Coxeter notation

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Fundamental domains of reflective 3D point groups
CDel node.png, [ ] = [1]
C1v
CDel node.pngCDel 2.pngCDel node.png, [2]
C2v
CDel node.pngCDel 3.pngCDel node.png, [3]
C3v
CDel node.pngCDel 4.pngCDel node.png, [4]
C4v
CDel node.pngCDel 5.pngCDel node.png, [5]
C5v
CDel node.pngCDel 6.pngCDel node.png, [6]
C6v
Spherical digonal hosohedron.png
Order 2
Spherical square hosohedron.png
Order 4
Spherical hexagonal hosohedron.png
Order 6
Spherical octagonal hosohedron.png
Order 8
Spherical decagonal hosohedron.png
Order 10
Spherical dodecagonal hosohedron.png
Order 12
CDel node.pngCDel 2.pngCDel node.png
[2] = [2,1]
D1h
CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
[2,2]
D2h
CDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.png
[2,3]
D3h
CDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.png
[2,4]
D4h
CDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.png
[2,5]
D5h
CDel node.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.png
[2,6]
D6h
Spherical digonal bipyramid.svg
Order 4
Spherical square bipyramid.svg
Order 8
Spherical hexagonal bipyramid.png
Order 12
Spherical octagonal bipyramid.png
Order 16
Spherical decagonal bipyramid.png
Order 20
Spherical dodecagonal bipyramid.png
Order 24
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png, [3,3], Td CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png, [4,3], Oh CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png, [5,3], Ih
Spherical tetrakis hexahedron-3edge-color.png
Order 24
Spherical disdyakis dodecahedron-3and1-color.png
Order 48
Spherical compound of five octahedra.png
Order 120
Coxeter notation expresses Coxeter groups as a list of branch orders of a Coxeter diagram, like the polyhedral groups, CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png = [p,q]. Dihedral groups, CDel node.pngCDel 2.pngCDel node.pngCDel n.pngCDel node.png, can be expressed as a product [ ]×[n] or in a single symbol with an explicit order 2 branch, [2,n].

In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a 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.

Contents

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 An group is represented by [3n−1], to imply n nodes connected by n−1 order-3 branches. Example A2 = [3,3] = [32] or [31,1] represents diagrams CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png or CDel node.pngCDel split1.pngCDel nodes.png.

Coxeter initially represented bifurcating diagrams with vertical positioning of numbers, but later abbreviated with an exponent notation, like [...,3p,q] or [3p,q,r], starting with [31,1,1] or [3,31,1] = CDel node.pngCDel splitsplit1.pngCDel branch3.pngCDel node.png or CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png as D4. Coxeter allowed for zeros as special cases to fit the An family, like A3 = [3,3,3,3] = [34,0,0] = [34,0] = [33,1] = [32,2], like CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = CDel node.pngCDel split1.pngCDel nodes.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png = CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png.

Coxeter groups formed by cyclic diagrams are represented by parentheseses inside of brackets, like [(p,q,r)] = CDel pqr.png 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 CDel node.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node.png or CDel branch.pngCDel 3ab.pngCDel branch.png. CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel branch.png 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 CDel node.pngCDel split1.pngCDel branch.pngCDel split2.pngCDel node.png 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 CDel tet.png or CDel branch.pngCDel splitcross.pngCDel branch.png, is represented as [3[3,3]] with the superscript [3,3] as the symmetry of its regular tetrahedron coxeter diagram.

The Coxeter diagram usually leaves order-2 branches undrawn, but the bracket notation includes an explicit 2 to connect the subgraphs. So the Coxeter diagram CDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.png = 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: CDel node.pngCDel 3.pngCDel node.pngCDel 2x.pngCDel node.pngCDel 3.pngCDel node.png or CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel 2c.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png, as an identical presentation as [3,2,3].

Finite groups
RankGroup
symbol
Bracket
notation
Coxeter
diagram
2A2[3]CDel node.pngCDel 3.pngCDel node.png
2B2[4]CDel node.pngCDel 4.pngCDel node.png
2H2[5]CDel node.pngCDel 5.pngCDel node.png
2G2[6]CDel node.pngCDel 6.pngCDel node.png
2I2(p)[p]CDel node.pngCDel p.pngCDel node.png
3 Ih, H3[5,3]CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
3 Td, A3[3,3]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
3 Oh, B3[4,3]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
4A4[3,3,3]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
4B4[4,3,3]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
4D4[31,1,1]CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
4 F4 [3,4,3]CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
4H4[5,3,3]CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
nAn[3n−1]CDel node.pngCDel 3.pngCDel node.pngCDel 3.png..CDel 3.pngCDel node.pngCDel 3.pngCDel node.png
nBn[4,3n−2]CDel node.pngCDel 4.pngCDel node.pngCDel 3.png...CDel 3.pngCDel node.pngCDel 3.pngCDel node.png
nDn[3n−3,1,1]CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.png...CDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6 E6 [32,2,1]CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
7 E7 [33,2,1]CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
8 E8 [34,2,1]CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Affine groups
Group
symbol
Bracket
notation
Coxeter diagram
[∞]CDel node.pngCDel infin.pngCDel node.png
[3[3]]CDel node.pngCDel split1.pngCDel branch.png
[4,4]CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
[6,3]CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
[3[4]]CDel node.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node.png
[4,31,1]CDel node.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png
[4,3,4]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
[3[5]]CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel branch.png
[4,3,31,1]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
[4,3,3,4]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
[ 31,1,1,1]CDel nodes.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes.png
[3,4,3,3]CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
[3[n+1]]CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.png...CDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png
or
CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.png...CDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png
[4,3n−3,31,1]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.png...CDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
[4,3n−2,4]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.png...CDel 3.pngCDel node.pngCDel 4.pngCDel node.png
[ 31,1,3n−4,31,1]CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.png...CDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
[32,2,2]CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branchbranch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
[33,3,1]CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
[35,2,1]CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Hyperbolic groups
Group
symbol
Bracket
notation
Coxeter
diagram
[p,q]
with 2(p + q) < pq
CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png
[(p,q,r)]
with
CDel pqr.png
[4,3,5]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
[5,3,5]CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
[3,5,3]CDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
[5,31,1]CDel node.pngCDel 5.pngCDel node.pngCDel split1.pngCDel nodes.png
[(3,3,3,4)]CDel label4.pngCDel branch.pngCDel 3ab.pngCDel branch.png 
[(3,3,3,5)]CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch.png 
[(3,4,3,4)]CDel label4.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label4.png
[(3,4,3,5)]CDel label4.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label5.png
[(3,5,3,5)]CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label5.png
[3,3,3,5]CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
[4,3,3,5]CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
[5,3,3,5]CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
[5,3,31,1]CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
[(3,3,3,3,4)]CDel label4.pngCDel branch.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png

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

Subgroups

Coxeter's notation represents rotational/translational symmetry by adding a + superscript operator outside the brackets, [X]+ which cuts the order of the group [X] in half, thus an index 2 subgroup. This operator implies an even number of operators must be applied, replacing reflections with rotations (or translations). When applied to a Coxeter group, this is called a direct subgroup because what remains are only direct isometries without reflective symmetry.

The + operators can also be applied inside of the brackets, like [X,Y+] or [X,(Y,Z)+], and creates "semidirect" subgroups that may include both reflective and nonreflective generators. Semidirect subgroups can only apply to Coxeter group subgroups that have even order branches adjacent to it. Elements by parentheses inside of a Coxeter group can be give a + superscript operator, having the effect of dividing adjacent ordered branches into half order, thus is usually only applied with even numbers. For example, [4,3+] and [4,(3,3)+] (CDel node.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png).

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
GroupOrderGeneratorsSubgroupOrderGeneratorsNotes
[p]CDel node n0.pngCDel p.pngCDel node n1.png2p{0,1}[p]+CDel node h2.pngCDel p.pngCDel node h2.pngp{01}Direct subgroup
[2p+] = [2p]+CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h2.png2p{01}[2p+]+ = [2p]+2 = [p]+CDel node h2.pngCDel p.pngCDel node h2.pngp{0101}
[2p]CDel node n0.pngCDel 2x.pngCDel p.pngCDel node n1.png4p{0,1}[1+,2p] = [p]CDel node h0.pngCDel 2x.pngCDel p.pngCDel node.png = CDel node h2.pngCDel 2x.pngCDel p.pngCDel node.png = CDel node.pngCDel p.pngCDel node.png2p{101,1}Half subgroups
[2p,1+] = [p]CDel node.pngCDel 2x.pngCDel p.pngCDel node h0.png = CDel node.pngCDel 2x.pngCDel p.pngCDel node h2.png = CDel node.pngCDel p.pngCDel node.png{0,010}
[1+,2p,1+] = [2p]+2 = [p]+CDel node h0.pngCDel 2x.pngCDel p.pngCDel node h0.png = CDel node h2.pngCDel 2c.pngCDel 2x.pngCDel p.pngCDel 2c.pngCDel node h2.png = CDel node h2.pngCDel p.pngCDel node h2.pngp{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, CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png has symmetry [4,3]+ (CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png), and the snub tetrahedron, CDel node.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png has symmetry [4,3+] (CDel node.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png), and a demicube, h{4,3} = {3,3} (CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png or CDel nodes 10ru.pngCDel split2.pngCDel node.png = CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png) has symmetry [1+,4,3] = [3,3] (CDel node h2.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png or CDel node h0.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png = CDel nodes.pngCDel split2.pngCDel node.png = CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png).

Note: Pyritohedral symmetry CDel node.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png can be written as CDel node.pngCDel 4.pngCDel 2c.pngCDel node h2.pngCDel 3.pngCDel node h2.png, separating the graph with gaps for clarity, with the generators {0,1,2} from the Coxeter group CDel node n0.pngCDel 4.pngCDel node n1.pngCDel 3.pngCDel node n2.png, producing pyritohedral generators {0,12}, a reflection and 3-fold rotation. And chiral tetrahedral symmetry can be written as CDel node h0.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png or CDel node h2.pngCDel 2c.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png, [1+,4,3+] = [3,3]+, with generators {12,0120}.

Halving subgroups and extended groups

Example halving operations
Dihedral symmetry domains 4.png Dihedral symmetry 4 half1.png
CDel node c1.pngCDel 4.pngCDel node c3.png
[1,4,1] = [4]
CDel node h0.pngCDel 4.pngCDel node c3.png = CDel node c3.pngCDel 2x.pngCDel node c3.png = CDel node c3.pngCDel 2.pngCDel node c3.png
[1+,4,1]=[2]=[ ]×[ ]
Dihedral symmetry 4 half2.png Cyclic symmetry 4 half.png
CDel node c1.pngCDel 4.pngCDel node h0.png = CDel node c1.pngCDel 2x.pngCDel node c1.png = CDel node c1.pngCDel 2.pngCDel node c1.png
[1,4,1+]=[2]=[ ]×[ ]
CDel node h0.pngCDel 4.pngCDel node h0.png = CDel node h0.pngCDel 4.pngCDel node h2.png = CDel node h2.pngCDel 4.pngCDel node h0.png = CDel node h2.pngCDel 2x.pngCDel node h2.png
[1+,4,1+] = [2]+

Johnson extends the + operator to work with a placeholder 1+ nodes, which removes mirrors, doubling the size of the fundamental domain and cuts the group order in half. [1] In general this operation only applies to individual mirrors bounded by even-order branches. The 1 represents a mirror so [2p] can be seen as [2p,1], [1,2p], or [1,2p,1], like diagram CDel node.pngCDel 2x.pngCDel p.pngCDel node.png or CDel node c1.pngCDel 2x.pngCDel p.pngCDel node c3.png, 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: CDel node h0.pngCDel 2x.pngCDel p.pngCDel node c3.png = CDel labelp.pngCDel branch c3.png, 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: CDel node h0.pngCDel 2x.pngCDel p.pngCDel node.png = CDel node.pngCDel 2x.pngCDel p.pngCDel node h0.png = CDel labelp.pngCDel branch.png.

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. CDel node h2.pngCDel 2c.pngCDel 2x.pngCDel p.pngCDel 2c.pngCDel node h2.png = CDel node h0.pngCDel 2x.pngCDel p.pngCDel node h0.png = CDel node h0.pngCDel 2x.pngCDel p.pngCDel node h2.png = CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h0.png = CDel labelp.pngCDel branch h2h2.png.

For example, (with p=2): [4,1+] = [1+,4] = [2] = [ ]×[ ], order 4. [1+,4,1+] = [2]+, order 2.

The opposite to halving is doubling [2] which adds a mirror, bisecting a fundamental domain, and doubling the group order.

[[p]] = [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: CDel node h0.pngCDel 2x.pngCDel p.pngCDel node c1.pngCDel 3.pngCDel node c2.png = CDel labelp.pngCDel branch c1.pngCDel split2.pngCDel node c2.png, h[2p,3] = [1+,2p,3] = [(p,3,3)].

If nodes are indexed, half subgroups can be labeled with new mirrors as composites. Like CDel node n0.pngCDel 2x.pngCDel p.pngCDel node n1.png, generators {0,1} has subgroup CDel node h0.pngCDel 2x.pngCDel p.pngCDel node n1.png = CDel 2 n0.pngCDel node n1.pngCDel 3 n0.pngCDel p.pngCDel node n1.png, generators {1,010}, where mirror 0 is removed, and replaced by a copy of mirror 1 reflected across mirror 0. Also given CDel node n0.pngCDel 4.pngCDel node n1.pngCDel 3.pngCDel node n2.png, generators {0,1,2}, it has half group CDel node h0.pngCDel 4.pngCDel node n1.pngCDel 3.pngCDel node n2.png = CDel node n1.pngCDel 3.pngCDel node n2.pngCDel 3.pngCDel 3 n0.pngCDel node n1.pngCDel 2 n0.png, generators {1,2,010}.

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

Tetrahedral symmetry Octahedral symmetry
Sphere symmetry group td.png
Td, [3,3] = [1+,4,3]
CDel node c1.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c1.png = CDel nodeab c1.pngCDel split2.pngCDel node c1.png = CDel node h0.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c1.png
(Order 24)
Sphere symmetry group oh.png
Oh, [4,3] = [[3,3]]
CDel node c2.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c1.png
(Order 48)

Radical subgroups

A radical subgroup is similar to an alternation, but removes the rotational generators. 43-radial subgroups.png
A radical subgroup is similar to an alternation, but removes the rotational generators.

Johnson also added an asterisk or star * operator for "radical" subgroups, [3] that acts similar to the + operator, but removes rotational symmetry. The index of the radical subgroup is the order of the removed element. For example, [4,3*] ≅ [2,2]. The removed [3] subgroup is order 6 so [2,2] is an index 6 subgroup of [4,3].

The radical subgroups represent the inverse operation to an extended symmetry operation. For example, [4,3*] ≅ [2,2], and in reverse [2,2] can be extended as [3[2,2]] ≅ [4,3]. The subgroups can be expressed as a Coxeter diagram: CDel node c1.pngCDel 4.pngCDel node.pngCDel 3s.pngCDel node.png or CDel node c1.pngCDel 4.pngCDel node x.pngCDel 3.pngCDel node x.pngCDel nodeab c1.pngCDel 2.pngCDel node c1.png. 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

Rank 2 example, [6] trionic subgroups with 3 colors of mirror lines Trionic subgroups hexagonal symmetry.png
Rank 2 example, [6] trionic subgroups with 3 colors of mirror lines
Example on octahedral symmetry: [4,3 ] = [2,4]. 432 trionic subgroups.png
Example on octahedral symmetry: [4,3 ] = [2,4].
Example trionic subgroup on hexagonal symmetry [6,3] maps onto a larger [6,3] symmetry. Trionic subgroups hexagonal.png
Example trionic subgroup on hexagonal symmetry [6,3] maps onto a larger [6,3] symmetry.
Rank 3 Trionic Coxeter groups rank 3.png
Rank 3
Example trionic subgroups on octagonal symmetry [8,3] maps onto larger [4,8] symmetries. Hyperbolic 832 trionic subgroup 842.png
Example trionic subgroups on octagonal symmetry [8,3] maps onto larger [4,8] symmetries.
Rank 4 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 CDel node n0.pngCDel 3x.pngCDel p.pngCDel node n1.png, 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] = CDel node n0.pngCDel 3x.pngCDel p.pngCDel node trionic.png = CDel node n0.pngCDel p.pngCDel 3 n1.pngCDel 3 n0.pngCDel node n1.pngCDel 2 n0.pngCDel 2 n1.png, CDel node trionic.pngCDel 3x.pngCDel p.pngCDel node n1.png = CDel 2 n0.pngCDel 2 n1.pngCDel node n0.pngCDel 3 n1.pngCDel 3 n0.pngCDel p.pngCDel node n1.png, and [3p] = CDel node n0.pngCDel 3x.pngCDel 3trionic.pngCDel p.pngCDel node n1.png = CDel 2 n0.pngCDel node n1.pngCDel 3 n0.pngCDel p.pngCDel 3 n1.pngCDel node n0.pngCDel 2 n1.png 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 CDel node n0.pngCDel 2x.pngCDel p.pngCDel node n1.pngCDel 3trionic.pngCDel node n2.png = CDel 2 n2.pngCDel 2 n1.pngCDel node n0.pngCDel 3 n1.pngCDel 3 n2.pngCDel p.pngCDel node n0.pngCDel 2x.pngCDel p.pngCDel node n1.png. 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))], CDel node.pngCDel q.pngCDel node.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 3trionic.pngCDel node.png = CDel labelq.pngCDel branch.pngCDel split2-pq.pngCDel node.pngCDel 2x.pngCDel p.pngCDel node.png.

For example, [3,4,3] = [4,3,3], or CDel node n0.pngCDel 3.pngCDel node n1.pngCDel 4.pngCDel node n2.pngCDel 3trionic.pngCDel node n3.png = CDel 2 n3.pngCDel 2 n2.pngCDel node n1.pngCDel 3 n2.pngCDel 3 n3.pngCDel 3.pngCDel node n0.pngCDel 3.pngCDel node n1.pngCDel 4.pngCDel node n2.png, 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

[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). 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).
Trionic relations of [3,3] 33-trionic subgroups.png
Trionic relations of [3,3]

Johnson identified two specific trionic subgroups [4] of [3,3], first an index 3 subgroup [3,3] ≅ [2+,4], with [3,3] (CDel node n0.pngCDel 3.pngCDel node n1.pngCDel 3.pngCDel node n2.png = CDel node.pngCDel split1.pngCDel nodes.png = CDel node.pngCDel split1.pngCDel branch.pngCDel label2.png) generators {0,1,2}. It can also be written as [(3,3,2)] (CDel node.pngCDel split1.pngCDel 2c.pngCDel branch h2h2.pngCDel label2.png) 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)]+ (CDel node h2.pngCDel split1.pngCDel 2c.pngCDel branch h2h2.pngCDel label2.png), 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.

Trionic subgroup relations of [3,3,4] 334 trionic subgroups2.png
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] ≅ [[4,2,4]] ≅ [8,2+,8], order 128, are {02,1,3} from [3,3,4] generators {0,1,2,3}. And [(3,3)Δ,4] ≅ [[4,2+,4]], order 64, has generators {02,1021,3}. As well, [3,4,3] ≅ [(3,3),4].

Also related [31,1,1] = [3,3,4,1+] has trionic subgroups: [31,1,1] = [(3,3),4,1+], order 64, and 1=[31,1,1]Δ = [(3,3)Δ,4,1+] ≅ [[4,2+,4]]+, order 32.

Central inversion

A 2D central inversion is a 180 degree rotation, [2] Point Reflection.png
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 . The order of the mirrors doesn't matter in the case of an inversion. The matrix of a central inversion is , the Identity matrix with negative one on the diagonal.

From that basis, the central inversion has a generator as the product of all the orthogonal mirrors. In Coxeter notation this inversion group is expressed by adding an alternation + to each 2 branch. The alternation symmetry is marked on Coxeter diagram nodes as open nodes.

A Coxeter-Dynkin diagram can be marked up with explicit 2 branches defining a linear sequence of mirrors, open-nodes, and shared double-open nodes to show the chaining of the reflection generators.

For example, [2+,2] and [2,2+] are subgroups index 2 of [2,2], CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png, and are represented as CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2.pngCDel node.png (or CDel node h2.pngCDel 3.pngCDel 2c.pngCDel 3.pngCDel node h2.pngCDel 2.pngCDel node.png) and CDel node.pngCDel 2.pngCDel node h2.pngCDel 2x.pngCDel node h2.png (or CDel node.pngCDel 2.pngCDel node h2.pngCDel 3.pngCDel 2c.pngCDel 3.pngCDel node h2.png) with generators {01,2} and {0,12} respectively. Their common subgroup index 4 is [2+,2+], and is represented by CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.png (or CDel node h2.pngCDel 3.pngCDel 2c.pngCDel 3.pngCDel node h4.pngCDel 3.pngCDel 2c.pngCDel 3.pngCDel node h2.png), with the double-open CDel node h4.png marking a shared node in the two alternations, and a single rotoreflection generator {012}.

DimensionCoxeter notationOrderCoxeter diagramOperationGenerator
2[2]+2CDel node h2.pngCDel 2x.pngCDel node h2.png180° rotation, C2{01}
3[2+,2+]2CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.png rotoreflection, Ci or S2{012}
4[2+,2+,2+]2CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.png double rotation {0123}
5[2+,2+,2+,2+]2CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.pngdouble rotary reflection{01234}
6[2+,2+,2+,2+,2+]2CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.pngtriple rotation{012345}
7[2+,2+,2+,2+,2+,2+]2CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.pngtriple 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 Zn, 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 ±[Cp×Cq], [5] order 2pq. From Coxeter diagram CDel node n0.pngCDel 2x.pngCDel p.pngCDel node n1.pngCDel 2.pngCDel node n2.pngCDel 2x.pngCDel q.pngCDel node n3.png, generators {0,1,2,3}, requires two generator for [2p+,2+,2q+], CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel q.pngCDel node h2.png as {0123,0132}. Half groups, [2p+,2+,2q+]+, or cyclic graph, [(2p+,2+,2q+,2+)], CDel 3.pngCDel node h4.pngCDel 2x.pngCDel p.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel q.pngCDel node h4.pngCDel 2x.png expressed by Conway is [Cp×Cq], order pq, with one generator, like {0123}.

If there is a common factor f, the double rotation can be written as 1f[2pf+,2+,2qf+] (with gcd(p,q)=1), generators {0123,0132}, order 2pqf. For example, p=q=1, f=2, 12[4+,2+,4+] is order 4. And 1f[2pf+,2+,2qf+]+, generator {0123}, is order pqf. For example, 12[4+,2+,4+]+ is order 2, a central inversion.

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

Examples
DimensionCoxeter notationOrderCoxeter diagramOperationGeneratorsDirect subgroup
2[2p]+2pCDel node h2.pngCDel 2x.pngCDel p.pngCDel node h2.png Rotation {01}[2p]+2 = [p]+Simple rotation:
[2p]+2 = [p]+
order p
3[2p+,2+]CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h4.pngCDel 2x.pngCDel node h2.png rotary reflection {012}[2p+,2+]+ = [p]+
4[2p+,2+,2+]CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.png double rotation {0123}[2p+,2+,2+]+ = [p]+
5[2p+,2+,2+,2+]CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.pngdouble rotary reflection{01234}[2p+,2+,2+,2+]+ = [p]+
6[2p+,2+,2+,2+,2+]CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.pngtriple rotation{012345}[2p+,2+,2+,2+,2+]+ = [p]+
7[2p+,2+,2+,2+,2+,2+]CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.pngtriple rotary reflection{0123456}[2p+,2+,2+,2+,2+,2+]+ = [p]+
4[2p+,2+,2q+]2pqCDel node h2.pngCDel 2x.pngCDel p.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel q.pngCDel node h2.pngdouble rotation{0123,
0132}
[2p+,2+,2q+]+Double rotation:
[2p+,2+,2q+]+
order pq
5[2p+,2+,2q+,2+]CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel q.pngCDel node h4.pngCDel 2x.pngCDel node h2.pngdouble rotary reflection{01234,
01243}
[2p+,2+,2q+,2+]+
6[2p+,2+,2q+,2+,2+]CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel q.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.pngtriple rotation{012345,
012354,
013245}
[2p+,2+,2q+,2+,2+]+
7[2p+,2+,2q+,2+,2+,2+]CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel q.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.pngtriple rotary reflection{0123456,
0123465,
0124356,
0124356}
[2p+,2+,2q+,2+,2+,2+]+
6[2p+,2+,2q+,2+,2r+]2pqrCDel node h2.pngCDel 2x.pngCDel p.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel q.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel r.pngCDel node h2.pngtriple 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+]CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel q.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel r.pngCDel node h4.pngCDel 2x.pngCDel node h2.pngtriple rotary reflection{0123456,
0123465,
0124356,
0213456}
[2p+,2+,2q+,2+,2r+,2+]+

Commutator subgroups

Subgroups of [4,4], down to its commutator subgroup, index 8 Subgroups of 442.png
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 2c, where c is the number of disconnected subgraphs when all the even-order branches are removed. [6]

For example, [4,4] has three independent nodes in the Coxeter diagram when the 4s are removed, so its commutator subgroup is index 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], CDel node n0.pngCDel 4.pngCDel node n1.png has two mirror generators 0, and 1. Each generate two virtual mirrors 101 and 010 by reflection across the other.

Rank 3 Euclidean example subgroups

The [4,4] group has 15 small index subgroups. This table shows them all, with a yellow fundamental domain for pure reflective groups, and alternating white and blue domains which are paired up to make rotational domains. Cyan, red, and green mirror lines correspond to the same colored nodes in the Coxeter diagram. Subgroup generators can be expressed as products of the original 3 mirrors of the fundamental domain, {0,1,2}, corresponding to the 3 nodes of the Coxeter diagram, CDel node n0.pngCDel 4.pngCDel node n1.pngCDel 4.pngCDel node n2.png. A product of two intersecting reflection lines makes a rotation, like {012}, {12}, or {02}. Removing a mirror causes two copies of neighboring mirrors, across the removed mirror, like {010}, and {212}. Two rotations in series cut the rotation order in half, like {0101} or {(01)2}, {1212} or {(02)2}. A product of all three mirrors creates a transreflection, like {012} or {120}.

Hyperbolic example subgroups

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

Extended symmetry

Wallpaper
group
Triangle
symmetry
Extended
symmetry
Extended
diagram
Extended
group
Honeycombs
p3m1 (*333)a1 Triangle symmetry1.png [3[3]]CDel node.pngCDel split1.pngCDel branch.png(none)
p6m (*632)i2 Triangle symmetry3.png [[3[3]]] ↔ [6,3]CDel node c1.pngCDel split1.pngCDel branch c2.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 6.pngCDel node.pngCDel node 1.pngCDel split1.pngCDel branch.png 1 , CDel node.pngCDel split1.pngCDel branch 11.png 2
p31m (3*3)g3 Triangle symmetry2.png [3+[3[3]]] ↔ [6,3+]CDel branch.pngCDel split2.pngCDel node.pngCDel node.pngCDel 6.pngCDel node h2.pngCDel 3.pngCDel node h2.png(none)
p6 (632)r6 Triangle symmetry4.png [3[3[3]]]+ ↔ [6,3]+CDel branch c1.pngCDel split2.pngCDel node c1.pngCDel node c1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel branch hh.pngCDel split2.pngCDel node h.png (1)
p6m (*632)[3[3[3]]] ↔ [6,3]CDel branch 11.pngCDel split2.pngCDel node 1.png 3
In the Euclidean plane, the , [3[3]] Coxeter group can be extended in two ways into the , [6,3] Coxeter group and relates uniform tilings as ringed diagrams.

Coxeter's notation includes double square bracket notation, [[X]] to express automorphic symmetry within a Coxeter diagram. Johnson added alternative 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 : CDel branch.pngCDel 3ab.pngCDel 3ab.pngCDel branch.png and CDel node.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node.png, the first doubled with square brackets, [[3[4]]] or twice doubled as [2[3[4]]], with [2], order 4 higher symmetry. To differentiate the second, angled brackets are used for doubling, <[3[4]]> and twice doubled as <2[3[4]]>, also with a different [2], order 4 symmetry. Finally a full symmetry where all 4 nodes are equivalent can be represented by [4[3[4]]], with the order 8, [4] symmetry of the square. But by considering the tetragonal disphenoid fundamental domain the [4] extended symmetry of the square graph can be marked more explicitly as [(2+,4)[3[4]]] or [2+,4[3[4]]].

Further symmetry exists in the cyclic and branching , , and diagrams. has order 2n symmetry of a regular n-gon, {n}, and is represented by [n[3[n]]]. and are represented by [3[31,1,1]] = [3,4,3] and [3[32,2,2]] respectively while by [(3,3)[31,1,1,1]] = [3,3,4,3], with the diagram containing the order 24 symmetry of the regular tetrahedron, {3,3}. The paracompact hyperbolic group = [31,1,1,1,1], CDel node.pngCDel branch3.pngCDel splitsplit2.pngCDel node.pngCDel split1.pngCDel nodes.png, contains the symmetry of a 5-cell, {3,3,3}, and thus is represented by [(3,3,3)[31,1,1,1,1]] = [3,4,3,3,3].

An asterisk * superscript is effectively an inverse operation, creating radical subgroups removing connected of odd-ordered mirrors. [7]

Examples:

Example Extended groups and radical subgroups
Extended groupsRadical subgroups Coxeter diagrams Index
[3[2,2]] = [4,3][4,3*] = [2,2]CDel node c1.pngCDel 4.pngCDel node.pngCDel 3s.pngCDel node.png = CDel node c1.pngCDel 2.pngCDel nodeab c1.png6
[(3,3)[2,2,2]] = [4,3,3][4,(3,3)*] = [2,2,2]CDel node c1.pngCDel 4.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 3g.pngCDel node g.png = CDel nodeab c1.pngCDel 2.pngCDel nodeab c1.png24
[1[31,1]] = [[3,3]] = [3,4][3,4,1+] = [3,3]CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 4.pngCDel node h0.png = CDel node c1.pngCDel split1.pngCDel nodeab c2.png2
[3[31,1,1]] = [3,4,3][3*,4,3] = [31,1,1]CDel node.pngCDel 3s.pngCDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.png = CDel node c1.pngCDel branch3 c1.pngCDel splitsplit2.pngCDel node c2.png6
[2[31,1,1,1]] = [4,3,3,4][1+,4,3,3,4,1+] = [31,1,1,1]CDel node h0.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 4.pngCDel node h0.png = CDel nodeab c1.pngCDel split2.pngCDel node c2.pngCDel split1.pngCDel nodeab c3.png4
[3[3,31,1,1]] = [3,3,4,3][3*,4,3,3] = [31,1,1,1]CDel node.pngCDel 3s.pngCDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.png = CDel node c1.pngCDel branch3 c1.pngCDel splitsplit2.pngCDel node c2.pngCDel 3.pngCDel node c3.png6
[(3,3)[31,1,1,1]] = [3,4,3,3][3,4,(3,3)*] = [31,1,1,1]CDel node c2.pngCDel 3.pngCDel node c1.pngCDel 4.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 3g.pngCDel node g.png = CDel nodeab c1.pngCDel split2.pngCDel node c2.pngCDel split1.pngCDel nodeab c1.png24
[2[3,31,1,1,1]] = [3,(3,4)1,1][3,(3,4,1+)1,1] = [3,31,1,1,1]CDel node c4.pngCDel 3.pngCDel node c3.pngCDel split1.pngCDel nodeab c1-2.pngCDel 4a4b.pngCDel nodes.png = CDel node c4.pngCDel branch3 c1.pngCDel splitsplit2.pngCDel node c3.pngCDel split1.pngCDel nodeab c2.png4
[(2,3)[1,131,1,1]] = [4,3,3,4,3][3*,4,3,3,4,1+] = [31,1,1,1,1]CDel node.pngCDel 3s.pngCDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 4.pngCDel node h0.png = CDel node c1.pngCDel branch3 c1.pngCDel splitsplit2.pngCDel node c2.pngCDel split1.pngCDel nodeab c3.png12
[(3,3)[3,31,1,1,1]] = [3,3,4,3,3][3,3,4,(3,3)*] = [31,1,1,1,1]CDel node c3.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c1.pngCDel 4.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 3g.pngCDel node g.png = CDel node c3.pngCDel branch3 c1.pngCDel splitsplit2.pngCDel node c2.pngCDel split1.pngCDel nodeab c1.png24
[(3,3,3)[31,1,1,1,1]] = [3,4,3,3,3][3,4,(3,3,3)*] = [31,1,1,1,1]CDel node c2.pngCDel 3.pngCDel node c1.pngCDel 4.pngCDel node g.pngCDel 3g.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 3g.pngCDel node g.png = CDel node c1.pngCDel branch3 c1.pngCDel splitsplit2.pngCDel node c2.pngCDel split1.pngCDel nodeab c1.png120
Extended groupsRadical subgroupsCoxeter diagramsIndex
[1[3[3]]] = [3,6][3,6,1+] = [3[3]]CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 6.pngCDel node h0.png = CDel node c1.pngCDel split1.pngCDel branch c2.png2
[3[3[3]]] = [6,3][6,3*] = [3[3]]CDel node c1.pngCDel 6.pngCDel node.pngCDel 3s.pngCDel node.png = CDel node c1.pngCDel split1.pngCDel branch c1.png6
[1[3,3[3]]] = [3,3,6][3,3,6,1+] = [3,3[3]]CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 6.pngCDel node h0.png = CDel node c1.pngCDel 3.pngCDel node c2.pngCDel split1.pngCDel branch c3.png2
[(3,3)[3[3,3]]] = [6,3,3][6,(3,3)*] = [3[3,3]]CDel node c1.pngCDel 6.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 3g.pngCDel node g.png = CDel node c1.pngCDel splitsplit1.pngCDel branch4 c1.pngCDel splitsplit2.pngCDel node c1.png24
[1[]2] = [4,4][4,1+,4] = []2 = [,2,]CDel node c1.pngCDel 4.pngCDel node h0.pngCDel 4.pngCDel node c2.png = CDel labelinfin.pngCDel branch c1-2.pngCDel 2.pngCDel branch c1-2.pngCDel labelinfin.png2
[2[]2] = [4,4][1+,4,4,1+] = [(4,4,2*)] = []2CDel node h0.pngCDel 4.pngCDel node c2.pngCDel 4.pngCDel node h0.png = CDel labelinfin.pngCDel branch c2.pngCDel 2.pngCDel branch c2.pngCDel labelinfin.png4
[4[]2] = [4,4][4,4*] = []2CDel node c1.pngCDel 4.pngCDel node g.pngCDel 4sg.pngCDel node g.png = CDel labelinfin.pngCDel branch c1.pngCDel 2.pngCDel branch c1.pngCDel labelinfin.png8
[2[3[4]]] = [4,3,4][1+,4,3,4,1+] = [(4,3,4,2*)] = [3[4]]CDel node h0.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 4.pngCDel node h0.png = CDel node c1.pngCDel split1.pngCDel nodeab c2.pngCDel split2.pngCDel node c1.png = CDel nodeab c1.pngCDel splitcross.pngCDel nodeab c2.png4
[3[]3] = [4,3,4][4,3*,4] = []3 = [,2,,2,]CDel node c1.pngCDel 4.pngCDel node.pngCDel 3s.pngCDel node.pngCDel 4.pngCDel node c2.png = CDel labelinfin.pngCDel branch c1-2.pngCDel 2.pngCDel labelinfin.pngCDel branch c1-2.pngCDel 2.pngCDel labelinfin.pngCDel branch c1-2.png6
[(3,3)[]3] = [4,31,1][4,(31,1)*] = []3CDel node c1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png = CDel labelinfin.pngCDel branch c1.pngCDel 2.pngCDel labelinfin.pngCDel branch c1.pngCDel 2.pngCDel labelinfin.pngCDel branch c1.png24
[(4,3)[]3] = [4,3,4][4,(3,4)*] = []3CDel node c1.pngCDel 4.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 4g.pngCDel node g.png = CDel labelinfin.pngCDel branch c1.pngCDel 2.pngCDel labelinfin.pngCDel branch c1.pngCDel 2.pngCDel labelinfin.pngCDel branch c1.png48
[(3,3)[]4] = [4,3,3,4][4,(3,3)*,4] = []4CDel node c1.pngCDel 4.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 3g.pngCDel node g.pngCDel 4.pngCDel node c2.png = CDel labelinfin.pngCDel branch c1-2.pngCDel 2.pngCDel labelinfin.pngCDel branch c1-2.pngCDel 2.pngCDel labelinfin.pngCDel branch c1-2.pngCDel 2.pngCDel labelinfin.pngCDel branch c1-2.png24
[(4,3,3)[]4] = [4,3,3,4][4,(3,3,4)*] = []4CDel node c1.pngCDel 4.pngCDel node g.pngCDel 3g.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 4g.pngCDel node g.png = CDel labelinfin.pngCDel branch c1.pngCDel 2.pngCDel labelinfin.pngCDel branch c1.pngCDel 2.pngCDel labelinfin.pngCDel branch c1.pngCDel 2.pngCDel labelinfin.pngCDel branch c1.png384

Looking at generators, the double symmetry is seen as adding a new operator that maps symmetric positions in the Coxeter diagram, making some original generators redundant. For 3D space groups, and 4D point groups, Coxeter defines an index two subgroup of [[X]], [[X]+], which he defines as the product of the original generators of [X] by the doubling generator. This looks similar to [[X]]+, which is the chiral subgroup of [[X]]. So for example the 3D space groups [[4,3,4]]+ (I432, 211) and [[4,3,4]+] (Pm3n, 223) are distinct subgroups of [[4,3,4]] (Im3m, 229).

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–Dynkin diagram with a single node, CDel node.png. 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, CDel node h2.png.

GroupCoxeter notation Coxeter diagram OrderDescription
C1[ ]+CDel node h2.png1Identity
D2[ ]CDel node.png2Reflection 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. Regular hexagon symmetries2.png
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, CDel node.pngCDel 2.pngCDel node.png, with order 4. The 2 in [2] comes from linearization of the orthogonal subgraphs in the Coxeter diagram, as CDel node.pngCDel 2x.pngCDel node.png with explicit branch order 2. The rhombic group, [2]+ (CDel node h2.pngCDel 2x.pngCDel node h2.png or CDel node h2.pngCDel 3.pngCDel 2c.pngCDel 3.pngCDel node h2.png), 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 CDel node h2.png.

The full p-gonal group [p], abstract dihedral group D2p, (nonabelian for p>2), of order 2p, is generated by two mirrors at angle π/p, represented by Coxeter diagram CDel node.pngCDel p.pngCDel node.png. The p-gonal subgroup [p]+, cyclic group Zp, 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 CDel node.pngCDel infin.pngCDel node.png. The apeirogonal group [∞]+, CDel node h2.pngCDel infin.pngCDel node h2.png, 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]+, CDel node h2.pngCDel ultra.pngCDel node h2.png. These groups exist in regular infinite-sided polygons, with edge length λ. The mirrors are all orthogonal to a single line.