, [ ]=[1] C_{1v} 
, [2] C_{2v} 
, [3] C_{3v} 
, [4] C_{4v} 
, [5] C_{5v} 
, [6] C_{6v} 

Order 2 
Order 4 
Order 6 
Order 8 
Order 10 
Order 12 
[2]=[2,1] D_{1h} 
[2,2] D_{2h} 
[2,3] D_{3h} 
[2,4] D_{4h} 
[2,5] D_{5h} 
[2,6] D_{6h} 
Order 4 
Order 8 
Order 12 
Order 16 
Order 20 
Order 24 
, [3,3], T_{d}  , [4,3], O_{h}  , [5,3], I_{h}  
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 fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a CoxeterDynkin 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.
For Coxeter groups, defined by pure reflections, there is a direct correspondence between the bracket notation and CoxeterDynkin 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 A_{n} group is represented by [3^{n1}], to imply n nodes connected by n1 order3 branches. Example A_{2} = [3,3] = [3^{2}] or [3^{1,1}] represents diagrams or .
Coxeter initially represented bifurcating diagrams with vertical positioning of numbers, but later abbreviated with an exponent notation, like [...,3^{p,q}] or [3^{p,q,r}], starting with [3^{1,1,1}] or [3,3^{1,1}] = or as D_{4}. Coxeter allowed for zeros as special cases to fit the A_{n} family, like A_{3} = [3,3,3,3] = [3^{4,0,0}] = [3^{4,0}] = [3^{3,1}] = [3^{2,2}], 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 order2 branches undrawn, but the bracket notation includes an explicit 2 to connect the subgraphs. So the Coxeter diagram = A_{2}×A_{2} = 2A_{2} can be represented by [3]×[3] = [3]^{2} = [3,2,3]. Sometimes explicit 2branches may be included either with a 2 label, or with a line with a gap: or , as an identical presentation as [3,2,3].



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.
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.
Group  Order  Generators  Subgroup  Order  Generators  Notes  

[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 CoxeterDynkin 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 3fold rotation. And chiral tetrahedral symmetry can be written as or , [1^{+},4,3^{+}] = [3,3]^{+}, with generators {12,0120}.
[ 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 evenorder 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 order2p 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:
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.
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 4branch. 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 

T_{d}, [3,3] = [1^{+},4,3] = = (Order 24) 
O_{h}, [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)^{2}0, (012)^{2}01}.
A trionic subgroup is an index 3 subgroups. There are many 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 oddorder 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].
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 [3^{1,1,1}] = [3,3,4,1^{+}] has trionic subgroups: [3^{1,1,1}]^{⅄} = [(3,3)^{⅄},4,1^{+}], order 64, and 1=[3^{1,1,1}]^{Δ} = [(3,3)^{Δ},4,1^{+}] ≅ [[4,2^{+},4]]^{+}, order 32.
A central inversion, order 2, is operationally differently by dimension. The group [ ]^{n} = [2^{n1}] represents n orthogonal mirrors in ndimensional space, or an nflat subspace of a higher dimensional space. The mirrors of the group [2^{n1}] are numbered . The order of the mirrors doesn't matter in the case of an inversion. The matrix of a central inversion is , 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 CoxeterDynkin diagram can be marked up with explicit 2 branches defining a linear sequence of mirrors, opennodes, and shared doubleopen 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 doubleopen 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, C_{2}  {01}  
3  [2^{+},2^{+}]  2  rotoreflection, C_{i} or S_{2}  {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 reflections are constructed by a single singlegenerator product of all the reflections of a prismatic group, [2p]×[2q]×... where gcd(p,q,...)=1, they are isomorphic to the abstract cyclic group Z_{n}, of order n=2pq.
The 4dimensional double rotations, [2p^{+},2^{+},2q^{+}] (with gcd(p,q)=1), which include a central group, and are expressed by Conway as ±[C_{p}×C_{q}],^{[5]} order 2pq. From Coxeter diagram , generators {0,1,2,3}, the single generator of [2p^{+},2^{+},2q^{+}], is {0123}. The half group, [2p^{+},2^{+},2q^{+}]^{+}, or cyclc graph, [(2p^{+},2^{+},2q^{+},2^{+})], expressed by Conway is [C_{p}×C_{q}], order pq, with generator {01230123}.
If there is a common factor f, the double rotation can be written as ^{1}⁄_{f}[2pf^{+},2^{+},2qf^{+}] (with gcd(p,q)=1), generator {0123}, order 2pqf. For example, p=q=1, f=2, ^{1}⁄_{2}[4^{+},2^{+},4^{+}] is order 4. And ^{1}⁄_{f}[2pf^{+},2^{+},2qf^{+}]^{+}, generator {01230123}, is order pqf. For example, ^{1}⁄_{2}[4^{+},2^{+},4^{+}]^{+} is order 2, a central inversion.
Dimension  Coxeter notation  Order  Coxeter diagram  Operation  Generator  Direct subgroup  

2  [2p]^{+}  2p  Rotation  {01}  [2p]^{+2}  Simple rotation: [2p]^{+2} = [p]^{+} order p  
3  [2p^{+},2^{+}]  rotary reflection  {012}  [2p^{+},2^{+}]^{+}  
4  [2p^{+},2^{+},2^{+}]  double rotation  {0123}  [2p^{+},2^{+},2^{+}]^{+}  
5  [2p^{+},2^{+},2^{+},2^{+}]  double rotary reflection  {01234}  [2p^{+},2^{+},2^{+},2^{+}]^{+}  
6  [2p^{+},2^{+},2^{+},2^{+},2^{+}]  triple rotation  {012345}  [2p^{+},2^{+},2^{+},2^{+},2^{+}]^{+}  
7  [2p^{+},2^{+},2^{+},2^{+},2^{+},2^{+}]  triple rotary reflection  {0123456}  [2p^{+},2^{+},2^{+},2^{+},2^{+},2^{+}]^{+}  
4  [2p^{+},2^{+},2q^{+}]  2pq  double rotation  {0123}  [2p^{+},2^{+},2q^{+}]^{+}  Double rotation: [2p^{+},2^{+},2q^{+}]^{+} order pq gcd(p,q)=1  
5  [2p^{+},2^{+},2q^{+},2^{+}]  double rotary reflection  {01234}  [2p^{+},2^{+},2q^{+},2^{+}]^{+}  
6  [2p^{+},2^{+},2q^{+},2^{+},2^{+}]  triple rotation  {012345}  [2p^{+},2^{+},2q^{+},2^{+},2^{+}]  
7  [2p^{+},2^{+},2q^{+},2^{+},2^{+},2^{+}]  triple rotary reflection  {0123456}  [2p^{+},2^{+},2q^{+},2^{+},2^{+},2^{+}]^{+}  
6  [2p^{+},2^{+},2q^{+},2^{+},2r^{+}]  2pqr  triple rotation  {012345}  [2p^{+},2^{+},2q^{+},2^{+},2r^{+}]^{+}  Triple rotation: [2p^{+},2^{+},2q^{+},2^{+},2r^{+}]^{+} order pqr gcd(p,q,r)=1  
7  [2p^{+},2^{+},2q^{+},2^{+},2r^{+},2^{+}]  triple rotary reflection  {0123456}  [2p^{+},2^{+},2q^{+},2^{+},2r^{+},2^{+}]^{+} 
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.^{[6]} For example, [4,4] has three independent nodes in the Coxeter diagram when the 4s are removed, so its commutator subgroup is index 2^{3}, 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}.
Dihedral symmetry groups with evenorders have a number of subgroups. This example shows two generator mirrors of [4] in red and green, and looks at all subgroups by halfing, rankreduction, 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]  

Index  1  2 (half)  4 (Rankreduction)  
Diagram  
Coxeter 
[1,4,1] = [4] 
= = [1^{+},4,1] = [1^{+},4] = [2] 
= = [1,4,1^{+}] = [4,1^{+}] = [2] 
[1] = [ ] 
[1] = [ ]  
Generators  {0,1}  {101,1}  {0,010}  {0}  {1}  
Direct subgroups  
Index  2  4  8  
Diagram  
Coxeter  [4]^{+} 
= = = [4]^{+2} = [1^{+},4,1^{+}] = [2]^{+} 
[ ]^{+}  
Generators  {01}  {(01)^{2}}  {0^{2}} = {1^{2}} = {(01)^{4}} = { } 
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}.
Small index subgroups of [4,4]  

Index  1  2  4  
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  *2222  22×  
Semidirect subgroups  
Index  2  4  
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}  
Orbifold  4*2  2*22  
Direct subgroups  
Index  2  4  8  
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)^{2}2}  
Orbifold  442  2222  
Radical subgroups  
Index  8  16  
Diagram  
Coxeter  [4,4*] = 
[4*,4] = 
[4,4*]^{+} = 
[4*,4]^{+} =  
Orbifold  *2222  2222 
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]  

Index  1  2  4  
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  *3232  32×  
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}  
Orbifold  4*3  6*2  2*32  2*33  3*22  
Direct subgroups  
Index  2  4  8  
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}  
Orbifold  642  443  662  3222  3232  
Radical subgroups  
Index  8  12  16  24  
Diagram  
Coxeter (orbifold) 
[6,4*] = (*3333) 
[6*,4] (*222222) 
[6,4*]^{+} = (3333) 
[6*,4]^{+} (222222) 
 
In the Euclidean plane, the , [3^{[3]}] Coxeter group can be extended in two ways into the , [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 angledbracket <[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 : 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 and branching , , and diagrams. 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}]] = [3,4,3] and [3[3^{2,2,2}]] respectively while by [(3,3)[3^{1,1,1,1}]] = [3,3,4,3], with the diagram containing the order 24 symmetry of the regular tetrahedron, {3,3}. The paracompact hyperbolic group = [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}]] = [3,4,3,3,3].
An asterisk * superscript is effectively an inverse operation, creating radical subgroups removing connected of oddordered mirrors.^{[7]}
Examples:
Example Extended groups and radical subgroups  



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).
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 ρ_{i} (and matrix R_{i}). The generators of this group [p,q] are reflections: ρ_{0}, ρ_{1}, and ρ_{2}. Rotational subsymmetry is given as products of reflections: By convention, σ_{0,1} (and matrix S_{0,1}) = ρ_{0}ρ_{1} represents a rotation of angle π/p, and σ_{1,2} = ρ_{1}ρ_{2} is a rotation of angle π/q, and σ_{0,2} = ρ_{0}ρ_{2} 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: σ_{0,1}, σ_{1,2}, 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 σ_{0,1}, and reflectional ρ_{2}.
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: ψ_{0,1,2} and ψ_{1,2,0}, 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 τ_{0,1} (and matrix T_{0,1}) 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 φ_{0,1,2} (and matrix V_{0,1,2}), 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 = (ρ_{0}ρ_{1}ρ_{2})^{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 CoxeterDynkin diagram. 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}.
Example, in 2D, the Coxeter group [p] () is represented by two reflection matrices R_{0} and R_{1}, The cyclic symmetry [p]^{+} () is represented by rotation generator of matrix S_{0,1}.

 


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 (which goes through the origin), one can use , where is the 3x3 identity matrix and is the threedimensional unit vector for the vector normal of the plane. If the L2 norm of and is unity, the transformation matrix can be expressed as:
The reducible 3dimensional finite reflective group is dihedral symmetry, [p,2], order 4p, . The reflection generators are matrices R_{0}, R_{1}, R_{2}. R_{0}^{2}=R_{1}^{2}=R_{2}^{2}=(R_{0}×R_{1})^{3}=(R_{1}×R_{2})^{3}=(R_{0}×R_{2})^{2}=Identity. [p,2]^{+} () is generated by 2 of 3 rotations: S_{0,1}, S_{1,2}, and S_{0,2}. An order p rotoreflection is generated by V_{0,1,2}, the product of all 3 reflections.
Reflections  Rotation  Rotoreflection  

Name  R_{0}  R_{1}  R_{2}  S_{0,1}  S_{1,2}  S_{0,2}  V_{0,1,2} 
Group  
Order  2  2  2  p  2  2p  
Matrix 







The simplest irreducible 3dimensional finite reflective group is tetrahedral symmetry, [3,3], order 24, . The reflection generators, from a D_{3}=A_{3} construction, are matrices R_{0}, R_{1}, R_{2}. R_{0}^{2}=R_{1}^{2}=R_{2}^{2}=(R_{0}×R_{1})^{3}=(R_{1}×R_{2})^{3}=(R_{0}×R_{2})^{2}=Identity. [3,3]^{+} () is generated by 2 of 3 rotations: S_{0,1}, S_{1,2}, and S_{0,2}. A trionic subgroup, isomorphic to [2^{+},4], order 8, is generated by S_{0,2} and R_{1}. An order 4 rotoreflection is generated by V_{0,1,2}, the product of all 3 reflections.
Reflections  Rotations  Rotoreflection  

Name  R_{0}  R_{1}  R_{2}  S_{0,1}  S_{1,2}  S_{0,2}  V_{0,1,2} 
Name  
Order  2  2  2  3  2  4  
Matrix 







(0,1,1)_{n}  (1,1,0)_{n}  (0,1,1)_{n}  (1,1,1)_{axis}  (1,1,1)_{axis}  (1,0,0)_{axis} 
Another irreducible 3dimensional finite reflective group is octahedral symmetry, [4,3], order 48, . The reflection generators matrices are R_{0}, R_{1}, R_{2}. R_{0}^{2}=R_{1}^{2}=R_{2}^{2}=(R_{0}×R_{1})^{4}=(R_{1}×R_{2})^{3}=(R_{0}×R_{2})^{2}=Identity. Chiral octahedral symmetry, [4,3]^{+}, () is generated by 2 of 3 rotations: S_{0,1}, S_{1,2}, and S_{0,2}. Pyritohedral symmetry [4,3^{+}], () is generated by reflection R_{0} and rotation S_{1,2}. A 6fold rotoreflection is generated by V_{0,1,2}, the product of all 3 reflections.
Reflections  Rotations  Rotoreflection  

Name  R_{0}  R_{1}  R_{2}  S_{0,1}  S_{1,2}  S_{0,2}  V_{0,1,2} 
Group  
Order  2  2  2  4  3  2  6 
Matrix 







(0,0,1)_{n}  (0,1,1)_{n}  (1,1,0)_{n}  (1,0,0)_{axis}  (1,1,1)_{axis}  (1,1,0)_{axis} 
A final irreducible 3dimensional finite reflective group is icosahedral symmetry, [5,3], order 120, . The reflection generators matrices are R_{0}, R_{1}, R_{2}. R_{0}^{2}=R_{1}^{2}=R_{2}^{2}=(R_{0}×R_{1})^{5}=(R_{1}×R_{2})^{3}=(R_{0}×R_{2})^{2}=Identity. [5,3]^{+} () is generated by 2 of 3 rotations: S_{0,1}, S_{1,2}, and S_{0,2}. A 10fold rotoreflection is generated by V_{0,1,2}, the product of all 3 reflections.
Reflections  Rotations  Rotoreflection  

Name  R_{0}  R_{1}  R_{2}  S_{0,1}  S_{1,2}  S_{0,2}  V_{0,1,2} 
Group  
Order  2  2  2  5  3  2  10 
Matrix  
(1,0,0)_{n}  _{n}  (0,1,0)_{n}  (φ,1,0)_{axis}  (1,1,1)_{axis}  (1,0,0)_{axis} 
A simple example affine group is [4,4] () (p4m), can be given by three reflection matrices, constructed as a reflection 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 S_{0,1} S_{1,2}, and S_{0,2}. [4^{+},4^{+}] () (pgg) is generated by 2fold rotation S_{0,2} and transreflection V_{0,1,2}. [4^{+},4] () (p4g) is generated by S_{0,1} and R_{3}. The group [(4,4,2^{+})] () (cmm), is generated by 2fold rotation S_{1,3} and reflection R_{2}.
Reflections  Rotations  Rotoreflection  

Name  R_{0}  R_{1}  R_{2}  S_{0,1}  S_{1,2}  S_{0,2}  V_{0,1,2} 
Group  
Order  2  2  2  4  2  ∞  
Matrix 







A irreducible 4dimensional finite reflective group is hyperoctahedral group (or hexadecachoric group (for 16cell), B_{4}=[4,3,3], order 384, . The reflection generators matrices are R_{0}, R_{1}, R_{2}, R_{3}. R_{0}^{2}=R_{1}^{2}=R_{2}^{2}=R_{3}^{2}=(R_{0}×R_{1})^{4}=(R_{1}×R_{2})^{3}=(R_{2}×R_{3})^{3}=(R_{0}×R_{2})^{2}=(R_{1}×R_{3})^{2}=(R_{0}×R_{3})^{2}=Identity.
Chiral hyperoctahedral symmetry, [4,3,3]^{+}, () is generated by 3 of 6 rotations: S_{0,1}, S_{1,2}, S_{2,3}, S_{0,2}, S_{1,3}, and S_{0,3}. Hyperpyritohedral symmetry [4,(3,3)^{+}], () is generated by reflection R_{0} and rotations S_{1,2} and S_{2,3}. An 8fold double rotation is generated by W_{0,1,2,3}, the product of all 4 reflections.
Reflections  Rotations  Rotoreflection  Double rotation  

Name  R_{0}  R_{1}  R_{2}  R_{3}  S_{0,1}  S_{1,2}  S_{2,3}  S_{0,2}  S_{1,3}  S_{0,3}  V_{1,2,3}  V_{0,1,3}  V_{0,1,2}  V_{0,2,3}  W_{0,1,2,3} 
Group  
Order  2  2  2  2  4  3  2  4  6  8  
Matrix 















(0,0,0,1)_{n}  (0,0,1,1)_{n}  (0,1,1,0)_{n}  (1,1,0,0)_{n} 
A irreducible 4dimensional finite reflective group is Icositetrachoric group (for 24cell), F_{4}=[3,4,3], order 1152, . The reflection generators matrices are R_{0}, R_{1}, R_{2}, R_{3}. R_{0}^{2}=R_{1}^{2}=R_{2}^{2}=R_{3}^{2}=(R_{0}×R_{1})^{3}=(R_{1}×R_{2})^{4}=(R_{2}×R_{3})^{3}=(R_{0}×R_{2})^{2}=(R_{1}×R_{3})^{2}=(R_{0}×R_{3})^{2}=Identity.
Chiral icositetrachoric symmetry, [3,4,3]^{+}, () is generated by 3 of 6 rotations: S_{0,1}, S_{1,2}, S_{2,3}, S_{0,2}, S_{1,3}, and S_{0,3}. Ionic diminished [3,4,3^{+}] group, () is generated by reflection R_{0} and rotations S_{1,2} and S_{2,3}. A 12fold double rotation is generated by W_{0,1,2,3}, the product of all 4 reflections.
Reflections  Rotations  Rotoreflection  Double rotation  

Name  R_{0}  R_{1}  R_{2}  R_{3}  S_{0,1}  S_{1,2}  S_{2,3}  S_{0,2}  S_{1,3}  S_{0,3}  V_{1,2,3}  V_{0,1,3}  V_{0,1,2}  V_{0,2,3}  W_{0,1,2,3} 
Group  
Order  2  2  2  2  3  4  3  2  6  12  
Matrix 















(1,1,1,1)_{n}  (0,0,1,0)_{n}  (0,1,1,0)_{n}  (1,1,0,0)_{n} 
The hypericosahedral symmetry, [5,3,3], order 14400, . The reflection generators matrices are R_{0}, R_{1}, R_{2}, R_{3}. R_{0}^{2}=R_{1}^{2}=R_{2}^{2}=R_{3}^{2}=(R_{0}×R_{1})^{5}=(R_{1}×R_{2})^{3}=(R_{2}×R_{3})^{3}=(R_{0}×R_{2})^{2}=(R_{0}×R_{3})^{2}=(R_{1}×R_{3})^{2}=Identity. [5,3,3]^{+} () is generated by 3 rotations: S_{0,1} = R_{0}×R_{1}, S_{1,2} = R_{1}×R_{2}, S_{2,3} = R_{2}×R_{3}, etc.
Reflections  

Name  R_{0}  R_{1}  R_{2}  R_{3} 
Group  
Order  2  2  2  2 
Matrix  
(1,0,0,0)_{n}  _{n}  (0,1,0,0)_{n}  _{n} 
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–Dynkin diagram 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. Coxeter used a single open node to represent an alternation, .
Group  Coxeter notation  Coxeter diagram  Order  Description 

C_{1}  [ ]^{+}  1  Identity  
D_{1}  [ ]  2  Reflection group 
In two dimensions, the rectangular group [2], abstract D_{1}^{2} or D_{2}, 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, Z_{2}, order 2.
Coxeter notation to allow a 1 placeholder for lower rank groups, so [1] is the same as [ ], and [1^{+}] or [1]^{+} is the same as [ ]^{+} and Coxeter diagram .
The full pgonal group [p], abstract dihedral group D_{p}, (nonabelian for p>2), of order 2p, is generated by two mirrors at angle π/p, represented by Coxeter diagram . 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 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 infinitesided polygons, with edge length λ. The mirrors are all orthogonal to a single line.
Example rank 2 finite and hyperbolic symmetries  

Type  Finite  Affine  Hyperbolic  
Geometry  ...  
Coxeter  [ ] 
= [2]=[ ]×[ ] 
[3] 
[4] 
[p] 
[∞] 
[∞] 
[iπ/λ]  
Order  2  4  6  8  2p  ∞  
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]^{+}  
Order  1  2  3  4  p  ∞  
Cyclic subgroups represent alternate reflections, all even (direct) images. 
Group  Intl  Orbifold  Coxeter  Coxeter diagram  Order  Description 

Finite  
Z_{n}  n  n•  [n]^{+}  n  Cyclic: nfold rotations. Abstract group Z_{n}, the group of integers under addition modulo n.  
D_{n}  nm  *n•  [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 
Point groups in 3 dimensions can be expressed in bracket notation related to the rank 3 Coxeter groups:
Finite groups of isometries in 3space^{[2]}  

Rotation groups  Extended groups  
Name  Bracket  Orb  Sch  Abstract  Order  Name  Bracket  Orb  Sch  Abstract  Order 
Identity  [ ]^{+}  11  C_{1}  Z_{1}  1  Bilateral  [1,1] = [ ]  *  D_{1}  D_{1}  2 
Central  [2^{+},2^{+}]  ×  C_{i}  2×Z_{1}  2  
Acrorhombic  [1,2]^{+} = [2]^{+}  22  C_{2}  Z_{2}  2  Acrorectangular  [1,2] = [2]  *22  C_{2v}  D_{2}  4 
Gyrorhombic  [2^{+},4^{+}]  2×  S_{4}  Z_{4}  4  
Orthorhombic  [2,2^{+}]  2*  D_{1d}  D_{1}×Z_{2}  4  
Pararhombic  [2,2]^{+}  222  D_{2}  D_{2}  4  Gyrorectangular  [2^{+},4]  2*2  D_{2d}  D_{4}  8 
Orthorectangular  [2,2]  *222  D_{2h}  D_{1}×D_{2}  8  
Acropgonal  [1,p]^{+} = [p]^{+}  pp  C_{p}  Z_{p}  p  Full acropgonal  [1,p] = [p]  *pp  C_{pv}  D_{p}  2p 
Gyropgonal  [2^{+},2p^{+}]  p×  S_{2p}  Z_{2p}  2p  
Orthopgonal  [2,p^{+}]  p*  C_{ph}  D_{1}×Z_{p}  2p  
Parapgonal  [2,p]^{+}  p22  D_{p}  D_{p}  2p  Full gyropgonal  [2^{+},2p]  2*p  D_{pd}  D_{2p}  4p 
Full orthopgonal  [2,p]  *p22  D_{ph}  D_{1}×D_{p}  4p  
Tetrahedral  [3,3]+  332  T  A_{4}  12  Full tetrahedral  [3,3]  *332  T_{d}  S_{4}  24 
Pyritohedral  [3^{+},4]  3*2  T_{h}  2×A_{4}  24  
Octahedral  [3,4]^{+}  432  O  S_{4}  24  Full octahedral  [3,4]  *432  O_{h}  2×S_{4}  48 
Icosahedral  [3,5]^{+}  532  I  A_{5}  60  Full icosahedral  [3,5]  *532  I_{h}  2×A_{5}  120 
In three dimensions, the full orthorhombic group or orthorectangular [2,2], abstractly D_{2}×D_{2}, order 8, represents three orthogonal mirrors, (also represented by Coxeter diagram 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^{+}] ( or ), abstractly D_{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 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 the pararhombic group [2,2]^{+} ( or ), also order 4, and finally the central group [2^{+},2^{+}] ( or ) of order 2.
Next there is the full orthopgonal group, [2,p] (), abstractly D_{1}×D_{p}=Z_{2}×D_{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 diagram as .
The direct subgroup is called the parapgonal group, [2,p]^{+} ( or ), abstractly D_{p}, of order 2p, and another subgroup is [2,p^{+}] () abstractly D_{1}×Z_{p}, also of order 2p.
The full gyropgonal group, [2^{+},2p] ( or ), abstractly D_{2p}, of order 4p. The gyropgonal group, [2^{+},2p^{+}] ( or ), 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: [3,3] (), [3,4] (), [3,5] () called full tetrahedral symmetry, octahedral symmetry, and icosahedral symmetry, with orders of 24, 48, and 120.
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 the pyritohedral 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 mirror 1 across 0, {010}, and 3fold 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 symmetry  Octahedral symmetry 
Icosahedral symmetry  
Finite (point groups in three dimensions)  


