Coxeter notation

In geometry, Coxeter notation is a system of classifying symmetry groups, describing the angles between with fundamental reflections of a Coxeter group. It uses a bracketed notation, with modifiers to indicate certain subgroups. The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson.

Reflectional groups

Further information: Point group

For Coxeter groups defined by pure reflections, there is a direct correspondence between the bracket notation and Coxeter-Dynkin diagram graphs. The numbers in the bracket notation represent the mirror reflection orders in the branches of the Coxeter graph.

The Coxeter notation is simplified with exponents to represent the number of branches in a row for linear graphs. So the A_n group is represented by [3^n-1], to imply n nodes connected by n-1 order-3 branches.

Coxeter initially represented bifurcating diagrams with vertical positioning of numbers, but later abbreviated with an exponent notation, like [3^p,q,r], starting with [3^1,1,1] as D₄. Coxeter allowed for zeros as special cases to fit the rectified n-simplex polytopes into the same notation, and also allowed one -1 index for sequences that remove the common node to all the branches.

Coxeter groups formed by cyclic graphs are represented by parenthesese inside of brackets, like [(a,b,c)] for the triangle group (a b c). If they are equal, they can be grouped as an exponent as the length the cycle in brackets, like [(3,3,3,3)] = [3^[4]].

Finite Coxeter groups Rank Group symbol Bracket notation Coxeter graph 2 A2 [3] 2 BC2 [4] 2 H2 [5] 2 G2 [6] 2 I2(p) [p] 3 H3 [5,3] 3 A3 [3,3] 3 BC3 [4,3] 4 A4 [3,3,3] 4 BC4 [4,3,3] 4 D4 [31,1,1] 4 F4 [3,4,3] 4 H4 [5,3,3] n An [3n-1] .. n BCn [4,3n-2] ... n Dn [3n-3,1,1] ... 6 E6 [32,2,1] 7 E7 [33,2,1] 8 E8 [34,2,1]

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

Compact Hyperbolic Coxeter groups Group symbol Bracket notation Coxeter graph [p,q] with 2(p+q)<pq [(p,q,r)] with p+q+r>9 ${\bar{BH}}_3$ [4,3,5] ${\bar{K}}_3$ [5,3,5] ${\bar{J}}_3$ [3,5,3] ${\bar{DH}}_3$ [5,31,1] ${\widehat{AB}}_3$ [(3,3,3,4)]   ${\widehat{AH}}_3$ [(3,3,3,5)]   ${\widehat{BB}}_3$ [(3,4,3,4)] ${\widehat{BH}}_3$ [(3,4,3,5)] ${\widehat{HH}}_3$ [(3,5,3,5)] ${\bar{H}}_4$ [3,3,3,5] ${\bar{BH}}_4$ [4,3,3,5] ${\bar{K}}_4$ [5,3,3,5] ${\bar{DH}}_4$ [5,3,31,1] ${\widehat{AF}}_4$ [(3,3,3,3,4)]

For the affine and hyperbolic groups, the subscript is one less than the number of nodes in each case, since each of these groups was obtained by adding a node to a finite group's graph.

The Coxeter graph usually leaves order-2 branches undrawn, but the bracket notation includes an explicit 2 to connect the subgraphs. So the Coxeter graph, , H₃×A₂ can be represented by [5,3]×[3] and [5,3,2,3].

Subsymmetry by even/odd alternation

Coxeter's notation represents rotational/translational symmetry by adding a + superscript operator outside the brackets which cuts the order of the group in half (called index 2 subgroup). Multiple + operators may exist if neighboring elements are all even order, and the subgroup index is 2ⁿ for n operators.

For Coxeter groups with even order branches, elements by parentheses inside of a Coxeter group can be give a + superscript operator, having the effect of dividing adjacent ordered branches into half order, thus is usually only applied with even numbers. Example [4,3⁺].

Groups without neighboring + elements can be seen in ringed Coxeter-Dynkin diagram for uniform polytopes and honeycomb are related to holes on the nodes around the + elements, empty circles with the alternated nodes removed. So , has symmetry in Coxeter notation of [4,3]⁺, and has group notation [4,3⁺].

Subsymmetry by mirror removal

Johnson extends the + operator to work with a placeholder 1 nodes. In general this operation only applies to mirrors bounded by all even-order branches.

The 1 represents a mirror so [2p] can be seen as [1,2p,1], like diagram , with 2 mirrors related by an order-2p dihedral angle. Each of these mirrors can be removed so [1⁺,2p,1] = [1,2p,1⁺] = [p], a reflective subgroup index 2.

If both mirrors are removed, the branch order becomes a gyration point of half the order: [1⁺,2p,1⁺] = [1⁺,2p]⁺ = [2p,1⁺]⁺ = [p]⁺, a rotational subgroup of index 4. Example (with p=1): [1⁺,2] = [ ], order 2. [1⁺,2,1⁺] = [ ]⁺, order 1.

Commutator subgroups

Simple groups with only odd-order branch elements have only a single rotational/translational subgroup of order 2, which is also the commutator subgroup, examples [3,3]⁺, [3,5]⁺, [3,3,3]⁺, [3,3,5]⁺. For other Coxeter groups with even-order branches, the commutator subgroup has index 2^c, where c is the number of disconnected subgraphs when all the even-order branches are removed. For example [4,4] has 3 independent nodes in the Coxeter graph when the 4s are removed, so its commutator subgroup is index 2³, and can have different representions, 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

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.

Small index subgroups of [4,4]

Subgroup index	1	2			4
Diagram
Coxeter	[4,4]	[1+,4,4]	[4,4,1+]	[4,1+,4]	[4,1+,4,1+]	[1+,4,4,1+]
Orbifold	(*442)			(*2222)	(2*22)	(*2222)
Diagram
Coxeter		[4,4+]	[4+,4]	[(4,4,2+)]	[1+,4,1+,4]	[4+,4+]
Orbifold		(4*2)		(2*22)		(22×)
Rotational subgroups
Index	2	4			8
Diagram
Coxeter	[4,4]+	[1+,4,4+]	[4+,4,1+]	[(4,1+,4,2+)]	[1+,4,1+,4,1+] = [(4+,4+,2+)]
Orbifold	(442)			(2222)

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

Small index subgroups of [6,4]

Subgroup index	1	2			4
Diagrams
Coxeter (orbifold)	[6,4] (*642)	[1+,6,4] (*443)	[6,4,1+] (*662)	[6,1+,4] (*3222)	[6,1+,4,1+] (2*33)	[1+,6,4,1+] (*3232)
Diagrams
Coxeter (orbifold)		[6,4+] (4*3)	[6+,4] (6*2)	[(6,4,2+)] (2*32)	[1+,6,1+,4] (3*22)	[6+,4+] (32×)
Rotational subgroups
Subgroup index	2	4			8
Diagrams
Coxeter (orbifold)	[6,4]+ (642)	[1+,6,4+] (443)	[6+,4,1+] (662)	[(6,1+,4,2+)] (3222)	[1+,6,1+,4,1+] = [(6+,4+,2+)] (3232)

Extended symmetry

Further information: Goursat tetrahedron#Extended symmetry

Coxeter's notation includes double square bracket notation, [[X]] to express isomorphic symmetry within a Coxeter diagram. Johnson added alternative of angled-bracket <[X]> option as equivalent to square brackets for doubling to distinguish diagram symmetry through the nodes versus through the branches. Johnson also added a prefix symmetry modifier [Y[X]], where Y is the symmetry of the fundamental domain of [X].

For example in these equivalent rectangle and rhombic geometry diagrams of ${\tilde{A}}_3$: and , the first doubled with square brackets, [[3^[4]]] or twice doubled as [2[3^[4]]], with [2], order 4 higher symmetry. To differentiate the second, angled brackets are used for doubling, <[3^[4]]> and twice doubled as <2[3^[4]]>, also with a different [2], order 4 symmetry. Finally a full symmetry where all 4 nodes are equivalent can be represented by [4[3^[4]]], with the order 8, [4] symmetry of the square.

Further symmetry exists in the cyclic ${\tilde{A}}_n$ and branching $D_3$, ${\tilde{E}}_6$, and ${\tilde{D}}_4$ diagrams. ${\tilde{A}}_n$ graph 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]] and [3[3^2,2,2]] respectively while ${\tilde{D}}_4$ by [3,3[3^1,1,1,1]], with the diagram containing the order 24 symmetry of the regular tetrahedron, {3,3}. The noncompact hyperbolic group graph ${\bar{L}}_5$ = [3^1,1,1,1,1], , contains the symmetry of a 5-cell, {3,3,3}, and thus is represented by [3,3,3[3^1,1,1,1,1]].

Computation with reflection matrices as symmetry generators

A Coxeter group, represented by Coxeter diagram , is given Coxeter notation [p,q] for the branch orders. Each node in the Coxeter diagram represents a mirror, by convention called ρ_i (and matrix R_i). The generators of this group [p,q] are reflections: ρ₁, ρ₂, and ρ₃. Rotational subsymmetry is given as products of reflections: By convention, σ_1,2 (and matrix S_1,2) = ρ₁ρ₂ represents a rotation of angle π/p, and σ_2,3 = ρ₂ρ₃ is a rotation of angle π/q, and σ_1,3 = ρ₁ρ₃ represents a rotation of angle π/2.

[p,q]⁺ is an index 2 subgroup represented by two rotation generators, each a products of two reflections: σ_1,2, σ_2,3, and representing rotations of π/p, and π/q angles respectively.

If q is even, [p⁺,q] is another subgroup of index 2, represented by rotation generator σ_1,2, and reflectional ρ₃.

If both p and q are even, [p⁺,q⁺] is a subgroup of index 4 with a single generator type, constructed as a product of all three reflection matrices: By convention as: ψ_1,2,3 (and matrix U_1,2,3) = σ_1,2ρ₃ = ρ₁σ_2,3 = ρ₁ρ₂ρ₃, which is an improper rotation or roto-reflection, representing a reflection and rotation.

In the case of affine Coxeter groups like , or , one mirror, usually the last, is translated off the origin. A translation generator τ_1,2 (and matrix T_1,2) is constructed as the product of two (or an even number of) reflections, including the affine reflection. A trans-reflection (reflection plus a translation) can be the product of an odd number of reflections φ_1,2,3 (and matrix V_1,2,3), like the index 4 subgroup : [4⁺,4⁺].

Another composite generator, by convention as ζ (and matrix Z), represents the inversion, mapping a point to its inverse. For [4,3] and [5,3], ζ = (ρ₁ρ₂ρ₃)^h/2, where h is 6 and 10 respectively, the Coxeter number for each family.

Example, in 2D, the Coxeter group [p] is represented by two reflection matrices R₁ and R₂, The cyclic symmetry [p]⁺ is represented by rotation generator of matrix S_1,2.

R1	R2	S1,2=R1xR2
$\left [\begin{smallmatrix} 1 & 0 \\ 0 & -1 \\ \end{smallmatrix}\right ]$	$\left [\begin{smallmatrix} \cos 2\pi/p & \sin 2\pi/p \\ \sin 2\pi/p & -\cos 2\pi/p \\ \end{smallmatrix}\right ]$	$\left [\begin{smallmatrix} \cos 2\pi/p & \sin 2\pi/p \\ -\sin 2\pi/p & \cos 2\pi/p \\ \end{smallmatrix}\right ]$

A simple example affine group is [4,4] (p4m), can be given by three reflection matrices, constructed as a reflection across the x axis (y=0), a diagonal (x=y), and the affine reflection across the line (x=1). [4,4]⁺ (p4) is generated by S_1,2 S_2,3, and S_1,3. [4⁺,4⁺] (pgg) is generated by 2-fold rotation S_1,3 and trans-reflection V_1,2,3. [4⁺,4] (p4g) is generated by S_1,2 and R₃. The group [(4,4,2⁺)] (cmm), is generated by 2-fold rotation S_1,3 and reflection R₂.

R1	R2	R3	S1,2	S2,3	S1,3	V1,2,3
$\left [\begin{smallmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \\ \end{smallmatrix}\right ]$	$\left [\begin{smallmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{smallmatrix}\right ]$	$\left [\begin{smallmatrix} -1 & 0 & 2 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{smallmatrix}\right ]$	$\left [\begin{smallmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{smallmatrix}\right ]$	$\left [\begin{smallmatrix} 0 & 1 & 0 \\ -1 & 0 & 2 \\ 0 & 0 & 1 \\ \end{smallmatrix}\right ]$	$\left [\begin{smallmatrix} -1 & 0 & 2 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \\ \end{smallmatrix}\right ]$	$\left [\begin{smallmatrix} 0 & 1 & 0 \\ 1 & 0 & -2 \\ 0 & 0 & 1 \\ \end{smallmatrix}\right ]$

Coxeter groups are categorized by their rank, being the number of nodes in its Coxeter graph. The structure of the groups are also given with their abstract group types: In this article, the abstract dihedral groups are represented as Dih_n, and cyclic groups are represented by Z_n, with Dih₁=Z₂.

Rank one groups

In one dimension, the bilateral group [ ] represents a single mirror symmetry, abstract Dih₁ or Z₂, symmetry order 2. It is represented as a Coxeter graph with a single node, . The identity group is the direct subgroup [ ]⁺, Z₁, symmetry order 1. The + superscript simply implies that alternate mirror reflections are ignored, leaving the identity group in this simplest case.

Group	Coxeter	Coxeter diagram	Order	Description
C1	[ ]+		1	Identity
D1	[ ]		2	Reflection group

Rank two groups

Family correspondence: A mirror added between two corresponding mirror doubles the symmetry order

In two dimensions, the rectangular group [2], abstract Dih₂, also can be represented as a direct product [ ]×[ ] or Z₂×Z₂, being the product of two bilateral groups, represents two orthogonal mirrors, with Coxeter graph, , with order 4. The 2 in [2] comes from linearization of the orthogonal subgraphs in the Coxeter graph, as , with explicit branch order 2. The rhombic group, [2]⁺, half of the rectangular group, the point reflection symmetry, Z₂, order 2.

Coxeter notation allows a 1 place-holder for lower rank groups, so [1] is the same as [ ], and [1]⁺ is the same as [ ]⁺. This may be done to imply the group exists in 2-dimensions rather than 1 dimension.

The full p-gonal group [p], abstract dihedral group Dih_p, (nonabelian for p>2), of order 2p, is generated by two mirrors at angle π/p, represented by Coxeter graph . The 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 Dih_∞, represents two parallel mirrors and has a Coxeter graph . The apeirogonal group [∞]⁺, abstractly the infinite cyclic group Z_∞, isomorphic to the additive group of the integers, is generated by a single nonzero translation.

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

Group	Intl	Orbifold	Coxeter	Order	Description
Finite
Zn	n	n•	[n]+	n	Cyclic: n-fold rotations. Abstract group Zn, the group of integers under addition modulo n.
Dn	nm	*n•	[n]	2n	Dihedral: cyclic with reflections. Abstract group Dihn, the dihedral group.
Affine
Z∞	∞	∞•	[∞]+	∞	Cyclic: apeirogonal group. Abstract group Z∞, the group of integers under addition.
Dih∞	∞m	*∞•	[∞]	∞	Dihedral: parallel reflections. Abstract infinite dihedral group Dih∞.
Hyperbolic
Z∞			[πi/λ]+	∞	pseudogonal group
Dih∞			[πi/λ]	∞	full pseudogonal group

Rank three groups

Finite family correspondence

Affine isomorphism and correspondences

Further information: List of spherical symmetry groups and List of planar symmetry groups

In three dimensions, the full orthorhombic group [2,2], astracttly Z₂×Dih₂, order 8, represents three orthogonal mirrors, and also can be represented by Coxeter graph as three separate dots . It can also can be represented as a direct product [ ]×[ ]×[ ], but the [2,2] expression allows subgroups to be defined:

First there is a semidirect subgroup, the orthorhombic group, [2,2⁺], abstractly Dih₁×Z₂=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 graph, ) are alternated. In general, the branch orders neighboring the + node must be even. In this case [2,2⁺] and [2⁺,2] represent two isomorphic subgroups that are geometrically distinct. The other subgroups are the pararhombic group [2,2]⁺, also order 4, and finally the central group [2⁺,2⁺] of order 2.

Next there is the full ortho-p-gonal group, [2,p], abstractly Dih₁×Dih_p=Z₂×Dih_p, of order 4p, representing two mirrors at a dihedral angle π/p, and both are orthogonal to a third mirror. It is also represented by Coxeter graph as .

The direct subgroup is called the para-p-gonal group, [2,p]⁺, abstractly Dih_p, of order 2p, and another subgroup is [2,p⁺] abstractly Z_p×Z₂, also of order 2p.

The full gyro-p-gonal group, [2⁺,2p], abstractly Dih_2p, of order 4p. The gyro-p-gonal group, [2⁺,2p⁺], abstractly Z_2p, of order 2p is a subgroup of both [2⁺,2p] and [2,2p⁺].

The polyhedral groups are based on the symmetry of platonic solids, the tetrahedron, octahedron, cube, icosahedron, and dodecahedron, with Schläfli symbols {3,3}, {3,4}, {4,3}, {3,5}, and {5,3} respectively. The Coxeter groups for these are called in Coxeter's bracket notation [3,3], [3,4], [3,5] called full tetrahedral symmetry, octahedral symmetry, and icosahedral symmetry, with orders of 24, 48, and 120. The front-to-back order can be reversed in the Coxeter notation, unlike the Schläfli symbol.

The tetrahedral group, [3,3], has a doubling [[3,3]] which maps the first and last mirrors onto each other, and this produces the [3,4] group.

In all these symmetries, alternate reflections can be removed producing the rotational tetrahedral, octahedral, and icosahedral groups of order 12, 24, and 60. The octahedral group also has a unique subgroup called the pyritohedral symmetry group, [3⁺,4], of order 12, with a mixture of rotational and reflectional symmetry.

In the Euclidean plane there's 3 fundamental reflective groups generated by 3 mirrors, represented by Coxeter graphs , , and , and are given Coxeter notation as [4,4], [6,3], and [(3,3,3)]. The parentheses of the last group imply the graph ic cyclic, and also has a shorthand notation [3^[3]].

[[4,4]] as a doubling of the [4,4] group produced the same symmetry rotated π/4 from the original set of mirrors.

Direct subgroups of rotational symmetry are: [4,4]⁺, [6,3]⁺, and [(3,3,3)]⁺. [4⁺,4] and [6,3⁺] represent mixed reflectional and rotational symmetry.

Finite
Intl* Geo Orbifold Schönflies Conway Coxeter Order 1 1 1 C1 C1 [ ]+ 1 1 22 ×1 Ci = S2 CC2 [2+,2+] 2 2 = m 1 *1 Cs = C1v = C1h ±C1 = CD2 [ ] 2 2 3 4 5 6 n 2 3 4 5 6 n 22 33 44 55 66 nn C2 C3 C4 C5 C6 Cn C2 C3 C4 C5 C6 Cn [2]+ [3]+ [4]+ [5]+ [6]+ [n]+ 2 3 4 5 6 n 2mm 3m 4mm 5m 6mm nmm nm 2 3 4 5 6 n *22 *33 *44 *55 *66 *nn C2v C3v C4v C5v C6v Cnv CD4 CD6 CD8 CD10 CD12 CD2n [2] [3] [4] [5] [6] [n] 4 6 8 10 12 2n 2/m 3/m 4/m 5/m 6/m n/m 2 2 3 2 4 2 5 2 6 2 n 2 2* 3* 4* 5* 6* n* C2h C3h C4h C5h C6h Cnh ±C2 CC6 ±C4 CC10 ±C6 ±Cn / CC2n [2,2+] [2,3+] [2,4+] [2,5+] [2,6+] [2,n+] 4 6 8 10 12 2n 4 3 8 5 12 2n n 4 2 6 2 8 2 10 2 12 2 2n 2 2× 3× 4× 5× 6× n× S4 S6 S8 S10 S12 S2n CC4 ±C3 CC8 ±C5 CC12 CC2n / ±Cn [2+,4+] [2+,6+] [2+,8+] [2+,10+] [2+,12+] [2+,2n+] 4 6 8 10 12 2n	Intl Geo Orbifold Schönflies Conway Coxeter Order 222 32 422 52 622 n22 n2 2 2 3 2 4 2 5 2 6 2 n 2 222 223 224 225 226 22n D2 D3 D4 D5 D6 Dn D4 D6 D8 D10 D12 D2n [2,2]+ [2,3]+ [2,4]+ [2,5]+ [2,6]+ [2,n]+ 4 6 8 10 12 2n mmm 6m2 4/mmm 10m2 6/mmm n/mmm 2nm2 2 2 3 2 4 2 5 2 6 2 n 2 *222 *223 *224 *225 *226 *22n D2h D3h D4h D5h D6h Dnh ±D4 DD12 ±D8 DD20 ±D12 ±D2n / DD4n [2,2] [2,3] [2,4] [2,5] [2,6] [2,n] 8 12 16 20 24 4n 42m 3m 82m 5m 122m 2n2m nm 4 2 6 2 8 2 10 2 12 2 n 2 2*2 2*3 2*4 2*5 2*6 2*n D2d D3d D4d D5d D6d Dnd ±D4 ±D6 DD16 ±D10 DD24 DD4n / ±D2n [2+,4] [2+,6] [2+,8] [2+,10] [2+,12] [2+,2n] 8 12 16 20 24 4n 23 3 3 332 T T [3,3]+ 12 m3 4 3 3*2 Th ±T [3+,4] 24 43m 3 3 *332 Td TO [3,3] 24 432 4 3 432 O O [3,4]+ 24 m3m 4 3 *432 Oh ±O [3,4] 48 532 5 3 532 I I [3,5]+ 60 53m 5 3 *532 Ih ±I [3,5] 120
Semiaffine
Intl (orbifold) Geo Schönflies Coxeter Order n nn n Cn [1,n]+ n nm *nn n Dn [1,n] 2n IUC (Orbifold) Geo Schönflies Coxeter p1 ∞∞ p1 C∞ [1,∞]+ p1m1 *∞∞ p1 C∞v [1,∞]	IUC (Orbifold) Geo Schönflies Coxeter p11g ∞x p.g1 S2∞ [∞+,2+] p11m ∞* p. 1 C∞h [∞+,2] p2 22∞ p2 D∞ [∞,2]+ p2mg 2*∞ p2g D∞d [∞,2+] p2mm *22∞ p2 D∞h [∞,2]
Affine
IUC (Orbifold) Geometric Coxeter p2 (2222) p2 [1+,4,4]+ p2gg (22x) pg2g [4+,4+] p2mm (*2222) p2 [1+,4,4] c2mm (2*22) c2 [[4+,4+]] p4 (442) p4 [4,4]+ p4gm (4*2) pg4 [4+,4] p4mm (*442) p4 [4,4]	IUC (Orbifold) Geometric Coxeter p3 (333) p3 [1+,6,3+] = [3[3]]+ p3m1 (*333) p3 [1+,6,3] = [3[3]] p31m (3*3) h3 [6,3+] = [3[3[3]]+] p6 (632) p6 [6,3]+ = [3[3[3]]]+ p6mm (*632) p6 [6,3] = [3[3[3]]]

Subgroups

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

Reflection	Reflection subgroups	Rotation subgroup	Mixed	Improper rotation	Commutator subgroup
C1v, [ ], (*)		C1, [ ]+, (11)		S2, [2+,2+], (×)	[ ]+
C2v, [2], (*22)	[1+,2]=[ ] (*)	C2, [2]+, (22)	C2h, [2+,2], (2*)	S4, [4+,2+], (2×)
Cnv, [n], (*nn)	[1+,2n]=[n] (*nn)	Cn, [n]+, (nn)	Cnh, [n+,2], (n*)	S2n, [2n+,2+], (n×)	[n]+, n odd [n/2]+, n even
Dnh, [2,n], (*22n)	[1+,2,n]=[n] (*nn)	Dn, [2,n]+, (22n)	Dnd, [2+,2n], (2*n)
Td, [3,3], (*332)		T, [3,3]+, (332)			[3,3]+, (332)
Oh, [4,3], (*432)	[1+,4,3]=[3,3] (*332)	O, [4,3]+, (432)	Th, [3+,4], (3*2)
Ih, [5,3], (*532)		I, [5,3]+, (532)			[5,3]+, (532)

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
[∞,2,∞], (*2222)	[1+,∞,2,∞], (*2222)	[∞+,2,∞], (**)	[∞,2,∞]+, (2222) [∞+,2+,∞+]+, (2222)	[∞+,2,∞+], (°) [∞+,2+,∞], (*×) [(∞,2)+,∞+], (××) [∞+,2+,∞+], (22×)	(2222)
		[∞,2+,∞], (2*22) [(∞,2)+,∞], (22*)
[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×)
[3[3]], (*333)		[3[3]]+, (333)			(333)
[6,3], (*632)	[1+,6,3], (*333)	[3+,6], (3*3)	[6,3]+, (632) [1+,6,3+], (333)

Rank four groups

Point groups

Finite isomorphism and correspondences

Subgroup relations

Rank four groups defined the 4-dimensional point groups:

Finite groups

[ ]: Symbol Order [1]+ 1.1 [1] = [ ] 2.1 [2,1,1]: Symbol Order [2+,1,1]+ 1.1 [2,1,1]+ 2.1 [2,1,1] 4.1 [2,2,1]: Symbol Order [2+,2+,1]+ = [(2+,2+,2+)] 1.1 [2+,2+,1] 2.1 [2,2,1]+ 4.1 [2+,2,1] 4.1 [2,2,1] 8.1 [2,2,2]: Symbol Order [(2+,2+,2+,2+)] = [2+,2+,2+]+ 1.1 [2+,2+,2+] 2.1 [2+,2,2+] 4.1 [(2,2)+,2+] 4 [[2+,2+,2+]] 4 [2,2,2]+ 8 [2+,2,2] 8.1 [(2,2)+,2] 8 [[2+,2,2+]] 8.1 [2,2,2] 16.1 [[2,2,2]]+ 16 [[2,2+,2]] 16 [[2,2,2]] 32 [p,1,1]: Symbol Order [3,1,1]+ 3.1 [4,1,1]+ 4.2 [5,1,1]+ 5.1 [6,1,1]+ 6.1 [p,1,1]+ p [3,1,1] 6.2 [4,1,1] 8.4 [5,1,1] 10.2 [6,1,1] 12.3 [p,1,1] 2p [p,2,1]: Symbol Order [3,2,1]+ 6.1 [4,2,1]+ 8.3 [5,2,1]+ 10.2 [6,2,1]+ 12.3 [p,2,1]+ 2p [3,2,1] 12.3 [4,2,1] 16.6 [5,2,1] 20.3 [6,2,1] 24.6 [p,2,1] 4p [2p,2+,1]: Symbol Order [6,2+,1] 12.3 [8,2+,1] 16.12 [10,2+,1] 20.3 [12,2+,1] 24.12 [2p,2+,1] 4p

[p,2,2]: Symbol Order [3+,2,2+] 6 [4+,2,2+] 8 [5+,2,2+] 10 [6+,2,2+] 12 [p+,2,2+] 2p [(3,2)+,2+] 6 [(4,2)+,2+] 8 [(5,2)+,2+] 10 [(6,2)+,2+] 12 [(p,2)+,2+] 2p [3,2,2]+ 12 [4,2,2]+ 16 [5,2,2]+ 20 [6,2,2]+ 24 [p,2,2]+ 4p [3,2,2+] 12 [4,2,2+] 16 [5,2,2+] 20 [6,2,2+] 24 [p,2,2+] 4p [3+,2,2] 12 [4+,2,2] 16 [5+,2,2] 20 [6+,2,2] 24 [p+,2,2] 4p [(3,2)+,2] 12 [(4,2)+,2] 16 [(5,2)+,2] 20 [(6,2)+,2] 24 [(p,2)+,2] 4p [3,2,2] 24 [4,2,2] 32 [5,2,2] 40 [6,2,2] 48 [p,2,2] 8p [2p,2+,2]: Symbol Order [6+,2+,2+] 6 [8+,2+,2+] 8 [10+,2+,2+] 10 [12+,2+,2+] 12 [2p+,2+,2+] 2p [6+,2+,2] 12 [8+,2+,2] 16 [10+,2+,2] 20 [12+,2+,2] 24 [2p+,2+,2] 4p [6+,(2,2)+] 12 [8+,(2,2)+] 16 [10+,(2,2)+] 20 [12+,(2,2)+] 24 [2p+,(2,2)+] 4p [6,(2,2)+] 24 [8,(2,2)+] 32 [10,(2,2)+] 40 [12,(2,2)+] 48 [2p,(2,2)+] 8p [6,2+,2] 24 [8,2+,2] 32 [10,2+,2] 40 [12,2+,2] 48 [2p,2+,2] 8p

[p,2,q]: Symbol Order [3+,2,3+] 9 [4+,2,3+] 12 [5+,2,3+] 15 [6+,2,3+] 18 [4+,2,4+] 16 [5+,2,4+] 20 [6+,2,4+] 24 [5+,2,5+] 25 [6+,2,5+] 30 [6+,2,6+] 36 [p+,2,q+] pq [3,2,3]+ 18 [4,2,3]+ 24 [5,2,3]+ 30 [6,2,3]+ 36 [4,2,4]+ 32 [5,2,4]+ 40 [6,2,4]+ 48 [5,2,5]+ 50 [6,2,5]+ 60 [6,2,6]+ 72 [p,2,q]+ 2pq [3+,2,3] 18 [4+,2,3] 24 [5+,2,3] 30 [6+,2,3] 36 [4+,2,4] 32 [5+,2,4] 40 [6+,2,4] 48 [5+,2,5] 50 [6+,2,5] 60 [6+,2,6] 72 [p+,2,q] 2pq [3,2,3] 36 [4,2,3] 48 [5,2,3] 60 [6,2,3] 72 [4,2,4] 64 [5,2,4] 80 [6,2,4] 96 [5,2,5] 100 [6,2,5] 120 [6,2,6] 144 [p,2,q] 4pq

[(p,2)+,2q]: Symbol Order [(3,2)+,6+] 18 [(4,2)+,6+] 24 [(5,2)+,6+] 30 [(6,2)+,6+] 36 [(4,2)+,8+] 32 [(5,2)+,8+] 40 [(6,2)+,8+] 48 [(5,2)+,10+] 50 [(6,2)+,10+] 60 [(6,2)+,12+] 72 [(p,2)+,2q+] 2pq [(3,2)+,6] 36 [(4,2)+,6] 48 [(5,2)+,6] 60 [(6,2)+,6] 72 [(4,2)+,8] 64 [(5,2)+,8] 80 [(6,2)+,8] 96 [(5,2)+,10] 100 [(6,2)+,10] 120 [(6,2)+,12] 144 [(p,2)+,2q] 4pq [2p,2+,2q]: Symbol Order [6+,2+,6+] 18 [8+,2+,6+] 24 [10+,2+,6+] 30 [12+,2+,6+] 36 [8+,2+,8+] 32 [10+,2+,8+] 40 [12+,2+,8+] 48 [10+,2+,10+] 50 [12+,2+,10+] 60 [12+,2+,12+] 72 [2p+,2+,2q+] 2pq [6,2+,6+] 36 [8,2+,6+] 48 [10,2+,6+] 60 [12,2+,6+] 72 [8,2+,8+] 64 [10,2+,8+] 80 [12,2+,8+] 96 [10,2+,10+] 100 [12,2+,10+] 120 [12,2+,12+] 144 [2p,2+,2q+] 4pq [6,2+,6] 72 [8,2+,6] 96 [10,2+,6] 120 [12,2+,6] 144 [8,2+,8] 128 [10,2+,8] 160 [12,2+,8] 192 [10,2+,10] 200 [12,2+,10] 240 [12,2+,12] 288 [2p,2+,2q] 8pq

[[p,2,p]]: Symbol Order [[3+,2,3+]] 18 [[4+,2,4+]] 32 [[5+,2,5+]] 50 [[6+,2,6+]] 72 [[p+,2,p+]] 2p2 [[3,2,3]]+ 36 [[4,2,4]]+ 64 [[5,2,5]]+ 100 [[6,2,6]]+ 144 [[p,2,p]]+ 4p2 [[3,2,3]] 72 [[4,2,4]] 128 [[5,2,5]] 200 [[6,2,6]] 288 [[p,2,p]] 8p2 [[2p,2+,2p]]: Symbol Order [[6+,2+,6+]] 36 [[8+,2+,8+]] 64 [[10+,2+,10+]] 100 [[12+,2+,12+]] 144 [[2p+,2+,2p+]] 4p2 [[6,2+,6]] 144 [[8,2+,8]] 256 [[10,2+,10]] 400 [[12,2+,12]] 576 [[2p,2+,2p]] 16p2

[3,3,1]: Symbol Order [3,3,1]+ 12.5 [3,3,1] 24 [4,3,1]: Symbol Order [4,3,1]+ 24.15 [3+,4,1] 24.10 [4,3,1] 48.36 [5,3,1]: Symbol Order [5,3,1]+ 60.13 [5,3,1] 120.2 [3,3,2]: Symbol Order [(3,3)+,2] 24 [3,3,2]+ 24 [3,3,2] 48 [4,3,2]: Symbol Order [(3,4)+,2+] 24 [3,4,2+] 48 [4,(3,2)+] 48 [(4,3)+,2] 48 [4,3,2]+ 48 [4,3+,2] 48.22 [4,3,2] 96.5 [5,3,2]: Symbol Order [(5,3)+,2] 120.2 [5,3,2]+ 120.2 [5,3,2] 240 [31,1,1]: Symbol Order [31,1,1]+ = [1+,4,3,3]+ 96.1 [31,1,1] = [1+,4,3,3] 192 <[3,31,1]> = [4,3,3] 384.1 [3[31,1,1]] = [3,4,3] 1152.1 [3,3,3]: Symbol Order [3,3,3]+ 60 [3,3,3] 120.1 [[3,3,3]]+ 120.2 [[3,3,3]+] 120.1 [[3,3,3]] 240.1 [4,3,3]: Symbol Order [4,(3,3)+]+, = [1+,4,(3,3)+] = [1+,4,3,3]+ = [31,1,1]+ 96.1 [1+,4,3,3] = [3,31,1] 192 [4,(3,3)+] 192 [4,3,3]+ 192 [4,3,3] 384 [3,4,3]: Symbol Order [3+,4,3]+ = [3+,4,3+] 288.1 [3,4,3]+ 576.2 [3+,4,3] 576.1 [[3+,4,3+]] 576 [3,4,3] 1152.1 [[3,4,3]]+ 1152 [[3,4,3]+] 1152 [[3,4,3]] 2304 [5,3,3]: Symbol Order [5,3,3]+ 7200 [5,3,3] 14400

Subgroups

1D-4D reflective point groups and subgroups

Order	Reflection		Mixed subgroups	Improper rotation subgroups	Rotation subgroups	Commutator subgroup
2	[ ]				[ ]+	[ ]+1	[ ]+
4	[2]				[2]+	[2]+2
8	[2,2]		[2+,2]	[2+,2+]	[2,2]+	[2,2]+3
16	[2,2,2]		[2+,2,2] [(2,2)+,2]	[2+,2+,2] [(2,2)+,2+] [2+,2+,2+]	[2,2,2]+ [2+,2,2+]	[2,2,2]+4
	[21,1,1]			[(2+)1,1,1]
2n	[n]				[n]+	[n]+1	[n]+
4n	[2n]				[2n]+	[2n]+2
4n	[2,n]		[2,n+]		[2,n]+	[2,n]+2
8n	[2,2n]		[2+,2n]	[2+,2n+]	[2,2n]+	[2,2n]+3
8n	[2,2,n]		[2+,2,n] [2,2,n+]	[2+,(2,n)+]	[2,2,n]+ [2+,2,n+]	[2,2,n]+3
16n	[2,2,2n]		[2,2+,2n]	[2+,2+,2n] [2,2+,2n+] [(2,2)+,2n+] [2+,2+,2n+]	[2,2,2n]+ [2+,2n,2+]	[2,2,2n]+4
	[2,2n,2]			[2+,2n+,2+]
	[2n,21,1]			[2n+,(2+)1,1]
24	[3,3]				[3,3]+	[3,3]+1	[3,3]+
48	[3,3,2]		[(3,3)+,2]		[3,3,2]+	[3,3,2]+2
48	[4,3]		[4,3+]		[4,3]+	[4,3]+2
96	[4,3,2]		[(4,3)+,2] [4,(3,2)+]		[4,3,2]+	[4,3,2]+3
	[3,4,2]		[3,4,2+] [3+,4,2]	[(3,4)+,2+]	[3+,4,2+]
120	[5,3]				[5,3]+	[5,3]+1	[5,3]+
240	[5,3,2]		[(5,3)+,2]		[5,3,2]+	[5,3,2]+2
4pq	[p,2,q]		[p+,2,q]		[p,2,q]+ [p+,2,q+]	[p,2,q]+2	[p+,2,q+]
8pq	[2p,2,q]		[2p,(2,q)+]	[2p+,(2,q)+]	[2p,2,q]+	[2p,2,q]+3
16pq	[2p,2,2q]		[2p,2+,2q]	[2p+,2+,2q] [2p+,2+,2q+]	[2p,2,2q]+	[2p,2,2q]+4
120	[3,3,3]				[3,3,3]+	[3,3,3]+1	[3,3,3]+
192	[31,1,1]				[31,1,1]+	[31,1,1]+1	[31,1,1]+
384	[4,3,3]		[4,(3,3)+]		[4,3,3]+	[4,3,3]+2
1152	[3,4,3]		[3+,4,3]		[3,4,3]+ [3+,4,3+]	[3,4,3]+2	[3+,4,3+]
14400	[5,3,3]				[5,3,3]+	[5,3,3]+1	[5,3,3]+

Space groups

Affine isomorphism and correspondences

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

Orthorhombic [∞,2,∞,2,∞] Symbol [∞,2,∞,2,∞] [∞,2,∞,2,∞]+ [∞+,2,∞,2,∞] [∞+,2,∞+,2,∞] [∞+,2,∞+,2,∞+] [∞,2,∞,2+,∞] [∞,2+,∞,2+,∞] [(∞,2,∞)+,2,∞] [(∞,2,∞)+,2,∞+]

Trigonal & hexagonal Group Symbol [3,6,2,∞] [6,3,2,∞] [6,3,2,∞]+ [6,3+,2,∞] [(6,3)+,2,∞] [6,3,2,∞+] [6,3+,2,∞+] [(6,3)+,2,∞+] [1+,6,3,2,∞] [1+,6,3,2,∞]+ [1+,6,3,2,∞+] [(1+,6,3)+,2,∞] [(1+,6,3)+,2,∞+] [3[3],2,∞] [3[3],2,∞] [3[3],2,∞]+ [3[3],2,∞+] [(3[3])+,2,∞] [(3[3])+,2,∞+]

Tetragonal [4,4,2,∞] [4,4,2,∞] [4,4,2,∞]+ [(4,4)+,2,∞] [4,4,2+,∞] [4,4,2,∞+] [(4,4)+,2+,∞] [(4,4)+,2,∞+] [4,4,2+,∞+] [(4,4)+,2+,∞+] [4+,4+,2+,∞] [4,4+,2,∞] [4,4+,2+,∞] [(4,2+,4),2,∞] [[4,2+,4],2,∞] [4,4+,2,∞+] [4,4+,2+,∞+] [(4,2+,4),2,∞+] [[4,2+,4],2,∞+]

Cubic Group Coxeter Space group Index [4,3,4] [4,3,4] (221) Pm3m 1 [4,3,4]+ (222) Pn3n 2 [4,3+,4] (223) Pm3n 2 [4,(3,4)+] (224) Pn3m 2 [4,3,4,1+] (225) Fm3m 2 [(4,3,4,2+)] 2 [4,(3,4,1+)+] (226) Fm3c 4 [1+,4,3,4,1+] (227) Fd3m 4 [4,3,4,1+]+ (228) Fd3c 4 [[4,3,4]] [[4,3,4]] (229) Im3m [[4,3,4]]+ (230) Ia3d [[4,3+,4]] [[4,3,4]]+ [[(4,3,4,2+)]] [4,31,1] [4,31,1] = [4,3,4,1+] 2 [4,(31,1)+] = [4,(3,4,1+)+] 4 [1+,4,31,1] = [1+,4,3,4,1+] 4 [4,31,1]+ = [4,3,4,1+]+ 4 [1+,4,31,1]+ = [1+,4,3,4,1+]+ 2 <[4,31,1]> = [4,3,4] 1 [3[4]] [3[4]] = [1+,4,31,1] 4 [3[4]]+ = [1+,4,31,1]+ 2 <[(3,3,3,3)]> = [4,31,1] 2 <<[3[4]]>> = [4,3,4] 1 [[3[4]]] [4[3[4]]] = [[4,3,4]] (229) Im3m

Line groups

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

Semiaffine (3D)

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+]
2n	n	S2n	P2n	Pn	None	p11g, pg(h)	xx		[(∞,2)+,2n+]
n/m	2n	Cnh	Pn/m	P2n	None	p11m, pm(h)	**		[∞+,2,n]
2n/m		C2nh	P2nn/m		Zigzag	c11m, cm(h)	*x		[∞+,2+,2n]
nmm	nm	Cnv	Pnmm	Pnm	None	p1m1, pm(v)	**		[∞,2,n+]
			Pncc	Pnc	Planar reflection	p1g1, pg(v)	xx		[∞+,(2,n)+]
2nmm		C2nv	P2nnmc		Zigzag	c1m1, cm(v)	*x		[∞,2+,2n+]
n22	n2	Dn	Pnq22	Pnq2	Helical: q	p2	2222		[∞,2,n]+
2n2m	nm	Dnd	P2n2m	Pnm	None	p2mg, pmg(h)	22*		[(∞,2)+,2n]
			P2n2c	Pnc	Planar reflection	p2gg, pgg	22x		[∞+,2+,2n+]
n/mmm	2n2m	Dnh	Pn/mmm	P2n2m	None	p2mm, pmm	*2222		[∞,2,n]
			Pn/mcc	P2n2c	Planar reflection	p2mg, pmg(v)	22*		[∞,(2,n)+]
2n/mmm		D2nh	P2nn/mcm		Zigzag	c2mm, cmm	2*22		[∞,2+,2n]

Wallpaper groups

Rank four groups also defined some of the 2-dimensional wallpaper groups:

Affine (2D plane)

IUC (Orbifold) Geo Coxeter p1 (o) p1 [∞+,2,∞+] p2 (2222) p2 [∞,2,∞]+ c2mm (2*22) c2 [∞,2+,∞] p11g (xx) pg1 h: [∞+,(2,∞)+] p1g1 (xx) pg1 v: [(∞,2)+,∞+] p2gm (22*) pg2 h: [(∞,2)+,∞] p2mg (22*) pg2 v: [∞,(2,∞)+]

IUC (Orbifold) Geo Coxeter p11m (**) p1 h: [∞+,2,∞] p1m1 (**) p1 v: [∞,2,∞+] p2mm (*2222) p2 [∞,2,∞] c11m (*x) c1 h: [∞+,2+,∞] c1m1 (*x) c1 v: [∞,2+,∞+] p2gg (22x) pg2g [∞+,2+,∞+] c2mm (2*22) c2 [∞,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, ISBN 978-0-471-01003-6 [2]
    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10]
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559–591]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
  • Coxeter, H. S. M. and Moser, W. O. J. (1980). Generators and Relations for Discrete Groups. New York: Springer-Verlag. ISBN 0-387-09212-9. 
• Norman Johnson Uniform Polytopes, Manuscript (1991)
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
  • N.W. Johnson: Geometries and Transformations, Manuscript, (2011) Chapter 11: Finite symmetry groups