Coxeter notation

Fundamental domains of reflective 3D point groups

Template:CDD, Template:Math	Template:CDD, Template:Math	Template:CDD, Template:Math	Template:CDD, Template:Math	Template:CDD, Template:Math	Template:CDD, Template:Math
File:Spherical digonal hosohedron.svg Order 2	Order 4	Order 6	File:Spherical octagonal hosohedron.svg Order 8	Error creating thumbnail: Order 10	File:Spherical dodecagonal hosohedron.svg Order 12
Template:CDD Template:Math	Template:CDD Template:Math	Template:CDD Template:Math	Template:CDD Template:Math	Template:CDD Template:Math	Template:CDD Template:Math
File:Spherical digonal bipyramid.svg Order 4	Error creating thumbnail: Order 8	File:Spherical hexagonal bipyramid.svg Order 12	Error creating thumbnail: Order 16	File:Spherical decagonal bipyramid.svg Order 20	File:Spherical dodecagonal bipyramid.svg Order 24
Template:CDD, Template:Math		Template:CDD, Template:Math		Template:CDD, Template:Math
Order 24		File:Spherical disdyakis dodecahedron-3and1-color.png Order 48		Order 120
Coxeter notation expresses Coxeter groups as a list of branch orders of a Coxeter diagram, like the polyhedral groups, Template:Math. Dihedral groups, Template:CDD, can be expressed as a product Template:Math or in a single symbol with an explicit order 2 branch, Template:Math.

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 [3ⁿ⁻¹], to imply n nodes connected by n−1 order-3 branches. Example A₂ = [3,3] = [3²] or [3^1,1] represents diagrams Template:CDD or Template:CDD.

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] = Template:CDD or Template:CDD as D₄. Coxeter allowed for zeros as special cases to fit the A_n family, like A₃ = [3,3,3,3] = [3^4,0,0] = [3^4,0] = [3^3,1] = [3^2,2], like Template:CDD = Template:CDD = Template:CDD.

Coxeter groups formed by cyclic diagrams are represented by parentheseses inside of brackets, like [(p,q,r)] = Template:CDD 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 Template:CDD or Template:CDD. Template:CDD 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 Template:CDD 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 Template:CDD or Template:CDD, is represented as [3^[3,3]] with the superscript [3,3] as the symmetry of its regular tetrahedron coxeter diagram.

Finite groups Rank Group symbol Bracket notation Coxeter diagram 2 A2 [3] Template:CDD 2 B2 [4] Template:CDD 2 H2 [5] Template:CDD 2 G2 [6] Template:CDD 2 I2(p) [p] Template:CDD 3 Ih, H3 [5,3] Template:CDD 3 Td, A3 [3,3] Template:CDD 3 Oh, B3 [4,3] Template:CDD 4 A4 [3,3,3] Template:CDD 4 B4 [4,3,3] Template:CDD 4 D4 [31,1,1] Template:CDD 4 F4 [3,4,3] Template:CDD 4 H4 [5,3,3] Template:CDD n An [3n−1] Template:CDD..Template:CDD n Bn [4,3n−2] Template:CDD...Template:CDD n Dn [3n−3,1,1] Template:CDD...Template:CDD 6 E6 [32,2,1] Template:CDD 7 E7 [33,2,1] Template:CDD 8 E8 [34,2,1] Template:CDD

Affine groups Group symbol Bracket notation Coxeter diagram ${\tilde{I}}_1,{\tilde{A}}_1$ [∞] Template:CDD ${\tilde{A}}_2$ [3[3]] Template:CDD ${\tilde{C}}_2$ [4,4] Template:CDD ${\tilde{G}}_2$ [6,3] Template:CDD ${\tilde{A}}_3$ [3[4]] Template:CDD ${\tilde{B}}_3$ [4,31,1] Template:CDD ${\tilde{C}}_3$ [4,3,4] Template:CDD ${\tilde{A}}_4$ [3[5]] Template:CDD ${\tilde{B}}_4$ [4,3,31,1] Template:CDD ${\tilde{C}}_4$ [4,3,3,4] Template:CDD ${\tilde{D}}_4$ [ 31,1,1,1] Template:CDD ${\tilde{F}}_4$ [3,4,3,3] Template:CDD ${\tilde{A}}_n$ [3[n+1]] Template:CDD...Template:CDD or Template:CDD...Template:CDD ${\tilde{B}}_n$ [4,3n−3,31,1] Template:CDD...Template:CDD ${\tilde{C}}_n$ [4,3n−2,4] Template:CDD...Template:CDD ${\tilde{D}}_n$ [ 31,1,3n−4,31,1] Template:CDD...Template:CDD ${\tilde{E}}_6$ [32,2,2] Template:CDD ${\tilde{E}}_7$ [33,3,1] Template:CDD ${\tilde{E}}_8=E_9$ [35,2,1] Template:CDD

Hyperbolic groups Group symbol Bracket notation Coxeter diagram [p,q] with Template:Nowrap Template:CDD [(p,q,r)] with $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}<1$ Template:CDD ${\overline{BH}}_3$ [4,3,5] Template:CDD ${\bar{K}}_3$ [5,3,5] Template:CDD ${\bar{J}}_3,{\tilde{H}}_3$ [3,5,3] Template:CDD ${\overline{DH}}_3$ [5,31,1] Template:CDD ${\widehat{AB}}_3$ [(3,3,3,4)] Template:CDD  ${\widehat{AH}}_3$ [(3,3,3,5)] Template:CDD  ${\widehat{BB}}_3$ [(3,4,3,4)] Template:CDD ${\widehat{BH}}_3$ [(3,4,3,5)] Template:CDD ${\widehat{HH}}_3$ [(3,5,3,5)] Template:CDD ${\bar{H}}_4,{\tilde{H}}_4,H_5$ [3,3,3,5] Template:CDD ${\overline{BH}}_4$ [4,3,3,5] Template:CDD ${\bar{K}}_4$ [5,3,3,5] Template:CDD ${\overline{DH}}_4$ [5,3,31,1] Template:CDD ${\widehat{AF}}_4$ [(3,3,3,3,4)] Template:CDD

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.

Unconnected groups

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 Template:CDD = 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: Template:CDD or Template:CDD, as an identical presentation as [3,2,3].

Rank and dimension

Coxeter point group rank is equal to the number of nodes which is also equal to the dimension. A single mirror exists in 1-dimension, [ ], Template:CDD, while in 2-dimensions [1], Template:CDD or [ ]×[ ]⁺. The 1 is a place-holder, not an actual branch order, but a marker for an orthogonal inactive mirror. The notation [n,1], represents a rank 3 group, as [n]×[ ]⁺ or Template:CDD. Similarly, [1,1] as [ ]×[ ]⁺×[ ]⁺ or Template:CDD order 2 and [1,1]⁺ as [ ]⁺×[ ]⁺×[ ]⁺ or Template:CDD, order 1!

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)⁺] (Template:CDD).

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

Group		Order	Generators	Subgroup		Order	Generators	Notes
[p]	Template:CDD	2p	{0,1}	[p]+	Template:CDD	p	{01}	Direct subgroup
[2p+] = [2p]+	Template:CDD	2p	{01}	[2p+]+ = [2p]+2 = [p]+	Template:CDD	p	{0101}
[2p]	Template:CDD	4p	{0,1}	[1+,2p] = [p]	Template:CDD = Template:CDD = Template:CDD	2p	{101,1}	Half subgroups
				[2p,1+] = [p]	Template:CDD = Template:CDD = Template:CDD		{0,010}
				[1+,2p,1+] = [2p]+2 = [p]+	Template:CDD = Template:CDD = Template:CDD	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, Template:CDD has symmetry [4,3]⁺ (Template:CDD), and the snub tetrahedron, Template:CDD has symmetry [4,3⁺] (Template:CDD), and a demicube, h{4,3} = {3,3} (Template:CDD or Template:CDD = Template:CDD) has symmetry [1⁺,4,3] = [3,3] (Template:CDD or Template:CDD = Template:CDD = Template:CDD).

Note: Pyritohedral symmetry Template:CDD can be written as Template:CDD, separating the graph with gaps for clarity, with the generators {0,1,2} from the Coxeter group Template:CDD, producing pyritohedral generators {0,12}, a reflection and 3-fold rotation. And chiral tetrahedral symmetry can be written as Template:CDD or Template:CDD, [1⁺,4,3⁺] = [3,3]⁺, with generators {12,0120}.

Halving subgroups and extended groups

Example halving operations

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Template:CDD [1,4,1] = [4]	Template:CDD = Template:CDD = Template:CDD [1+,4,1]=[2]=[ ]×[ ]
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Template:CDD = Template:CDD = Template:CDD [1,4,1+]=[2]=[ ]×[ ]	Template:CDD = Template:CDD = Template:CDD = Template:CDD [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. 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 Template:CDD or Template:CDD, 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: Template:CDD = Template:CDD, 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: Template:CDD = Template:CDD = Template:CDD.

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. Template:CDD = Template:CDD = Template:CDD = Template:CDD = Template:CDD.

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.

Template:Brackets = [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: Template:CDD = Template:CDD, h[2p,3] = [1⁺,2p,3] = [(p,3,3)].

If nodes are indexed, half subgroups can be labeled with new mirrors as composites. Like Template:CDD, generators {0,1} has subgroup Template:CDD = Template:CDD, generators {1,010}, where mirror 0 is removed, and replaced by a copy of mirror 1 reflected across mirror 0. Also given Template:CDD, generators {0,1,2}, it has half group Template:CDD = Template:CDD, generators {1,2,010}.

Doubling by adding a mirror also applies in reversing the halving operation: Template:Brackets = [4,3], or more generally Template:Brackets = [2p,q].

Tetrahedral symmetry	Octahedral symmetry
File:Sphere symmetry group td.png Td, [3,3] = [1+,4,3] Template:CDD = Template:CDD = Template:CDD (Order 24)	Oh, [4,3] = Template:Brackets Template:CDD (Order 48)

Radical subgroups

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

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: Template:CDD or Template:CDD ≅ Template:CDD. 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

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Rank 2 example, [6] trionic subgroups with 3 colors of mirror lines

Example on octahedral symmetry: [4,3^⅄] = [2,4].

Example trionic subgroup on hexagonal symmetry [6,3] maps onto a larger [6,3] symmetry.

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Rank 3

File:Hyperbolic 832 trionic subgroup 842.png

Example trionic subgroups on octagonal symmetry [8,3] maps onto larger [4,8] symmetries.

File:Trionic subgroups rank 4b.png

Rank 4

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 Template:CDD, 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^⅄] = Template:CDD = Template:CDD, Template:CDD = Template:CDD, and [3p^⅄] = Template:CDD = Template:CDD 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 Template:CDD = Template:CDD. 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))], Template:CDD = Template:CDD.

For example, [3,4,3^⅄] = [4,3,3], or Template:CDD = Template:CDD, 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

File:Trionic subgroups of tetrahedral symmetry stereographic projection.png

[3,3]^⅄ ≅ [2⁺,4] as one of 3 sets of 2 orthogonal mirrors in stereographic projection. The red, green, and blue represent 3 sets of mirrors, and the gray lines are removed mirrors, leaving 2-fold gyrations (purple diamonds).

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Trionic relations of [3,3]

Johnson identified two specific trionic subgroups of [3,3], first an index 3 subgroup [3,3]^⅄ ≅ [2⁺,4], with [3,3] (Template:CDD = Template:CDD = Template:CDD) generators {0,1,2}. It can also be written as [(3,3,2^⅄)] (Template:CDD) 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^⅄)]⁺ (Template:CDD), 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.

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Trionic subgroup relations of [3,3,4]

For example, [(3,3)⁺,4], [(3,3)^⅄,4], and [(3,3)^Δ,4] are subgroups of [3,3,4], index 2, 3 and 6 respectively. The generators of [(3,3)^⅄,4] ≅ Template:Brackets ≅ [8,2⁺,8], order 128, are {02,1,3} from [3,3,4] generators {0,1,2,3}. And [(3,3)^Δ,4] ≅ Template:Brackets, 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.

Central inversion

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

A central inversion, order 2, is operationally differently by dimension. The group [ ]ⁿ = [2ⁿ⁻¹] represents n orthogonal mirrors in n-dimensional space, or an n-flat subspace of a higher dimensional space. The mirrors of the group [2ⁿ⁻¹] are numbered Template:Tmath. The order of the mirrors doesn't matter in the case of an inversion. The matrix of a central inversion is Template:Tmath, 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], Template:CDD, and are represented as Template:CDD (or Template:CDD) and Template:CDD (or Template:CDD) with generators {01,2} and {0,12} respectively. Their common subgroup index 4 is [2⁺,2⁺], and is represented by Template:CDD (or Template:CDD), with the double-open Template:CDD 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	Template:CDD	180° rotation, C2	{01}
3	[2+,2+]	2	Template:CDD	rotoreflection, Ci or S2	{012}
4	[2+,2+,2+]	2	Template:CDD	double rotation	{0123}
5	[2+,2+,2+,2+]	2	Template:CDD	double rotary reflection	{01234}
6	[2+,2+,2+,2+,2+]	2	Template:CDD	triple rotation	{012345}
7	[2+,2+,2+,2+,2+,2+]	2	Template:CDD	triple 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 Z_n, 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 ±[C_p×C_q], order 2pq. From Coxeter diagram Template:CDD, generators {0,1,2,3}, requires two generator for [2p⁺,2⁺,2q⁺], Template:CDD as {0123,0132}. Half groups, [2p⁺,2⁺,2q⁺]⁺, or cyclic graph, [(2p⁺,2⁺,2q⁺,2⁺)], Template:CDD expressed by Conway is [C_p×C_q], order pq, with one generator, like {0123}.

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

In general a n-rotation group, [2p₁⁺,2,2p₂⁺,2,...,p_n⁺] may require up to n generators if gcd(p₁,..,p_n)>1, as a product of all mirrors, and then swapping sequential pairs. The half group, [2p₁⁺,2,2p₂⁺,2,...,p_n⁺]⁺ has generators squared. n-rotary reflections are similar.

Examples

Dimension	Coxeter notation	Order	Coxeter diagram	Operation	Generators	Direct subgroup
2	[2p]+	2p	Template:CDD	Rotation	{01}	[2p]+2 = [p]+	Simple rotation: [2p]+2 = [p]+ order p
3	[2p+,2+]		Template:CDD	rotary reflection	{012}	[2p+,2+]+ = [p]+
4	[2p+,2+,2+]		Template:CDD	double rotation	{0123}	[2p+,2+,2+]+ = [p]+
5	[2p+,2+,2+,2+]		Template:CDD	double rotary reflection	{01234}	[2p+,2+,2+,2+]+ = [p]+
6	[2p+,2+,2+,2+,2+]		Template:CDD	triple rotation	{012345}	[2p+,2+,2+,2+,2+]+ = [p]+
7	[2p+,2+,2+,2+,2+,2+]		Template:CDD	triple rotary reflection	{0123456}	[2p+,2+,2+,2+,2+,2+]+ = [p]+
4	[2p+,2+,2q+]	2pq	Template:CDD	double rotation	{0123, 0132}	[2p+,2+,2q+]+	Double rotation: [2p+,2+,2q+]+ order pq
5	[2p+,2+,2q+,2+]		Template:CDD	double rotary reflection	{01234, 01243}	[2p+,2+,2q+,2+]+
6	[2p+,2+,2q+,2+,2+]		Template:CDD	triple rotation	{012345, 012354, 013245}	[2p+,2+,2q+,2+,2+]+
7	[2p+,2+,2q+,2+,2+,2+]		Template:CDD	triple rotary reflection	{0123456, 0123465, 0124356, 0124356}	[2p+,2+,2q+,2+,2+,2+]+
6	[2p+,2+,2q+,2+,2r+]	2pqr	Template:CDD	triple 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+]		Template:CDD	triple rotary reflection	{0123456, 0123465, 0124356, 0213456}	[2p+,2+,2q+,2+,2r+,2+]+

Commutator subgroups

File:Subgroups of 442.png

Hasse diagram subgroups of [4,4], down to its commutator subgroup, index 8

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], Template:CDD 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 (Rank-reduction)
Diagram			Error creating thumbnail:
Coxeter Template:CDD	Template:CDD [1,4,1] = [4]	Template:CDD = Template:CDD = Template:CDD [1+,4,1] = [1+,4] = [2]	Template:CDD = Template:CDD = Template:CDD [1,4,1+] = [4,1+] = [2]	Template:CDD [2,1+] = [1] = [ ]	Template:CDD [1+,2] = [1] = [ ]
Generators	{0,1}	{101,1}	{0,010}	{0}	{1}
Direct subgroups
Index	2	4		8
Diagram	File:Cyclic symmetry 4.png	File:Cyclic symmetry 4 half.png
Coxeter	Template:CDD [4]+	Template:CDD = Template:CDD = Template:CDD = Template:CDD [4]+2 = [1+,4,1+] = [2]+		Template:CDD [ ]+
Generators	{01}	{(01)2}		{02} = {12} = {(01)4} = { }

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, Template:CDD. 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			File:442 symmetry 00a.png		File:442 symmetry a0b.png	File:442 symmetry xxx.png
Coxeter Template:CDD	[1,4,1,4,1] = [4,4] Template:CDD	[1+,4,4] Template:CDD = Template:CDD	[4,4,1+] Template:CDD = Template:CDD	[4,1+,4] Template:CDD = Template:CDD	[1+,4,4,1+] Template:CDD = Template:CDD	[4+,4+] Template:CDD
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		Error creating thumbnail:	File:442 symmetry aa0.png	Error creating thumbnail:	File:442 symmetry 0ab.png	File:442 symmetry ab0.png
Coxeter		[4,4+] Template:CDD	[4+,4] Template:CDD	[(4,4,2+)] Template:CDD = Template:CDD	[4,1+,4,1+] Template:CDD = Template:CDD = Template:CDD	[1+,4,1+,4] Template:CDD = Template:CDD = Template:CDD
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	Error creating thumbnail:	File:442 symmetry abb.png	File:442 symmetry aab.png	Error creating thumbnail:	File:442 symmetry abc.png
Coxeter	[4,4]+ Template:CDD = Template:CDD	[4,4+]+ Template:CDD = Template:CDD	[4+,4]+ Template:CDD = Template:CDD	[(4,4,2+)]+ Template:CDD = Template:CDD	[4,4]+3 = [(4+,4+,2+)] = [1+,4,1+,4,1+] = [4+,4+]+ Template:CDD =Template:CDD = Template:CDD = Template:CDD = Template:CDD
Generators	{01,12}	{(01)2,12}	{01,(12)2}	{02,(01)2,(12)2}	{(01)2,(12)2,2(01)22}
Orbifold	442			2222
Radical subgroups
Index		8		16
Diagram		File:442 symmetry 0dd.png		File:442 symmetry add.png	File:442 symmetry dda.png
Coxeter		[4,4*] Template:CDD = Template:CDD	[4*,4] Template:CDD = Template:CDD	[4,4*]+ Template:CDD = Template:CDD	[4*,4]+ Template:CDD = Template:CDD
Orbifold		*2222		2222

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:

Small index subgroups of [6,4]
Index	1	2			4
Diagram	File:642 symmetry 000.png	File:642 symmetry a00.png	File:642 symmetry 00a.png	File:642 symmetry 0a0.png	Error creating thumbnail:
Coxeter Template:CDD	[1,6,1,4,1] = [6,4] Template:CDD	[1+,6,4] Template:CDD = Template:CDD	[6,4,1+] Template:CDD = Template:CDD	[6,1+,4] Template:CDD = Template:CDD	[1+,6,4,1+] Template:CDD = Template:CDD	[6+,4+] Template:CDD
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		File:642 symmetry 0aa.png	Error creating thumbnail:	File:642 symmetry a0a.png	File:642 symmetry 0ab.png	File:642 symmetry ab0.png
Coxeter		[6,4+] Template:CDD	[6+,4] Template:CDD	[(6,4,2+)] Template:CDD	[6,1+,4,1+] Template:CDD = Template:CDD = Template:CDD = Template:CDD = Template:CDD	[1+,6,1+,4] Template:CDD = Template:CDD = Template:CDD = Template:CDD = Template:CDD
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	File:642 symmetry aaa.png		File:642 symmetry aab.png	File:642 symmetry aba.png	Error creating thumbnail:
Coxeter	[6,4]+ Template:CDD = Template:CDD	[6,4+]+ Template:CDD = Template:CDD	[6+,4]+ Template:CDD = Template:CDD	[(6,4,2+)]+ Template:CDD = Template:CDD	[6+,4+]+ = [1+,6,1+,4,1+] Template:CDD = Template:CDD = Template:CDD = Template:CDD
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		File:642 symmetry 0zz.png		File:642 symmetry azz.png	File:642 symmetry zza.png
Generators		{0,101,21012,1210121}	{2,121,101020101,0102010, 010101020101010, 10101010201010101}
Coxeter (orbifold)		[6,4*] Template:CDD = Template:CDD (*3333)	[6*,4] Template:CDD (*222222)	[6,4*]+ Template:CDD = Template:CDD (3333)	[6*,4]+ Template:CDD (222222)

Parabolic subgroups

A parabolic subgroup of a Coxeter group can be identified by removing one or more generator mirrors represented with a Coxeter diagram. For example the octahedral group Template:CDD has parabolic subgroups Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD. In bracket notation [4,3] has parabolic subgroups [4],[2],[3], and a single mirror []. The order of the subgroup is known, and always an integer divisor group order, or index. Parabolic subgroups can also be written with x nodes, like Template:CDD=[4,3] subgroup by removing second mirror: Template:CDD or Template:CDD = Template:CDD = [4,1^×,3] = [2].

Petrie subgroup

A petrie subgroup of an irreducible coxeter group can be created by the product of all of the generators. It can be seen in the skew regular petrie polygon of a regular polytope. The order of the new group is called the Coxeter number of the original Coxeter group. The Coxeter number of a Coxeter group is 2m/n, where n is the rank, and m is the number of reflections. A petrie subgroup can be written with a Template:Pi superscript. For example, [3,3]^Template:Pi is the petrie subgroup of a tetrahedral group, cyclic group order 4, generated by a rotoreflection. A rank 4 Coxeter group will have a double rotation generator, like [4,3,3]^Template:Pi is order 8.

Extended symmetry

Wallpaper group Triangle symmetry Extended symmetry Extended diagram Extended group Honeycombs p3m1 (*333) a1 File:Triangle symmetry1.png [3[3]] Template:CDD ${\tilde{A}}_2$ Template:CNone p6m (*632) i2 File:Triangle symmetry3.png Template:Brackets ↔ [6,3] Template:CDD ↔ Template:CDD ${\tilde{A}}_2\times 2\leftrightarrow {\tilde{G}}_2$ Template:CDD 1, Template:CDD 2 p31m (3*3) g3 File:Triangle symmetry2.png [3+[3[3]]] ↔ [6,3+] Template:CDD ↔ Template:CDD ${\tilde{A}}_2\times 3\leftrightarrow \frac{1}{2}{\tilde{G}}_2$ Template:CNone p6 (632) r6 Error creating thumbnail: [3[3[3]]]+ ↔ [6,3]+ Template:CDD ↔ Template:CDD $\frac{1}{2}{\tilde{A}}_2\times 6\leftrightarrow \frac{1}{2}{\tilde{G}}_2$ Template:CDD (1) p6m (*632) [3[3[3]]] ↔ [6,3] ${\tilde{A}}_2\times 6\leftrightarrow {\tilde{G}}_2$ Template:CDD 3

In the Euclidean plane, the ${\tilde{A}}_2$, [3[3]] Coxeter group can be extended in two ways into the ${\tilde{G}}_2$, [6,3] Coxeter group and relates uniform tilings as ringed diagrams.

Coxeter's notation includes double square bracket notation, Template:Brackets 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 ${\tilde{A}}_3$: Template:CDD and Template:CDD, the first doubled with square brackets, Template: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 ${\tilde{A}}_n$ and branching $D_3$, ${\tilde{E}}_6$, and ${\tilde{D}}_4$ diagrams. ${\tilde{A}}_n$ has order 2n symmetry of a regular n-gon, {n}, and is represented by [n[3^[n]]]. $D_3$ and ${\tilde{E}}_6$ are represented by [3[3^1,1,1]] = [3,4,3] and [3[3^2,2,2]] respectively while ${\tilde{D}}_4$ 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 ${\overline{L}}_5$ = [3^1,1,1,1,1], Template:CDD, contains the symmetry of a 5-cell, {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 odd-ordered mirrors. Template:Clear Examples:

Example Extended groups and radical subgroups

Extended groups Radical subgroups Coxeter diagrams Index [3[2,2]] = [4,3] [4,3*] = [2,2] Template:CDD = Template:CDD 6 [(3,3)[2,2,2]] = [4,3,3] [4,(3,3)*] = [2,2,2] Template:CDD = Template:CDD 24 [1[31,1]] = Template:Brackets = [3,4] [3,4,1+] = [3,3] Template:CDD = Template:CDD 2 [3[31,1,1]] = [3,4,3] [3*,4,3] = [31,1,1] Template:CDD = Template:CDD 6 [2[31,1,1,1]] = [4,3,3,4] [1+,4,3,3,4,1+] = [31,1,1,1] Template:CDD = Template:CDD 4 [3[3,31,1,1]] = [3,3,4,3] [3*,4,3,3] = [31,1,1,1] Template:CDD = Template:CDD 6 [(3,3)[31,1,1,1]] = [3,4,3,3] [3,4,(3,3)*] = [31,1,1,1] Template:CDD = Template:CDD 24 [2[3,31,1,1,1]] = [3,(3,4)1,1] [3,(3,4,1+)1,1] = [3,31,1,1,1] Template:CDD = Template:CDD 4 [(2,3)[1,131,1,1]] = [4,3,3,4,3] [3*,4,3,3,4,1+] = [31,1,1,1,1] Template:CDD = Template:CDD 12 [(3,3)[3,31,1,1,1]] = [3,3,4,3,3] [3,3,4,(3,3)*] = [31,1,1,1,1] Template:CDD = Template:CDD 24 [(3,3,3)[31,1,1,1,1]] = [3,4,3,3,3] [3,4,(3,3,3)*] = [31,1,1,1,1] Template:CDD = Template:CDD 120

Extended groups Radical subgroups Coxeter diagrams Index [1[3[3]]] = [3,6] [3,6,1+] = [3[3]] Template:CDD = Template:CDD 2 [3[3[3]]] = [6,3] [6,3*] = [3[3]] Template:CDD = Template:CDD 6 [1[3,3[3]]] = [3,3,6] [3,3,6,1+] = [3,3[3]] Template:CDD = Template:CDD 2 [(3,3)[3[3,3]]] = [6,3,3] [6,(3,3)*] = [3[3,3]] Template:CDD = Template:CDD 24 [1[∞]2] = [4,4] [4,1+,4] = [∞]2 = [∞,2,∞] Template:CDD = Template:CDD 2 [2[∞]2] = [4,4] [1+,4,4,1+] = [(4,4,2*)] = [∞]2 Template:CDD = Template:CDD 4 [4[∞]2] = [4,4] [4,4*] = [∞]2 Template:CDD = Template:CDD 8 [2[3[4]]] = [4,3,4] [1+,4,3,4,1+] = [(4,3,4,2*)] = [3[4]] Template:CDD = Template:CDD = Template:CDD 4 [3[∞]3] = [4,3,4] [4,3*,4] = [∞]3 = [∞,2,∞,2,∞] Template:CDD = Template:CDD 6 [(3,3)[∞]3] = [4,31,1] [4,(31,1)*] = [∞]3 Template:CDD = Template:CDD 24 [(4,3)[∞]3] = [4,3,4] [4,(3,4)*] = [∞]3 Template:CDD = Template:CDD 48 [(3,3)[∞]4] = [4,3,3,4] [4,(3,3)*,4] = [∞]4 Template:CDD = Template:CDD 24 [(4,3,3)[∞]4] = [4,3,3,4] [4,(3,3,4)*] = [∞]4 Template:CDD = Template:CDD 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 Template:Brackets, Template:Brackets, which he defines as the product of the original generators of [X] by the doubling generator. This looks similar to Template:Brackets⁺, which is the chiral subgroup of Template:Brackets. So for example the 3D space groups Template:Brackets⁺ (I432, 211) and Template:Brackets (PmTemplate:Overlinen, 223) are distinct subgroups of Template:Brackets (ImTemplate:Overlinem, 229).

Rank one groups

Template:Further 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, Template:CDD. 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, Template:CDD.

Group	Coxeter notation	Coxeter diagram	Order	Description
C1	[ ]+	Template:CDD	1	Identity
D2	[ ]	Template:CDD	2	Reflection group

Rank two groups

A regular hexagon, with markings on edges and vertices has 8 symmetries: [6], [3], [2], [1], [6]⁺, [3]⁺, [2]⁺, [1]⁺, with [3] and [1] existing in two forms, depending whether the mirrors are on the edges or vertices.

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, Template:CDD, with order 4. The 2 in [2] comes from linearization of the orthogonal subgraphs in the Coxeter diagram, as Template:CDD with explicit branch order 2. The rhombic group, [2]⁺ (Template:CDD or Template:CDD), 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 Template:CDD.

The full p-gonal group [p], abstract dihedral group D_2p, (nonabelian for p>2), of order 2p, is generated by two mirrors at angle π/p, represented by Coxeter diagram Template:CDD. 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 Template:CDD. The apeirogonal group [∞]⁺, Template:CDD, 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]⁺, Template:CDD. These groups exist in regular infinite-sided 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	File:Dihedral symmetry domains 1.png	Error creating thumbnail:	File:Dihedral symmetry domains 3.png	Error creating thumbnail:	...	File:Dihedral symmetry domains infinity.png	File:Horocycle mirrors.png	Error creating thumbnail:
Coxeter	Template:CDD [ ]	Template:CDD = Template:CDD [2]=[ ]×[ ]	Template:CDD [3]	Template:CDD [4]	Template:CDD [p]	Template:CDD [∞]	Template:CDD [∞]	Template:CDD [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)		Error creating thumbnail:	File:Cyclic symmetry 3.png	File:Cyclic symmetry 4.png	...	File:Cyclic symmetry infin.png		File:Cyclic symmetry ultra.png
Odd images (inverted)	File:Cyclic symmetry 1b.svg	File:Cyclic symmetry 2b.svg	File:Cyclic symmetry 3b.png	File:Cyclic symmetry 4b.png		File:Cyclic symmetry infinb.png		Error creating thumbnail:
Coxeter	Template:CDD [ ]+	Template:CDD [2]+	Template:CDD [3]+	Template:CDD [4]+	Template:CDD [p]+	Template:CDD [∞]+	Template:CDD [∞]+	Template:CDD [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
Zn	n	n•	[n]+	Template:CDD	n	Cyclic: n-fold rotations. Abstract group Zn, the group of integers under addition modulo n.
D2n	nm	*n•	[n]	Template:CDD	2n	Dihedral: cyclic with reflections. Abstract group Dihn, the dihedral group.
Affine
Z∞	∞	∞•	[∞]+	Template:CDD	∞	Cyclic: apeirogonal group. Abstract group Z∞, the group of integers under addition.
Dih∞	∞m	*∞•	[∞]	Template:CDD	∞	Dihedral: parallel reflections. Abstract infinite dihedral group Dih∞.
Hyperbolic
Z∞			[πi/λ]+	Template:CDD	∞	pseudogonal group
Dih∞			[πi/λ]	Template:CDD	∞	full pseudogonal group

Rank three groups

Template:Further Point groups in 3 dimensions can be expressed in bracket notation related to the rank 3 Coxeter groups:

Finite groups of isometries in 3-space
Rotation groups						Extended groups
Name	Bracket	Orb	Sch	Abstract	Order	Name	Bracket	Orb	Sch	Abstract	Order
Identity	[ ]+	11	C1	Z1	1	Bilateral	[1,1] = [ ]	*	D2	Z2	2
						Central	[2+,2+]	×	Ci	2×Z1	2
Acrorhombic	[1,2]+ = [2]+	22	C2	Z2	2	Acrorectangular	[1,2] = [2]	*22	C2v	D4	4
						Gyrorhombic	[2+,4+]	2×	S4	Z4	4
						Orthorhombic	[2,2+]	2*	D1d	Z2×Z2	4
Pararhombic	[2,2]+	222	D2	D4	4	Gyrorectangular	[2+,4]	2*2	D2d	D8	8
						Orthorectangular	[2,2]	*222	D2h	Z2×D4	8
Acro-p-gonal	[1,p]+ = [p]+	pp	Cp	Zp	p	Full acro-p-gonal	[1,p] = [p]	*pp	Cpv	D2p	2p
						Gyro-p-gonal	[2+,2p+]	p×	S2p	Z2p	2p
						Ortho-p-gonal	[2,p+]	p*	Cph	Z2×Zp	2p
Para-p-gonal	[2,p]+	p22	D2p	D2p	2p	Full gyro-p-gonal	[2+,2p]	2*p	Dpd	D4p	4p
						Full ortho-p-gonal	[2,p]	*p22	Dph	Z2×D2p	4p
Tetrahedral	[3,3]+	332	T	A4	12	Full tetrahedral	[3,3]	*332	Td	S4	24
						Pyritohedral	[3+,4]	3*2	Th	2×A4	24
Octahedral	[3,4]+	432	O	S4	24	Full octahedral	[3,4]	*432	Oh	2×S4	48
Icosahedral	[3,5]+	532	I	A5	60	Full icosahedral	[3,5]	*532	Ih	2×A5	120

In three dimensions, the full orthorhombic group or orthorectangular [2,2], abstractly Z₂³, order 8, represents three orthogonal mirrors, (also represented by Coxeter diagram as three separate dots Template:CDD). 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⁺] (Template:CDD or Template:CDD), abstractly Z₂×Z₂, 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, Template:CDD) 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]⁺ (Template:CDD or Template:CDD), also order 4, and finally the central group [2⁺,2⁺] (Template:CDD or Template:CDD) of order 2.

Next there is the full ortho-p-gonal group, [2,p] (Template:CDD), abstractly Z₂×D_2p, 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 Template:CDD.

The direct subgroup is called the para-p-gonal group, [2,p]⁺ (Template:CDD or Template:CDD), abstractly D_2p, of order 2p, and another subgroup is [2,p⁺] (Template:CDD) abstractly Z₂×Z_p, also of order 2p.

The full gyro-p-gonal group, [2⁺,2p] (Template:CDD or Template:CDD), abstractly D_4p, of order 4p. The gyro-p-gonal group, [2⁺,2p⁺] (Template:CDD or Template:CDD), 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] (Template:CDD), [3,4] (Template:CDD), [3,5] (Template:CDD) called full tetrahedral symmetry, octahedral symmetry, and icosahedral symmetry, with orders of 24, 48, and 120.

File:Pyritohedral in icosahedral symmetry.png

Pyritohedral symmetry, [3+,4] is an index 5 subgroup of icosahedral symmetry, [5,3].

In all these symmetries, alternate reflections can be removed producing the rotational tetrahedral [3,3]⁺(Template:CDD), octahedral [3,4]⁺ (Template:CDD), and icosahedral [3,5]⁺ (Template:CDD) groups of order 12, 24, and 60. The octahedral group also has a unique index 2 subgroup called the pyritohedral symmetry group, [3⁺,4] (Template:CDD or Template:CDD), of order 12, with a mixture of rotational and reflectional symmetry. Pyritohedral symmetry is also an index 5 subgroup of icosahedral symmetry: Template:CDD --> Template:CDD, with virtual mirror 1 across 0, {010}, and 3-fold rotation {12}.

The tetrahedral group, [3,3] (Template:CDD), has a doubling Template:Brackets (which can be represented by colored nodes Template:CDD), mapping the first and last mirrors onto each other, and this produces the [3,4] (Template:CDD or Template:CDD) group. The subgroup [3,4,1⁺] (Template:CDD or Template:CDD) is the same as [3,3], and [3⁺,4,1⁺] (Template:CDD or Template:CDD) 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 Template:CDD, Template:CDD, and Template:CDD, 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]].

Template:Brackets 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.

Semiaffine (frieze groups) IUC Orb. Geo Sch. Coxeter p1 ∞∞ pTemplate:Overline C∞ [∞] = [∞,1]+ = [∞+,2,1+] Template:CDD = Template:CDD p1m1 *∞∞ p1 C∞v [∞] = [∞,1] = [∞,2,1+] Template:CDD = Template:CDD p11g ∞× p.g1 S2∞ [∞+,2+] Template:CDD p11m ∞* p. 1 C∞h [∞+,2] Template:CDD p2 22∞ pTemplate:Overline D∞ [∞,2]+ Template:CDD p2mg 2*∞ p2g D∞d [∞,2+] Template:CDD p2mm *22∞ p2 D∞h [∞,2] Template:CDD

Affine (Wallpaper groups) IUC Orb. Geo. Coxeter p2 2222 pTemplate:Overline [4,1+,4]+ Template:CDD p2gg 22× pg2g [4+,4+] Template:CDD p2mm *2222 p2 [4,1+,4] Template:CDD Template:CDD c2mm 2*22 c2 Template:Brackets Template:CDD p4 442 pTemplate:Overline [4,4]+ Template:CDD p4gm 4*2 pg4 [4+,4] Template:CDD p4mm *442 p4 [4,4] Template:CDD p3 333 pTemplate:Overline [1+,6,3+] = [3[3]]+ Template:CDD = Template:CDD p3m1 *333 p3 [1+,6,3] = [3[3]] Template:CDD = Template:CDD p31m 3*3 h3 [6,3+] = [3[3[3]]+] Template:CDD p6 632 pTemplate:Overline [6,3]+ = [3[3[3]]]+ Template:CDD p6mm *632 p6 [6,3] = [3[3[3]]] Template:CDD

Given in Coxeter notation (orbifold notation), some low index affine subgroups are:

Reflective group	Reflective subgroup	Mixed subgroup	Rotation subgroup	Improper rotation/ translation	Commutator subgroup
[4,4], (*442)	[1+,4,4], (*442) [4,1+,4], (*2222) [1+,4,4,1+], (*2222)	[4+,4], (4*2) [(4,4,2+)], (2*22) [1+,4,1+,4], (2*22)	[4,4]+, (442) [1+,4,4+], (442) [1+,4,1+4,1+], (2222)	[4+,4+], (22×)	[4+,4+]+, (2222)
[6,3], (*632)	[1+,6,3] = [3[3]], (*333)	[3+,6], (3*3)	[6,3]+, (632) [1+,6,3+], (333)		[1+,6,3+], (333)

Rank four groups

File:Polychoral group tree.png Hasse diagram subgroup relations (partial!)

Point groups

Rank four groups defined the 4-dimensional point groups: Template:3-sphere symmetry groups2

Subgroups

1D-4D reflective point groups and subgroups
Order	Reflection		Semidirect subgroups				Direct subgroups		Commutator subgroup
2	[ ]	Template:CDD					[ ]+	Template:CDD	[ ]+1	[ ]+
4	[2]	Template:CDD					[2]+	Template:CDD	[2]+2
8	[2,2]	Template:CDD	[2+,2]	Template:CDD	[2+,2+]	Template:CDD	[2,2]+	Template:CDD	[2,2]+3
16	[2,2,2]	Template:CDD	[2+,2,2] [(2,2)+,2]	Template:CDD Template:CDD	[2+,2+,2] [(2,2)+,2+] [2+,2+,2+]	Template:CDD Template:CDD Template:CDD	[2,2,2]+ [2+,2,2+]	Template:CDD Template:CDD	[2,2,2]+4
	[21,1,1]	Template:CDD			[(2+)1,1,1]	Template:CDD
2n	[n]	Template:CDD					[n]+	Template:CDD	[n]+1	[n]+
4n	[2n]	Template:CDD					[2n]+	Template:CDD	[2n]+2
4n	[2,n]	Template:CDD	[2,n+]	Template:CDD			[2,n]+	Template:CDD	[2,n]+2
8n	[2,2n]	Template:CDD	[2+,2n]	Template:CDD	[2+,2n+]	Template:CDD	[2,2n]+	Template:CDD	[2,2n]+3
8n	[2,2,n]	Template:CDD	[2+,2,n] [2,2,n+]	Template:CDD Template:CDD	[2+,(2,n)+]	Template:CDD	[2,2,n]+ [2+,2,n+]	Template:CDD Template:CDD	[2,2,n]+3
16n	[2,2,2n]	Template:CDD	[2,2+,2n]	Template:CDD	[2+,2+,2n] [2,2+,2n+] [(2,2)+,2n+] [2+,2+,2n+]	Template:CDD Template:CDD Template:CDD Template:CDD	[2,2,2n]+ [2+,2n,2+]	Template:CDD Template:CDD	[2,2,2n]+4
	[2,2n,2]	Template:CDD			[2+,2n+,2+]	Template:CDD
	[2n,21,1]	Template:CDD			[2n+,(2+)1,1]	Template:CDD
24	[3,3]	Template:CDD					[3,3]+	Template:CDD	[3,3]+1	[3,3]+
48	[3,3,2]	Template:CDD	[(3,3)+,2]	Template:CDD			[3,3,2]+	Template:CDD	[3,3,2]+2
48	[4,3]	Template:CDD	[4,3+]	Template:CDD			[4,3]+	Template:CDD	[4,3]+2
96	[4,3,2]	Template:CDD	[(4,3)+,2] [4,(3,2)+]	Template:CDD Template:CDD			[4,3,2]+	Template:CDD	[4,3,2]+3
	[3,4,2]	Template:CDD	[3,4,2+] [3+,4,2]	Template:CDD Template:CDD	[(3,4)+,2+]	Template:CDD	[3+,4,2+]	Template:CDD
120	[5,3]	Template:CDD					[5,3]+	Template:CDD	[5,3]+1	[5,3]+
240	[5,3,2]	Template:CDD	[(5,3)+,2]	Template:CDD			[5,3,2]+	Template:CDD	[5,3,2]+2
4pq	[p,2,q]	Template:CDD	[p+,2,q]	Template:CDD			[p,2,q]+ [p+,2,q+]	Template:CDD Template:CDD	[p,2,q]+2	[p+,2,q+]
8pq	[2p,2,q]	Template:CDD	[2p,(2,q)+]	Template:CDD	[2p+,(2,q)+]	Template:CDD	[2p,2,q]+	Template:CDD	[2p,2,q]+3
16pq	[2p,2,2q]	Template:CDD	[2p,2+,2q]	Template:CDD	[2p+,2+,2q] [2p+,2+,2q+] [(2p,(2,2q)+,2+)]	Template:CDD Template:CDD -	[2p,2,2q]+	Template:CDD	[2p,2,2q]+4
120	[3,3,3]	Template:CDD					[3,3,3]+	Template:CDD	[3,3,3]+1	[3,3,3]+
192	[31,1,1]	Template:CDD					[31,1,1]+	Template:CDD	[31,1,1]+1	[31,1,1]+
384	[4,3,3]	Template:CDD	[4,(3,3)+]	Template:CDD			[4,3,3]+	Template:CDD	[4,3,3]+2
1152	[3,4,3]	Template:CDD	[3+,4,3]	Template:CDD			[3,4,3]+ [3+,4,3+]	Template:CDD Template:CDD	[3,4,3]+2	[3+,4,3+]
14400	[5,3,3]	Template:CDD					[5,3,3]+	Template:CDD	[5,3,3]+1	[5,3,3]+

Space groups

Space groups
File:Coxeter diagram affine rank4 correspondence.png Affine isomorphism and correspondences	File:Cubic space subgroup tree coxeter notation.png 8 cubic space groups as extended symmetry from [3[4]], with square Coxeter diagrams and reflective fundamental domains	File:35 cubic fibrifold groups.png 35 cubic space groups in International, Fibrifold notation, and Coxeter notation

Line groups

Rank four groups also defined the 3-dimensional line groups:

Semiaffine (3D) groups
Point group			Line group
Hermann-Mauguin		Schönflies	Hermann-Mauguin		Offset type	Wallpaper			Coxeter [∞h,2,pv]
Even n	Odd n		Even n	Odd n		IUC	Orbifold	Diagram
n		Cn	Pnq		Helical: q	p1	o		[∞+,2,n+]	Template:CDD
Template:Overline	Template:Overline	S2n	PTemplate:Overline	PTemplate:Overline	None	p11g, pg(h)	××		[(∞,2)+,2n+]	Template:CDD
n/m	Template:Overline	Cnh	Pn/m	PTemplate:Overline	None	p11m, pm(h)	**	File:Wallpaper group diagram pm.svg	[∞+,2,n]	Template:CDD
2n/m		C2nh	P2nn/m		Zigzag	c11m, cm(h)	*×		[∞+,2+,2n]	Template:CDD
nmm	nm	Cnv	Pnmm	Pnm	None	p1m1, pm(v)	**	Error creating thumbnail:	[∞,2,n+]	Template:CDD
			Pncc	Pnc	Planar reflection	p1g1, pg(v)	××	File:Wallpaper group diagram pg rotated.svg	[∞+,(2,n)+]	Template:CDD
2nmm		C2nv	P2nnmc		Zigzag	c1m1, cm(v)	*×	Error creating thumbnail:	[∞,2+,2n+]	Template:CDD
n22	n2	Dn	Pnq22	Pnq2	Helical: q	p2	2222	File:Wallpaper group diagram p2.svg	[∞,2,n]+	Template:CDD
Template:Overline2m	Template:Overlinem	Dnd	PTemplate:Overline2m	PTemplate:Overlinem	None	p2mg, pmg(h)	22*	Error creating thumbnail:	[(∞,2)+,2n]	Template:CDD
			PTemplate:Overline2c	PTemplate:Overlinec	Planar reflection	p2gg, pgg	22×	File:Wallpaper group diagram pgg rhombic.svg	[+(∞,(2),2n)+]
n/mmm	Template:Overline2m	Dnh	Pn/mmm	PTemplate:Overline2m	None	p2mm, pmm	*2222	Error creating thumbnail:	[∞,2,n]	Template:CDD
			Pn/mcc	PTemplate:Overline2c	Planar reflection	p2mg, pmg(v)	22*	Error creating thumbnail:	[∞,(2,n)+]	Template:CDD
2n/mmm		D2nh	P2nn/mcm		Zigzag	c2mm, cmm	2*22	File:Wallpaper group diagram cmm.svg	[∞,2+,2n]	Template:CDD

Duoprismatic group

Extended duoprismatic symmetry
Error creating thumbnail:
Extended duoprismatic groups, [p]×[p] or [p,2,p] or Template:CDD, expressed in relation to its tetragonal disphenoid fundamental domain symmetry.

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.

Duoprismatic groups (4D)
Wallpaper			Coxeter [p,2,q] Template:CDD	Coxeter Template:Brackets Template:CDD	Wallpaper
IUC	Orbifold	Diagram			IUC	Orbifold	Diagram
p1	o		[p+,2,q+]	Template:Brackets	p1	o	File:Wallpaper group diagram p1 rhombic.svg
pg	××		[(p,2)+,2q+]	-
pm	**	File:Wallpaper group diagram pm.svg	[p+,2,q]	-
cm	*×	Error creating thumbnail:	[2p+,2+,2q]	-
p2	2222	File:Wallpaper group diagram p2 rect.svg	[p,2,q]+	Template:Brackets+	p4	442	File:Wallpaper group diagram p4 square.svg
pmg	22*	Error creating thumbnail:	[(p,2)+,2q]	-
pgg	22×	File:Wallpaper group diagram pgg rhombic.svg	[+(2p,(2),2q)+]	Template:Brackets	cmm	2*22	File:Wallpaper group diagram cmm square.svg
pmm	*2222	Error creating thumbnail:	[p,2,q]	Template:Brackets	p4m	*442	Error creating thumbnail:
cmm	2*22	File:Wallpaper group diagram cmm.svg	[2p,2+,2q]	Template:Brackets	p4g	4*2	File:Wallpaper group diagram p4g square.svg

Wallpaper groups

Rank four groups also defined some of the 2-dimensional wallpaper groups, as limiting cases of the four-dimensional duoprism groups:

Affine (2D plane)

IUC Orb. Geo Coxeter Diagram p1 o pTemplate:Overline [∞+,2,∞+] Template:CDD Error creating thumbnail: [∞+,2+,∞+] Template:CDD File:Wallpaper group diagram p1 rhombic.svg [(∞+,2+,∞+,2+)] Template:CDD Error creating thumbnail: p2 2222 pTemplate:Overline [∞,2,∞]+ Template:CDD File:Wallpaper group diagram p2 rect.svg [(∞,2+,∞,2+)] Template:CDD p11g ×× pg1 h: [∞+,(2,∞)+] Template:CDD p1g1 v: [(∞,2)+,∞+] Template:CDD File:Wallpaper group diagram pg rotated.svg p2gm 22* pg2 h: [(∞,2)+,∞] Template:CDD p2mg v: [∞,(2,∞)+] Template:CDD

IUC Orb. Geo Coxeter Diagram p11m ** p1 h: [∞+,2,∞] Template:CDD File:Wallpaper group diagram pm.svg p1m1 v: [∞,2,∞+] Template:CDD Error creating thumbnail: p2mm *2222 p2 [∞,2,∞] Template:CDD Error creating thumbnail: c11m *× c1 h: [∞+,2+,∞] Template:CDD c1m1 v: [∞,2+,∞+] Template:CDD p2gg 22× pg2g [+(∞,(2),∞)+] [((∞,2)+)[2]] Template:CDD File:Wallpaper group diagram pgg rhombic.svg c2mm 2*22 c2 [∞,2+,∞] Template:CDD File:Wallpaper group diagram cmm.svg

Subgroups of [∞,2,∞], (*2222) can be expressed down to its index 16 commutator subgroup:

Subgroups of [∞,2,∞]
Reflective group	Reflective subgroup	Mixed subgroup	Rotation subgroup	Improper rotation/ translation	Commutator subgroup
[∞,2,∞], (*2222)	[1+,∞,2,∞], (*2222)	[∞+,2,∞], (**)	[∞,2,∞]+, (2222)	[∞,2+,∞]+, (°) [∞+,2+,∞+], (°) [∞+,2,∞+], (°) [∞+,2+,∞], (*×) [(∞,2)+,∞+], (××) [((∞,2)+,(∞,2)+)], (22×)	[(∞+,2+,∞+,2+)], (°)
		[∞,2+,∞], (2*22) [(∞,2)+,∞], (22*)

Complex reflections

File:Rank2 shephard subgroups3.png

Hasse diagram with all subgroup relations on rank 2 Shephard groups.

Coxeter notation has been extended to Complex space, Cⁿ where nodes are unitary reflections of period 2 or greater. 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 ${ℂ}^1$, a rank 1 Shephard group Template:CDD, order p, is represented as _p[ ], [ ]_p or ]_p[. It has a single generator, representing a 2π/p radian rotation in the Complex plane: $e^{2\pi i/p}$.

Coxeter writes the rank 2 complex group, _p[q]_r represents Coxeter diagram Template:CDD. 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 $g=8/q(1/p+2/q+1/r-1{)}^{-2}$.

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 Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD.

Some subgroup relations among infinite Shephard groups

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 Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD.

Index 2 subgroups exists by removing a real reflection: _p[2q]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. Template:Clear

Computation with reflection matrices as symmetry generators

A Coxeter group, represented by Coxeter diagram Template:CDD, 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: ρ₀, ρ₁, and ρ₂. Rotational subsymmetry is given as products of reflections: By convention, σ_0,1 (and matrix S_0,1) = ρ₀ρ₁ represents a rotation of angle π/p, and σ_1,2 = ρ₁ρ₂ is a rotation of angle π/q, and σ_0,2 = ρ₀ρ₂ represents a rotation of angle π/2.

[p,q]⁺, Template:CDD, 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], Template:CDD or Template:CDD, is another subgroup of index 2, represented by rotation generator σ_0,1, and reflectional ρ₂.

With even branches, [2p⁺,2q⁺], Template:CDD, 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 Template:CDD, or Template:CDD, 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 Template:CDD: [4⁺,4⁺] = Template:CDD.

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] (Template:CDD), 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 Dih₁=Z₂.

Rank 2

Dihedral groups	Cyclic groups
[2]	Error creating thumbnail: [2]+
File:Dihedral symmetry domains 3.png [3]	File:Cyclic symmetry 3.png [3]+
Error creating thumbnail: [4]	File:Cyclic symmetry 4.png [4]+
File:Dihedral symmetry domains 6.png [6]	[6]+

Example, in 2D, the Coxeter group [p] (Template:CDD) is represented by two reflection matrices R₀ and R₁, The cyclic symmetry [p]⁺ (Template:CDD) is represented by rotation generator of matrix S_0,1.

[p], Template:CDD Reflections Rotation Name R0 Template:CDD R1 Template:CDD S0,1=R0×R1 Template:CDD Order 2 2 p Matrix $\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$ $\left[\begin{array}{cc}\cos 2\pi /p& \sin 2\pi /p\\ \sin 2\pi /p& -\cos 2\pi /p\end{array}\right]$ $\left[\begin{array}{cc}\cos 2\pi /p& \sin 2\pi /p\\ -\sin 2\pi /p& \cos 2\pi /p\end{array}\right]$	[2], Template:CDD Reflections Rotation Name R0 Template:CDD R1 Template:CDD S0,1=R0×R1 Template:CDD Order 2 2 2 Matrix $\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$ $\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]$ $\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$
[3], Template:CDD Reflections Rotation Name R0 Template:CDD R1 Template:CDD S0,1=R0×R1 Template:CDD Order 2 2 3 Matrix $\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$ $\left[\begin{array}{cc}1/2& \sqrt{3}/2\\ \sqrt{3}/2& -1/2\end{array}\right]$ $\left[\begin{array}{cc}1/2& \sqrt{3}/2\\ -\sqrt{3}/2& 1/2\end{array}\right]$	[4], Template:CDD Reflections Rotation Name R0 Template:CDD R1 Template:CDD S0,1=R0×R1 Template:CDD Order 2 2 4 Matrix $\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$ $\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$ $\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$
[6], Template:CDD Reflections Rotation Name R0 Template:CDD R1 Template:CDD S0,1=R0×R1 Template:CDD Order 2 2 6 Matrix $\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$ $\left[\begin{array}{cc}\sqrt{3}/2& 1/2\\ 1/2& -\sqrt{3}/2\end{array}\right]$ $\left[\begin{array}{cc}\sqrt{3}/2& 1/2\\ -1/2& \sqrt{3}/2\end{array}\right]$	[8], Template:CDD Reflections Rotation Name R0 Template:CDD R1 Template:CDD S0,1=R0×R1 Template:CDD Order 2 2 8 Matrix $\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$ $\left[\begin{array}{cc}\sqrt{2}/2& \sqrt{2}/2\\ \sqrt{2}/2& -\sqrt{2}/2\end{array}\right]$ $\left[\begin{array}{cc}\sqrt{2}/2& \sqrt{2}/2\\ -\sqrt{2}/2& \sqrt{2}/2\end{array}\right]$

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

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

[p,2]

File:Spherical decagonal bipyramid.svg

Example fundamental domains, [5,2], as spherical triangles

The reducible 3-dimensional finite reflective group is dihedral symmetry, [p,2], order 4p, Template:CDD. The reflection generators are matrices R₀, R₁, R₂. R₀²=R₁²=R₂²=(R₀×R₁)³=(R₁×R₂)³=(R₀×R₂)²=Identity. [p,2]⁺ (Template:CDD) 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.

[p,2], Template:CDD

	Reflections			Rotation			Rotoreflection
Name	R0	R1	R2	S0,1	S1,2	S0,2	V0,1,2
Group	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD
Order	2	2	2	p	2		2p
Matrix	$\left[\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc}\cos 2\pi /p& \sin 2\pi /p& 0\\ \sin 2\pi /p& -\cos 2\pi /p& 0\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& -1\end{array}\right]$	$\left[\begin{array}{ccc}\cos 2\pi /p& \sin 2\pi /p& 0\\ -\sin 2\pi /p& \cos 2\pi /p& 0\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc}\cos 2\pi /p& \sin 2\pi /p& 0\\ -\sin 2\pi /p& \cos 2\pi /p& 0\\ 0& 0& -1\end{array}\right]$	$\left[\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right]$	$\left[\begin{array}{ccc}\cos 2\pi /p& -\sin 2\pi /p& 0\\ -\sin 2\pi /p& -\cos 2\pi /p& 0\\ 0& 0& 1\end{array}\right]$

[3,3]

File:Sphere symmetry group td.png

reflection lines for [3,3] = Template:CDD

The simplest irreducible 3-dimensional finite reflective group is tetrahedral symmetry, [3,3], order 24, Template:CDD. 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]⁺ (Template:CDD) 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 R₁. An order 4 rotoreflection is generated by V_0,1,2, the product of all 3 reflections.

[3,3], Template:CDD

	Reflections			Rotations			Rotoreflection
Name	R0	R1	R2	S0,1	S1,2	S0,2	V0,1,2
Name	Template:CDD	Template:CDD	Template:CDD	Template:CDD		Template:CDD
Order	2	2	2	3		2	4
Matrix	$\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right]$	$\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& -1\\ 0& -1& 0\end{array}\right]$	$\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right]$	$\left[\begin{array}{ccc}0& 0& -1\\ 1& 0& 0\\ 0& -1& 0\end{array}\right]$	$\left[\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right]$	$\left[\begin{array}{ccc}0& 0& -1\\ 0& -1& 0\\ 1& 0& 0\end{array}\right]$
	(0,1,−1)n	(1,−1,0)n	(0,1,1)n	(1,1,1)axis	(1,1,−1)axis	(1,0,0)axis

[4,3]

Reflection lines for [4,3] = Template:CDD

Another irreducible 3-dimensional finite reflective group is octahedral symmetry, [4,3], order 48, Template:CDD. 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]⁺, (Template:CDD) is generated by 2 of 3 rotations: S_0,1, S_1,2, and S_0,2. Pyritohedral symmetry [4,3⁺], (Template:CDD) is generated by reflection R₀ and rotation S_1,2. A 6-fold rotoreflection is generated by V_0,1,2, the product of all 3 reflections.

[4,3], Template:CDD

	Reflections			Rotations			Rotoreflection
Name	R0	R1	R2	S0,1	S1,2	S0,2	V0,1,2
Group	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD
Order	2	2	2	4	3	2	6
Matrix	$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& -1\end{array}\right]$	$\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right]$	$\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& -1& 0\end{array}\right]$	$\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right]$	$\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& -1\end{array}\right]$	$\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -1& 0& 0\end{array}\right]$
	(0,0,1)n	(0,1,−1)n	(1,−1,0)n	(1,0,0)axis	(1,1,1)axis	(1,−1,0)axis

[5,3]

File:Sphere symmetry group ih.png

Reflection lines for [5,3] = Template:CDD

A final irreducible 3-dimensional finite reflective group is icosahedral symmetry, [5,3], order 120, Template:CDD. The reflection generators matrices are R₀, R₁, R₂. R₀²=R₁²=R₂²=(R₀×R₁)⁵=(R₁×R₂)³=(R₀×R₂)²=Identity. [5,3]⁺ (Template:CDD) is generated by 2 of 3 rotations: S_0,1, S_1,2, and S_0,2. A 10-fold rotoreflection is generated by V_0,1,2, the product of all 3 reflections.

[5,3], Template:CDD

	Reflections			Rotations			Rotoreflection
Name	R0	R1	R2	S0,1	S1,2	S0,2	V0,1,2
Group	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD
Order	2	2	2	5	3	2	10
Matrix	$\left[\begin{array}{ccc}-1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc}\frac{1-\varphi }{2}& \frac{-\varphi }{2}& \frac{-1}{2}\\ \frac{-\varphi }{2}& \frac{1}{2}& \frac{1-\varphi }{2}\\ \frac{-1}{2}& \frac{1-\varphi }{2}& \frac{\varphi }{2}\end{array}\right]$	$\left[\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc}\frac{\varphi -1}{2}& \frac{\varphi }{2}& \frac{1}{2}\\ \frac{-\varphi }{2}& \frac{1}{2}& \frac{1-\varphi }{2}\\ \frac{-1}{2}& \frac{1-\varphi }{2}& \frac{\varphi }{2}\end{array}\right]$	$\left[\begin{array}{ccc}\frac{1-\varphi }{2}& \frac{\varphi }{2}& \frac{-1}{2}\\ \frac{-\varphi }{2}& \frac{-1}{2}& \frac{1-\varphi }{2}\\ \frac{-1}{2}& \frac{\varphi -1}{2}& \frac{\varphi }{2}\end{array}\right]$	$\left[\begin{array}{ccc}-1& 0& 0\\ 0& -1& 0\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc}\frac{\varphi -1}{2}& \frac{-\varphi }{2}& \frac{1}{2}\\ \frac{-\varphi }{2}& \frac{-1}{2}& \frac{1-\varphi }{2}\\ \frac{-1}{2}& \frac{\varphi -1}{2}& \frac{\varphi }{2}\end{array}\right]$
	(1,0,0)n	(φ,1,φ−1)n	(0,1,0)n	(φ,1,0)axis	(1,1,1)axis	(1,0,0)axis

Rank 4

Template:See There are 4 irreducible Coxeter groups in 4 dimensions: [3,3,3], [4,3,3], [3^1,1,1], [3,4,4], [5,3,3], as well as an infinite family of duoprismatic groups [p,2,q].

[p,2,q]

The duprismatic group, [p,2,q], has order 4pq.

[p,2,q], Template:CDD

	Reflections
Name	R0	R1	R2	R3
Group element	Template:CDD	Template:CDD	Template:CDD	Template:CDD
Order	2	2	2	2
Matrix	$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}\cos 2\pi /p& \sin 2\pi /p& 0& 0\\ \sin 2\pi /p& -\cos 2\pi /p& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& -1\end{array}\right]$	$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& \cos 2\pi /q& \sin 2\pi /q\\ 0& 0& \sin 2\pi /q& -\cos 2\pi /q\end{array}\right]$

[[p,2,p]]

The duoprismatic group can double in order, to 8p², with a 2-fold rotation between the two planes.

[[p,2,p]], Template:CDD

	Rotation	Reflections
Name	T	R0	R1	R2=TR1T	R3=TR0T
Element	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD
Order	2	2	2
Matrix	$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}\cos 2\pi /p& \sin 2\pi /p& 0& 0\\ \sin 2\pi /p& -\cos 2\pi /p& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}0& 0& 0& -1\\ 0& 0& -1& 0\\ 0& -1& 0& 0\\ -1& 0& 0& 0\end{array}\right]$	$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& -1\end{array}\right]$	$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& \cos 2\pi /p& \sin 2\pi /p\\ 0& 0& \sin 2\pi /p& -\cos 2\pi /p\end{array}\right]$

[3,3,3]

Hypertetrahedral symmetry, [3,3,3], order 120, is easiest to represent with 4 mirrors in 5-dimensions, as a subgroup of [4,3,3,3].

[3,3,3], Template:CDD

	Reflections				Rotations						Rotoreflections		Double rotation
Name	R0	R1	R2	R3	S0,1	S1,2	S2,3	S0,2	S1,3	S2,3	V0,1,2	V0,1,3	W0,1,2,3
Element group	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD			Template:CDD			Template:CDD	Template:CDD	Template:CDD
Order	2	2	2	2	3			2			4	6	5
Matrix	$\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1\\ 0& 0& 0& 1& 0\end{array}\right]$	$\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccccc}0& 1& 0& 0& 0\\ 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\\ 0& 0& 1& 0& 0\end{array}\right]$	$\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 1\\ 0& 0& 0& 1& 0\end{array}\right]$	$\left[\begin{array}{ccccc}0& 1& 0& 0& 0\\ 1& 0& 0& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccccc}0& 1& 0& 0& 0\\ 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1\\ 0& 0& 0& 1& 0\end{array}\right]$	$\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\\ 0& 1& 0& 0& 0\end{array}\right]$	$\left[\begin{array}{ccccc}0& 1& 0& 0& 0\\ 1& 0& 0& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\\ 0& 0& 1& 0& 0\end{array}\right]$	$\left[\begin{array}{ccccc}0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\\ 1& 0& 0& 0& 0\end{array}\right]$
	(0,0,0,1,-1)n	(0,0,1,−1,0)n	(0,1,−1,0,0)n	(1,−1,0,0,0)n

[[3,3,3]]

The extended group [[3,3,3]], order 240, is doubled by a 2-fold rotation matrix T, here reversing coordinate order and sign: There are 3 generators {T, R₀, R₁}. Since T is self-reciprocal R₃=TR₀T, and R₂=TR₁T.

[[3,3,3]], Template:CDD

	Rotation	Reflections
Name	T	R0	R1	TR1T=R2	TR0T=R3
Element group	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD
Order	2	2	2	2	2
Matrix	$\left[\begin{array}{ccccc}0& 0& 0& 0& -1\\ 0& 0& 0& -1& 0\\ 0& 0& -1& 0& 0\\ 0& -1& 0& 0& 0\\ -1& 0& 0& 0& 0\end{array}\right]$	$\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1\\ 0& 0& 0& 1& 0\end{array}\right]$	$\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccccc}0& 1& 0& 0& 0\\ 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]$
		(0,0,0,1,-1)n	(0,0,1,−1,0)n	(0,1,−1,0,0)n	(1,−1,0,0,0)n

[4,3,3]

A irreducible 4-dimensional finite reflective group is hyperoctahedral group (or hexadecachoric group (for 16-cell), B₄=[4,3,3], order 384, Template:CDD. 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]⁺, (Template:CDD) 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)⁺], (Template:CDD) is generated by reflection R₀ and rotations S_1,2 and S_2,3. An 8-fold double rotation is generated by W_0,1,2,3, the product of all 4 reflections.

[4,3,3], Template:CDD

	Reflections				Rotations						Rotoreflection				Double rotation
Name	R0	R1	R2	R3	S0,1	S1,2	S2,3	S0,2	S1,3	S0,3	V1,2,3	V0,1,3	V0,1,2	V0,2,3	W0,1,2,3
Group	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD		Template:CDD			Template:CDD		Template:CDD		Template:CDD
Order	2	2	2	2	4	3		2			4		6		8
Matrix	$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& -1\end{array}\right]$	$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right]$	$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\\ 0& 0& -1& 0\end{array}\right]$	$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ 0& 1& 0& 0\end{array}\right]$	$\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ 0& 0& 0& -1\end{array}\right]$	$\left[\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right]$	$\left[\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& -1\end{array}\right]$	$\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ 1& 0& 0& 0\end{array}\right]$	$\left[\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& -1& 0\end{array}\right]$	$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ 0& 0& -1& 0\end{array}\right]$	$\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 1& 0& 0& 0\\ 0& 0& 0& -1\end{array}\right]$	$\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ -1& 0& 0& 0\end{array}\right]$
	(0,0,0,1)n	(0,0,1,−1)n	(0,1,−1,0)n	(1,−1,0,0)n

[3,3^1,1]

A half group of [4,3,3] is [3,3^1,1], Template:CDD, order 192. It shares 3 generators with [4,3,3] group, but has two copies of an adjacent generator, one reflected across the removed mirror.

[3,3^1,1], Template:CDD

	Reflections
Name	R0	R1	R2	R3
Group	Template:CDD	Template:CDD	Template:CDD	Template:CDD
Order	2	2	2	2
Matrix	$\left[\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right]$	$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& -1\\ 0& 0& -1& 0\end{array}\right]$
	(1,−1,0,0)n	(0,1,−1,0)n	(0,0,1,−1)n	(0,0,1,1)n

[3,4,3]

A irreducible 4-dimensional finite reflective group is Icositetrachoric group (for 24-cell), F₄=[3,4,3], order 1152, Template:CDD. 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]⁺, (Template:CDD) 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, (Template:CDD) is generated by reflection R₀ and rotations S_1,2 and S_2,3. A 12-fold double rotation is generated by W_0,1,2,3, the product of all 4 reflections.

[3,4,3], Template:CDD

	Reflections				Rotations
Name	R0	R1	R2	R3	S0,1	S1,2	S2,3	S0,2	S1,3	S0,3
Element group	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD		Template:CDD
Order	2	2	2	2	3	4	3	2
Matrix	$\left[\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}1/2& -1/2& -1/2& -1/2\\ -1/2& 1/2& -1/2& -1/2\\ -1/2& -1/2& 1/2& -1/2\\ -1/2& -1/2& -1/2& 1/2\end{array}\right]$	$\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& -1& 0& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}1/2& -1/2& 1/2& -1/2\\ -1/2& 1/2& 1/2& -1/2\\ -1/2& -1/2& -1/2& -1/2\\ -1/2& -1/2& 1/2& 1/2\end{array}\right]$	$\left[\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}1/2& -1/2& -1/2& -1/2\\ -1/2& -1/2& 1/2& -1/2\\ -1/2& 1/2& -1/2& -1/2\\ -1/2& -1/2& -1/2& 1/2\end{array}\right]$	$\left[\begin{array}{cccc}-1/2& 1/2& -1/2& -1/2\\ 1/2& -1/2& -1/2& -1/2\\ -1/2& -1/2& 1/2& -1/2\\ -1/2& 1/2& -1/2& 1/2\end{array}\right]$
	(1,−1,0,0)n	(0,1,−1,0)n	(0,0,1,0)n	(−1,−1,−1,−1)n

[3,4,3], Template:CDD

	Rotoreflection				Double rotation
Name	V1,2,3	V0,1,3	V0,1,2	V0,2,3	W0,1,2,3
Element group	Template:CDD				Template:CDD
Order	6				12
Matrix	$\left[\begin{array}{cccc}-1/2& 1/2& -1/2& -1/2\\ -1/2& -1/2& 1/2& -1/2\\ 1/2& -1/2& -1/2& -1/2\\ -1/2& -1/2& -1/2& 1/2\end{array}\right]$	$\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ -1& 0& 0& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}1/2& 1/2& -1/2& -1/2\\ -1/2& 1/2& 1/2& -1/2\\ -1/2& -1/2& -1/2& -1/2\\ -1/2& 1/2& -1/2& 1/2\end{array}\right]$	$\left[\begin{array}{cccc}-1/2& 1/2& 1/2& -1/2\\ 1/2& -1/2& 1/2& -1/2\\ -1/2& -1/2& -1/2& -1/2\\ -1/2& -1/2& 1/2& 1/2\end{array}\right]$	$\left[\begin{array}{cccc}1/2& 1/2& -1/2& -1/2\\ 1/2& -1/2& 1/2& -1/2\\ -1/2& -1/2& -1/2& -1/2\\ 1/2& -1/2& -1/2& 1/2\end{array}\right]$

[[3,4,3]]

The group [[3,4,3]] extends [3,4,3] by a 2-fold rotation, T, doubling order to 2304.

[[3,4,3]], Template:CDD

	Rotation	Reflections
Name	T	R0	R1	R2 = TR1T	R3 = TR0T
Element group	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD
Order	2	2	2	2	2
Matrix		$\left[\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}1/2& -1/2& -1/2& -1/2\\ -1/2& 1/2& -1/2& -1/2\\ -1/2& -1/2& 1/2& -1/2\\ -1/2& -1/2& -1/2& 1/2\end{array}\right]$
		(1,−1,0,0)n	(0,1,−1,0)n	(0,0,1,0)n	(−1,−1,−1,−1)n

[5,3,3]

Stereographic projections

Error creating thumbnail: [5,3,3]+ 72 order-5 gyrations	File:Coxeter 533 order-3 gyration axes.png [5,3,3]+ 200 order-3 gyrations
File:Coxeter 533 order-2 gyration axes.png [5,3,3]+ 450 order-2 gyrations	[5,3,3]+ all gyrations

The hyper-icosahedral symmetry, [5,3,3], order 14400, Template:CDD. 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]⁺ (Template:CDD) is generated by 3 rotations: S_0,1 = R₀×R₁, S_1,2 = R₁×R₂, S_2,3 = R₂×R₃, etc.

[5,3,3], Template:CDD

	Reflections
Name	R0	R1	R2	R3
Element group	Template:CDD	Template:CDD	Template:CDD	Template:CDD
Order	2	2	2	2
Matrix	$\left[\begin{array}{cccc}-1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}\frac{1-\varphi }{2}& \frac{-\varphi }{2}& \frac{-1}{2}& 0\\ \frac{-\varphi }{2}& \frac{1}{2}& \frac{1-\varphi }{2}& 0\\ \frac{-1}{2}& \frac{1-\varphi }{2}& \frac{\varphi }{2}& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \frac{1}{2}& \frac{\varphi }{2}& \frac{1-\varphi }{2}\\ 0& \frac{\varphi }{2}& \frac{1-\varphi }{2}& \frac{1}{2}\\ 0& \frac{1-\varphi }{2}& \frac{1}{2}& \frac{\varphi }{2}\end{array}\right]$
	(1,0,0,0)n	(φ,1,φ−1,0)n	(0,1,0,0)n	(0,−1,φ,1−φ)n

Rank 8

[3^4,2,1]

The E8 Coxeter group, [3^4,2,1], Template:CDD, has 8 mirror nodes, order 696729600 (192x10!). E7 and E6, [3^3,2,1], Template:CDD, and [3^2,2,1], Template:CDD can be constructed by ignoring the first mirror or the first two mirrors respectively.

E8=[3^4,2,1], Template:CDD

	Reflections
Name	R0	R1	R2	R3	R4	R5	R6	R7
Element group	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD
Order	2	2	2	2	2	2	2	2
Matrix	$\left[\begin{array}{cccccccc}0& 1& 0& 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccccccc}1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccccccc}1& 0& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccccccc}1& 0& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccccccc}1& 0& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccccccc}1& 0& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccccccc}1& 0& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& -1& 0\\ 0& 0& 0& 0& 0& -1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccccccc}3/4& -1/4& -1/4& -1/4& -1/4& -1/4& -1/4& -1/4\\ -1/4& 3/4& -1/4& -1/4& -1/4& -1/4& -1/4& -1/4\\ -1/4& -1/4& 3/4& -1/4& -1/4& -1/4& -1/4& -1/4\\ -1/4& -1/4& -1/4& 3/4& -1/4& -1/4& -1/4& -1/4\\ -1/4& -1/4& -1/4& -1/4& 3/4& -1/4& -1/4& -1/4\\ -1/4& -1/4& -1/4& -1/4& -1/4& 3/4& -1/4& -1/4\\ -1/4& -1/4& -1/4& -1/4& -1/4& -1/4& 3/4& -1/4\\ -1/4& -1/4& -1/4& -1/4& -1/4& -1/4& -1/4& 3/4\end{array}\right]$
	(1,-1,0,0,0,0,0,0)n	(0,1,-1,0,0,0,0,0)n	(0,0,1,-1,0,0,0,0)n	(0,0,0,1,-1,0,0,0)n	(0,0,0,0,1,-1,0,0)n	(0,0,0,0,0,1,-1,0)n	(0,0,0,0,0,1,1,0)n	(1,1,1,1,1,1,1,1)n

Affine rank 2

Affine matrices are represented by adding an extra row and column, the last row being zero except last entry 1. The last column represents a translation vector.

[∞]

The affine group [∞], Template:CDD, can be given by two reflection matrices, x=0 and x=1.

[∞], Template:CDD

	Reflections		Translation
Name	R0	R1	S0,1
Element group	Template:CDD	Template:CDD	Template:CDD
Order	2	2	∞
Matrix	$\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]$	$\left[\begin{array}{cc}-1& 1\\ 0& 1\end{array}\right]$	$\left[\begin{array}{cc}1& -1\\ 0& 1\end{array}\right]$
Hyperplane	x=0	x=1

Affine rank 3

[4,4]

The affine group [4,4], Template:CDD, (p4m), can be given by three reflection matrices, reflections across the x axis (y=0), a diagonal (x=y), and the affine reflection across the line (x=1). [4,4]⁺ (Template:CDD) (p4) is generated by S_0,1 S_1,2, and S_0,2. [4⁺,4⁺] (Template:CDD) (pgg) is generated by 2-fold rotation S_0,2 and glide reflection (transreflection) V_0,1,2. [4⁺,4] (Template:CDD) (p4g) is generated by S_0,1 and R₃. The group [(4,4,2⁺)] (Template:CDD) (cmm), is generated by 2-fold rotation S_1,3 and reflection R₂.

[4,4], Template:CDD

	Reflections			Rotations			Glides
Name	R0	R1	R2	S0,1	S1,2	S0,2	V0,1,2	V0,2,1
Element group	Template:CDD	Template:CDD	Template:CDD	Template:CDD		Template:CDD	Template:CDD
Order	2	2	2	4		2	∞ (2)
Matrix	$\left[\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc}-1& 0& 2\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc}0& 1& 0\\ -1& 0& 0\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc}0& 1& 0\\ -1& 0& 2\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc}-1& 0& 2\\ 0& -1& 0\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& -2\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc}0& -1& 2\\ -1& 0& 0\\ 0& 0& 1\end{array}\right]$
Hyperplane	y=0	x=y	x=1

[3,6]

The affine group [3,6], Template:CDD, (p6m), can be given by three reflection matrices, reflections across the x axis (y=0), line y=(√3/2)x, and vertical line x=1.

[3,6], Template:CDD

	Reflections			Rotations			Glides
Name	R0	R1	R2	S0,1	S1,2	S0,2	V0,1,2	V0,2,1
Element group	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD
Order	2	2	2	3	6	2	∞ (2)
Matrix	$\left[\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc}-1/2& \sqrt{3}/2& 0\\ \sqrt{3}/2& 1/2& 0\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc}-1& 0& 2\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc}-1/2& \sqrt{3}/2& 0\\ -\sqrt{3}/2& -1/2& 0\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc}1/2& \sqrt{3}/2& -1\\ -\sqrt{3}/2& 1/2& \sqrt{3}\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc}-1& 0& 2\\ 0& -1& 0\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc}1/2& \sqrt{3}/2& -1\\ \sqrt{3}/2& -1/2& -\sqrt{3}\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc}1/2& -\sqrt{3}/2& 2\\ -\sqrt{3}/2& -1/2& 0\\ 0& 0& 1\end{array}\right]$
Hyperplane	y=0	y=(√3/2)x	x=1

[3^[3]]

The affine group [3^[3]] can be constructed as a half group of Template:CDD. R₂ is replaced by R'₂ = R₂×R₁×R₂, presented by the hyperplane: y+(√3/2)x=2. The fundamental domain is an equilateral triangle with edge length 2.

[3^[3]], Template:CDD

	Reflections			Rotations			Glides
Name	R0	R1	R'2 = R2×R1×R2	S0,1	S1,2	S0,2	V0,1,2	V0,2,1
Element group	Template:CDD	Template:CDD	Template:CDD	Template:CDD			Template:CDD
Order	2	2	2	3			∞ (2)
Matrix	$\left[\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc}-1/2& \sqrt{3}/2& 0\\ \sqrt{3}/2& 1/2& 0\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc}-1/2& -\sqrt{3}/2& 3\\ -\sqrt{3}/2& 1/2& \sqrt{3}\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc}0& 1& 0\\ -1& 0& 0\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc}-1/2& \sqrt{3}/2& 0\\ -\sqrt{3}/2& -1/2& 2\sqrt{3}\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc}-1/2& -\sqrt{3}/2& 3\\ \sqrt{3}/2& -1/2& -\sqrt{3}\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc}-1/2& \sqrt{3}/2& 0\\ \sqrt{3}/2& 1/2& -2\sqrt{3}\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc}-1/2& -\sqrt{3}/2& 3\\ -\sqrt{3}/2& 1/2& -\sqrt{3}\\ 0& 0& 1\end{array}\right]$
Hyperplane	y=0	y=(√3/2)x	y+(√3/2)x=2

Affine rank 4

[4,3,4]

Error creating thumbnail:

[4,3,4] fundamental domain

The affine group is [4,3,4] (Template:CDD), can be given by four reflection matrices. Mirror R₀ can be put on z=0 plane. Mirror R₁ can be put on plane y=z. Mirror R₂ can be put on x=y plane. Mirror R₃ can be put on x=1 plane. [4,3,4]⁺ (Template:CDD) is generated by S_0,1, S_1,2, and S_2,3.

[4,3,4], Template:CDD

	Reflections				Rotations						Transflections		Screw axis
Name	R0	R1	R2	R3	S0,1	S1,2	S2,3	S0,2	S0,3	S1,3	T0,1,2	T1,2,3	U0,1,2,3
Element group	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD		Template:CDD
Order	2	2	2	2	4	3	4	2			6		∞ (3)
Matrix	$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}-1& 0& 0& 2\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& -1& 0& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}0& 1& 0& 0\\ -1& 0& 0& 2\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}-1& 0& 0& 2\\ 0& 1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}-1& 0& 0& 2\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ -1& 0& 0& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ -1& 0& 0& 2\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 1& 0& 0& -2\\ 0& 0& 0& 1\end{array}\right]$
Hyperplane	z=0	y=z	x=y	x=1

[[4,3,4]]

The extended group [[4,3,4]] doubles the group order, adding with a 2-fold rotation matrix T, with a fixed axis through points (1,1/2,0) and (1/2,1/2,1/2). The generators are {R₀,R₁,T}. R₂ = T×R₁×T and R₃ = T×R₀×T.

[[4,3,4]], Template:CDD

	Rotation	Reflections
Name	T	R0	R1	R2 = T×R1×T	R3 = T×R0×T
Element group	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD
Order	2	2	2	2	2
Matrix	$\left[\begin{array}{cccc}0& 0& -1& 1\\ 0& -1& 0& 1\\ -1& 0& 0& 1\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}-1& 0& 0& 2\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$
Hyperplane	Point (1/2,1/2,1/2) Axis (-1,0,1)	z=0	y=z	x=y	x=1

[4,3^1,1]

File:Triangular pyramidille cell1.png

[4,3^1,1] fundamental domain

The group [4,3^1,1] can be constructed from [4,3,4], by computing [4,3,4,1⁺], Template:CDD, as R'₃=R₃×R₂×R₃, with new R'₃ as an image of R₂ across R₃.

[4,3^1,1], Template:CDD

	Reflections				Rotations
Name	R0	R1	R2	R'3	S0,1	S1,2	S1,3	S0,2	S0,3	S2,3
Element group	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD
Order	2	2	2	2	3	3	3	2
Matrix	$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}0& -1& 0& 2\\ -1& 0& 0& 2\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& -1& 0& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}0& -1& 0& 2\\ 0& 0& 1& 0\\ -1& 0& 0& 2\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}0& -1& 0& 2\\ -1& 0& 0& 2\\ 0& 0& -1& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}-1& 0& 0& 2\\ 0& -1& 0& 2\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$
Hyperplane	z=0	y=z	x=y	x+y=2

[3^[4]]

File:Oblate tetrahedrille cell.png

[3^[4]] fundamental domain

The group [3^[4]] can be constructed from [4,3,4], by removing first and last mirrors, [1⁺,4,3,4,1⁺], Template:CDD, by R'₁=R₀×R₁×R₀ and R'₃=R₃×R₂×R₃.

[3^[4]] Template:CDD

	Reflections				Rotations
Name	R'0	R1	R2	R'3	S0,1	S1,2	S1,3	S0,2	S0,3	S2,3
Element group	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD
Order	2	2	2	2	3	3	3	2
Matrix	$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& -1& 0\\ 0& -1& 0& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}0& -1& 0& 2\\ -1& 0& 0& 2\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}0& -1& 0& 2\\ 0& 0& 1& 0\\ -1& 0& 0& 2\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& -1& 0\\ -1& 0& 0& 0\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}0& -1& 0& 2\\ 0& 0& -1& 0\\ 1& 0& 0& -2\\ 0& 0& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}-1& 0& 0& 2\\ 0& -1& 0& 2\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$
Hyperplane	y=-z	y=z	x=y	x+y=2

References

• H.S.M. Coxeter:
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, Template:ISBN [1]
    • (Paper 22) Template:Citation
    • (Paper 23) Template:Citation
    • (Paper 24) Template:Citation
• Template:Cite book
• Norman Johnson Uniform Polytopes, Manuscript (1991)
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
  • Norman W. Johnson and Asia Ivic Weiss Quadratic Integers and Coxeter Groups Template:Webarchive PDF Can. J. Math. Vol. 51 (6), 1999 pp. 1307–1336
  • N. W. Johnson: Geometries and Transformations, (2018) Template:ISBN [2] PDF
• Template:Citation
• John H. Conway and Derek A. Smith, On Quaternions and Octonions, 2003, Template:ISBN
• The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, Template:ISBN Ch.22 35 prime space groups, ch.25 184 composite space groups, ch.26 Higher still, 4D point groups