Coxeter notation

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 "A""n" group is represented by [3"n"−1], to imply "n" nodes connected by "n−1" order-3 branches. Example "A"2 = [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 "A""n" family, like "A"3 = [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 = "A"2×"A"2 = 2"A"2 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].
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.
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.
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. 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: 
For example, (with p=2): [4,1+] = [1+,4] = [2] = [ ]×[ ], order 4. [1+,4,1+] = [2]+, order 2.
The opposite to halving is doubling 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 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: = [4,3], or more generally = [2p,q].
Radical subgroups.
Johnson also added an asterisk or star * operator for "radical" subgroups, 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. 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 [3"p"], the trionic subgroup is [3"p"]⅄ ≅ ["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: [3"p",1⅄] = = , = , and [3"p"⅄] = = 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].
For rank 4, ["q",2"p",3⅄] = [2"p",((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 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] ≅ ≅ [8,2+,8], order 128, are {02,1,3} from [3,3,4] generators {0,1,2,3}. And [(3,3)Δ,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" = [2"n"−1] represents "n" orthogonal mirrors in n-dimensional space, or an n-flat subspace of a higher dimensional space. The mirrors of the group [2"n"−1] 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 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}.
Rotations and rotary reflections.
Rotations and rotary reflections are constructed by a single single-generator product of all the reflections of a prismatic group, [2"p"]×[2"q"]×... where gcd("p","q"...)=1, they are isomorphic to the abstract cyclic group Zn, of order "n"=2"pq".
The 4-dimensional double rotations, [2"p"+,2+,2"q"+] (with gcd("p","q")=1), which include a central group, and are expressed by Conway as ±[C"p"×C"q"], order 2"pq". From Coxeter diagram , generators {0,1,2,3}, the single generator of [2"p"+,2+,2"q"+], is {0123}. The half group, [2"p"+,2+,2"q"+]+, or cyclic graph, [(2"p"+,2+,2"q"+,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 [2"pf"+,2+,2"qf"+] (with gcd("p","q")=1), generator {0123}, order 2"pqf". For example, "p"="q"=1, "f"=2, [4+,2+,4+] is order 4. And [2"pf"+,2+,2"qf"+]+, generator {01230123}, is order "pqf". For example, [4+,2+,4+]+ is order 2, a central inversion.
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 2"c", where c is the number of disconnected subgraphs when all the even-order branches are removed. 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.
Coxeter's notation includes double square bracket notation, 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 formula_1: and , the first doubled with square brackets, 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 formula_2 and branching formula_3, formula_4, and formula_5 diagrams. formula_2 has order 2"n" symmetry of a regular "n"-gon, {"n"}, and is represented by ["n"[3["n"]]]. formula_3 and formula_4 are represented by [3[31,1,1]] = [3,4,3] and [3[32,2,2]] respectively while formula_5 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 formula_10 = [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.
Examples:
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 , , which he defines as the product of the original generators of [X] by the doubling generator. This looks similar to +, which is the chiral subgroup of . So for example the 3D space groups + (I432, 211) and (Pmn, 
223) are distinct subgroups of (Imm, 229).
Computation with reflection matrices as symmetry generators.
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.
With one even branch, ["p"+,2"q"], or , is another subgroup of index 2, represented by rotation generator σ0,1, and reflectional ρ2.
With even branches, [2"p"+,2"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 "Dih"n, and cyclic groups are represented by "Z"n, with "Dih"1="Z"2.
Rank 2.
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.
Rank 3.
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 formula_11 (which goes through the origin), one can use formula_12, where formula_13 is the 3×3 identity matrix and formula_14 is the three-dimensional unit vector for the vector normal of the plane. If the L2 norm of formula_15 and formula_16 is unity, the transformation matrix can be expressed as:
Dihedral symmetry.
The reducible 3-dimensional finite reflective group is dihedral symmetry, ["p",2], order 4"p", . 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.
Tetrahedral symmetry.
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.
Octahedral symmetry.
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.
Icosahedral symmetry.
A final irreducible 3-dimensional finite reflective group is icosahedral symmetry, [5,3], order 120, . The reflection generators matrices are R0, R1, R2. R02=R12=R22=(R0×R1)5=(R1×R2)3=(R0×R2)2=Identity. [5,3]+ () is generated by 2 of 3 rotations: S0,1, S1,2, and S0,2. A 10-fold rotoreflection is generated by V0,1,2, the product of all 3 reflections.
Affine rank 3.
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 S0,1 S1,2, and S0,2. [4+,4+] () (pgg) is generated by 2-fold rotation S0,2 and transreflection V0,1,2. [4+,4] () (p4g) is generated by S0,1 and R3. The group [(4,4,2+)] () (cmm), is generated by 2-fold rotation S1,3 and reflection R2.
Rank 4.
Hyperoctahedral or hexadecachoric symmetry.
A irreducible 4-dimensional finite reflective group is hyperoctahedral group (or hexadecachoric group (for 16-cell), B4=[4,3,3], order 384, . The reflection generators matrices are R0, R1, R2, R3. R02=R12=R22=R32=(R0×R1)4=(R1×R2)3=(R2×R3)3=(R0×R2)2=(R1×R3)2=(R0×R3)2=Identity.
Chiral hyperoctahedral symmetry, [4,3,3]+, () is generated by 3 of 6 rotations: S0,1, S1,2, S2,3, S0,2, S1,3, and S0,3. Hyperpyritohedral symmetry [4,(3,3)+], () is generated by reflection R0 and rotations S1,2 and S2,3. An 8-fold double rotation is generated by W0,1,2,3, the product of all 4 reflections.
Hyperoctahedral subgroup D4 symmetry.
A half group of the Hyperoctahedral group is D4, [3,31,1], , order 192. It shares 3 generators with Hyperoctahedral group, but has two copies of an adjacent generator, one reflected across the removed mirror.
Icositetrachoric symmetry.
A irreducible 4-dimensional finite reflective group is Icositetrachoric group (for 24-cell), F4=[3,4,3], order 1152, . The reflection generators matrices are R0, R1, R2, R3. R02=R12=R22=R32=(R0×R1)3=(R1×R2)4=(R2×R3)3=(R0×R2)2=(R1×R3)2=(R0×R3)2=Identity.
Chiral icositetrachoric symmetry, [3,4,3]+, () is generated by 3 of 6 rotations: S0,1, S1,2, S2,3, S0,2, S1,3, and S0,3. Ionic diminished [3,4,3+] group, () is generated by reflection R0 and rotations S1,2 and S2,3. A 12-fold double rotation is generated by W0,1,2,3, the product of all 4 reflections.
Hypericosahedral symmetry.
The hyper-icosahedral symmetry, [5,3,3], order 14400, . The reflection generators matrices are R0, R1, R2, R3. R02=R12=R22=R32=(R0×R1)5=(R1×R2)3=(R2×R3)3=(R0×R2)2=(R0×R3)2=(R1×R3)2=Identity. [5,3,3]+ () is generated by 3 rotations: S0,1 = R0×R1, S1,2 = R1×R2, S2,3 = R2×R3, etc.
Rank one groups.
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 [ ]+, 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, .
Rank two groups.
In two dimensions, the "rectangular group" [2], abstract "D"12 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, 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" Dp, (nonabelian for p>2), of order 2"p", is generated by two mirrors at angle "π"/"p", represented by Coxeter diagram . The "p-gonal" subgroup [p]+, "cyclic group" "Z""p", 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.
Rank three groups.
Point groups in 3 dimensions can be expressed in bracket notation related to the rank 3 Coxeter groups:
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 ortho-"p"-gonal 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 para-"p"-gonal 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 gyro-p-gonal group", [2+,2"p"] ( or ), abstractly "D"2"p", of order 4"p". The gyro-"p"-gonal group, [2+,2p+] ( or ), abstractly "Z"2"p", of order 2"p" is a subgroup of both [2+,2"p"] and [2,2"p"+].
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 3-fold rotation {12}.
The tetrahedral group, [3,3] (), has a doubling (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]+.
Affine.
In the Euclidean plane there's 3 fundamental reflective groups generated by 3 mirrors, represented by Coxeter diagrams , , and , and are given Coxeter notation as [4,4], [6,3], and [(3,3,3)]. The parentheses of the last group imply the diagram cycle, and also has a shorthand notation [3[3]].
 as a doubling of the [4,4] group produced the same symmetry rotated π/4 from the original set of mirrors.
Direct subgroups of rotational symmetry are: [4,4]+, [6,3]+, and [(3,3,3)]+. [4+,4] and [6,3+] are semidirect subgroups.
Given in Coxeter notation (orbifold notation), some low index affine subgroups are:
Rank four groups.
Point groups.
Rank four groups defined the 4-dimensional point groups:
Line groups.
Rank four groups also defined the 3-dimensional line groups:
Duoprismatic group.
Rank four groups defined the 4-dimensional duoprismatic groups. In the limit as p and q go to infinity, they degenerate into 2 dimensions and the wallpaper groups.
Wallpaper groups.
Rank four groups also defined some of the 2-dimensional wallpaper groups, as limiting cases of the four-dimensional duoprism groups:
Subgroups of [∞,2,∞], (*2222) can be expressed down to its index 16 commutator subgroup:
Complex reflections.
Coxeter notation has been extended to Complex space, Cn where nodes are unitary reflections of period greater than 2. Nodes are labeled by an index, assumed to be 2 for ordinary real reflection if suppressed. Complex reflection groups are called Shephard groups rather than Coxeter groups, and can be used to construct complex polytopes.
In formula_18, a rank 1 shephard group , order "p", is represented as "p"[], []"p" or ]"p"[. It has a single generator, representing a 2"π"/"p" radian rotation in the Complex plane: formula_19.
Coxeter writes the rank 2 complex group, "p"["q"]"r" represents Coxeter diagram . The "p" and "r" should only be suppressed if both are 2, which is the real case ["q"]. The order of a rank 2 group "p"["q"]"r" is formula_20.
The rank 2 solutions that generate complex polygons are: "p"[4]2 ("p" is 2,3,4...), 3[3]3, 3[6]2, 3[4]3, 4[3]4, 3[8]2, 4[6]2, 4[4]3, 3[5]3, 5[3]5, 3[10]2, 5[6]2, and 5[4]3 with Coxeter diagrams , , , , , , , , , , , , . 
Infinite groups are 3[12]2, 4[8]2, 6[6]2, 3[6]3, 6[4]3, 4[4]4, and 6[3]6 or , , , , , , .
Index 2 subgroups exists by removing a real reflection: "p"[2"q"]2 → "p"["q"]"p". Also index "r" subgroups exist for 4 branches: "p"[4]"r" → "p"["r"]"p".
For the infinite family "p"[4]2, for any "p" = 2, 3, 4..., there are two subgroups: "p"[4]2 → ["p"], index "p", while and "p"[4]2 → "p"[]×"p"[], index 2.