, symmetry order 1. The + superscript simply implies that alternate mirror reflections are ignored, leaving the identity group in this simplest case.
Group  Coxeter  Coxeter diagram  Order  Description 

C_{1}  [ ]^{+}  1  Identity  
D_{1}  [ ]  2  Reflection group 
In two dimensions, the rectangular group [2], abstract Dih_{2}, also can be represented as a direct product [ ]×[ ] or Z_{2}×Z_{2}, being the product of two bilateral groups, represents two orthogonal mirrors, with Coxeter graph, , with order 4. The 2 in [2] comes from linearization of the orthogonal subgraphs in the Coxeter graph, as , with explicit branch order 2. The rhombic group, [2]^{+}, half of the rectangular group, the point reflection symmetry, Z_{2}, order 2.
Coxeter notation allows a 1 placeholder for lower rank groups, so [1] is the same as [ ], and [1]^{+} is the same as [ ]^{+}. This may be done to imply the group exists in 2dimensions rather than 1 dimension.
The full pgonal group [p], abstract dihedral group Dih_{p}, (nonabelian for p>2), of order 2p, is generated by two mirrors at angle π/p, represented by Coxeter graph . The pgonal subgroup [p]^{+}, cyclic group Z_{}p, of order p, generated by a rotation angle of π/p.
Coxeter notation uses doublebracking 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 Dih_{∞}, represents two parallel mirrors and has a Coxeter graph . 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's a full pseudogonal group [πi/λ], and pseudogonal subgroup [πi/λ]^{+}. These groups exist in regular infinite sided polygons, with edge length λ. The mirrors are all orthogonal to a single line.
Group  Intl  Orbifold  Coxeter  Order  Description 

Finite  
Z_{n}  n  nn  [n]^{+}  n  Cyclic: nfold rotations. Abstract group Z_{n}, the group of integers under addition modulo n. 
D_{n}  nm  *nn  [n]  2n  Dihedral: cyclic with reflections. Abstract group Dih_{n}, the dihedral group. 
Affine  
Z_{∞}  ∞  ∞∞  [∞]^{+}  ∞  Cyclic: apeirogonal group. Abstract group Z_{∞}, the group of integers under addition. 
Dih_{∞}  ∞m  *∞∞  [∞]  ∞  Dihedral: parallel reflections. Abstract infinite dihedral group Dih_{∞}. 
Hyperbolic  
Z_{∞}  [πi/λ]^{+}  ∞  pseudogonal group  
Dih_{∞}  [πi/λ]  ∞  full pseudogonal group 
Further information: List of spherical symmetry groups and List of planar symmetry groups 
In three dimensions, the full orthorhombic group [2,2], astracttly Z_{2}×Dih_{2}, order 8, represents three orthogonal mirrors, and also can be represented by Coxeter graph as three separate dots . It can also can be represented as a direct product [ ]×[ ]×[ ], but the [2,2] expression allows subgroups to be defined:
First there is a semidirect subgroup, the orthorhombic group, [2,2^{+}], abstractly Dih_{1}×Z_{2}=Z_{2}×Z_{2}, 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 graph, ) 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 the pararhombic group [2,2]^{+}, also order 4, and finally the central group [2^{+},2^{+}] of order 2.
Next there is the full orthopgonal group, [2,p], abstractly Dih_{1}×Dih_{p}=Z_{2}×Dih_{p}, of order 4p, representing two mirrors at a dihedral angle π/p, and both are orthogonal to a third mirror. It is also represented by Coxeter graph as .
The direct subgroup is called the parapgonal group, [2,p]^{+}, abstractly Dih_{p}, of order 2p, and another subgroup is [2,p^{+}] abstractly Z_{p}×Z_{2}, also of order 2p.
The full gyropgonal group, [2^{+},2p], abstractly Dih_{2p}, of order 4p. The gyropgonal group, [2^{+},2p^{+}], abstractly Z_{2p}, of order 2p is a subgroup of both [2^{+},2p] and [2,2p^{+}].
The polyhedral groups are based on the symmetry of platonic solids, the tetrahedron, octahedron, cube, icosahedron, and dodecahedron, with Schläfli symbols {3,3}, {3,4}, {4,3}, {3,5}, and {5,3} respectively. The Coxeter groups for these are called in Coxeter's bracket notation [3,3], [3,4], [3,5] called full tetrahedral symmetry, octahedral symmetry, and icosahedral symmetry, with orders of 24, 48, and 120. The order of the number in the Coxeter notation don't make a difference, unlike the Schläfli symbol.
The tetrahedral group, [3,3], has a doubling [[3,3]] which maps the first and last mirrors onto each other, and this produces the [3,4] group.
In all these symmetries, alternate reflections can be removed producing the rotational tetrahedral, octahedral, and icosahedral groups of order 12, 24, and 60. The octahedral group also has a unique subgroup called the pyritohedral symmetry group, [3^{+},4], of order 12, with a mixture of rotational and reflectional symmetry.
In the Euclidean plane there's 3 fundamental reflective groups generated by 3 mirrors, represented by Coeter graphs , , and , and are given Coxeter notation as [4,4], [6,3], and [(3,3,3)]. The parentheses of the last group imply the graph ic cyclic, 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^{+}] represent mixed reflectional and rotational symmetry.
Finite  


 
Semiaffine  

 
Affine  


Rank four groups defined the 4dimensional point groups: Template:3sphere symmetry groups
Rank four groups also defined the 3dimensional space groups:




Rank four groups also defined the 3dimensional line groups:
Point group  Line group  

HermannMauguin  Schönflies  HermannMauguin  Offset type  Wallpaper  Coxeter [∞_{h},2,p_{v}]  
Even n  Odd n  Even n  Odd n  IUC  Orbifold  Diagram  
n  C_{n}  Pn_{q}  Helical: q  p1  o  [∞^{+},2,n^{+}]  
2n  n  S_{2n}  P2n  Pn  None  p11g, pg(h)  xx  [(∞,2)^{+},2n^{+}]  
n/m  2n  C_{nh}  Pn/m  P2n  None  p11m, pm(h)  **  [∞^{+},2,n]  
2n/m  C_{2nh}  P2n_{n}/m  Zigzag  c11m, cm(h)  *x  [∞^{+},2^{+},2n]  
nmm  nm  C_{nv}  Pnmm  Pnm  None  p1m1, pm(v)  **  [∞,2,n^{+}]  
Pncc  Pnc  Planar reflection  p1g1, pg(v)  xx  [∞^{+},(2,n)^{+}]  
2nmm  C_{2nv}  P2n_{n}mc  Zigzag  c1m1, cm(v)  *x  [∞,2^{+},2n^{+}]  
n22  n2  D_{n}  Pn_{q}22  Pn_{q}2  Helical: q  p2  2222  [∞,2,n]^{+}  
2n2m  nm  D_{nd}  P2n2m  Pnm  None  p2mg, pmg(h)  22*  [(∞,2)^{+},2n]  
P2n2c  Pnc  Planar reflection  p2gg, pgg  22x  [∞^{+},2^{+},2n^{+}]  
n/mmm  2n2m  D_{nh}  Pn/mmm  P2n2m  None  p2mm, pmm  *2222  [∞,2,n]  
Pn/mcc  P2n2c  Planar reflection  p2mg, pmg(v)  22*  [∞,(2,n)^{+}]  
2n/mmm  D_{2nh}  P2n_{n}/mcm  Zigzag  c2mm, cmm  2*22  [∞,2^{+},2n] 
Rank four groups also defined some of the 2dimensional wallpaper groups:

