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 hosohedron2.png
Order 2
Spherical square hosohedron2.png
Order 4
Spherical hexagonal hosohedron2.png
Order 6
Spherical octagonal hosohedron2.png
Order 8
Spherical decagonal hosohedron2.png
Order 10
Spherical dodecagonal hosohedron2.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 bipyramid2.png
Order 4
Spherical square bipyramid2.png
Order 8
Spherical hexagonal bipyramid2.png
Order 12
Spherical octagonal bipyramid2.png
Order 16
Spherical decagonal bipyramid2.png
Order 20
Spherical dodecagonal bipyramid2.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
Tetrahedral reflection domains.png
Order 24
Octahedral reflection domains.png
Order 48
Icosahedral reflection domains.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 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 with fundamental reflections of a Coxeter group in 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[edit]

Further information: Point group

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].

Coxeter initially represented bifurcating diagrams with vertical positioning of numbers, but later abbreviated with an exponent notation, like [3p,q,r], starting with [31,1,1] = 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] = [33,1,0] = [32,2,0], 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 parenthesese 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 [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 = A22 can be represented by [3]×[3] = [3]2 = [3,2,3].

Finite Coxeter groups
Rank Group
symbol
Bracket
notation
Coxeter
diagram
2 A2 [3] CDel node.pngCDel 3.pngCDel node.png
2 BC2 [4] CDel node.pngCDel 4.pngCDel node.png
2 H2 [5] CDel node.pngCDel 5.pngCDel node.png
2 G2 [6] CDel node.pngCDel 6.pngCDel node.png
2 I2(p) [p] CDel node.pngCDel p.pngCDel node.png
3 H3 [5,3] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
3 A3 [3,3] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
3 BC3 [4,3] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
4 A4 [3,3,3] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
4 BC4 [4,3,3] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
4 D4 [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
4 H4 [5,3,3] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
n An [3n-1] CDel node.pngCDel 3.pngCDel node.pngCDel 3.png..CDel 3.pngCDel node.pngCDel 3.pngCDel node.png
n BCn [4,3n-2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.png...CDel 3.pngCDel node.pngCDel 3.pngCDel node.png
n Dn [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 Coxeter groups
Group
symbol
Bracket
notation
Coxeter
diagram
{\tilde{I}}_1 [∞] CDel node.pngCDel infin.pngCDel node.png
{\tilde{A}}_2 [3[3]] CDel node.pngCDel split1.pngCDel branch.png
{\tilde{C}}_2 [4,4] CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
{\tilde{G}}_2 [6,3] CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
{\tilde{A}}_3 [3[4]] CDel node.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node.png
{\tilde{B}}_3 [4,31,1] CDel node.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png
{\tilde{C}}_3 [4,3,4] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
{\tilde{A}}_4 [3[5]] CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel branch.png
{\tilde{B}}_4 [4,3,31,1] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
{\tilde{C}}_4 [4,3,3,4] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
{\tilde{D}}_4 [ 31,1,1,1] CDel nodes.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes.png
{\tilde{F}}_4 [3,4,3,3] CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
{\tilde{A}}_n [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
{\tilde{B}}_n [4,3n-2,31,1] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.png...CDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
{\tilde{C}}_n [4,3n-1,4] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.png...CDel 3.pngCDel node.pngCDel 4.pngCDel node.png
{\tilde{D}}_n [ 31,1,3n-3,31,1] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.png...CDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
{\tilde{E}}_6 [32,2,2] CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
{\tilde{E}}_7 [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
{\tilde{E}}_8 [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
Compact Hyperbolic Coxeter 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 p+q+r>9
CDel pqr.png
{\bar{BH}}_3 [4,3,5] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
{\bar{K}}_3 [5,3,5] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
{\bar{J}}_3 [3,5,3] CDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
{\bar{DH}}_3 [5,31,1] CDel node.pngCDel 5.pngCDel node.pngCDel split1.pngCDel nodes.png
{\widehat{AB}}_3 [(3,3,3,4)] CDel label4.pngCDel branch.pngCDel 3ab.pngCDel branch.png 
{\widehat{AH}}_3 [(3,3,3,5)] CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch.png 
{\widehat{BB}}_3 [(3,4,3,4)] CDel label4.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label4.png
{\widehat{BH}}_3 [(3,4,3,5)] CDel label4.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label5.png
{\widehat{HH}}_3 [(3,5,3,5)] CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label5.png
{\bar{H}}_4 [3,3,3,5] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
{\bar{BH}}_4 [4,3,3,5] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
{\bar{K}}_4 [5,3,3,5] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
{\bar{DH}}_4 [5,3,31,1] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
{\widehat{AF}}_4 [(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[edit]

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). This is called a direct subgroup because what remains are only direct isometries without reflective symmetry.

+ operators can also be applied inside of the brackets, and creates "semidirect" subgroups that include both reflective and nonreflective generators. Semidirect subgroups can only apply to Coxeter group subgroups that have even order branches next 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+] (CDel node.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png) and [4,(3,3)+] (CDel node.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png). The subgroup index is 2n for n + operators.

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 CDel node h.pngCDel 4.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).

Halving subgroups[edit]

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. 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 [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: [2p] = [1+,2p, 1] = [1,p,1] = [p].

Each of these mirrors can be removed so [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, the branch order becomes a gyration point of half the order:

[1+,2p,1+] = [p]+, a rotational subgroup of index 4. 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 which adds a mirror, bisecting a fundamental domain, and doubling the group order.

[[p]] = [2p]

Halving operations apply for higher rank groups, like [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, [1+,2p,3] = [(p,3,3)].

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[edit]

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

Johnson also added an asterisk or star * operator, 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 g.pngCDel 3sg.pngCDel node g.png CDel node c1.pngCDel 2.pngCDel node 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], index 2 has generators {010,1,2}; while radical subgroup [4,3*], index 6, has generators {01210, 2, (012)3}; and finally [1+,4,3*], index 12 has generators {0(12)20, (012)201}.

Trionic subgroups[edit]

Trionic subgroup relations of [3,3]
Trionic subgroup relations of [3,3,4]

Johnson identified two specific subgroups 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) generators {0,1,2}. It can also be written as [(3,3,2)] (CDel node.pngCDel split1.pngCDel 2.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)]+, index 3 from [3,3]+ [2,2]+, with generators {02,1021}, from [3,3] and its generators {0,1,2}.

These subgroups also apply within larger Coxeter groups with [3,3] subgroup with neighboring branches all even order.

For example [(3,3)+,4], [(3,3),4], and [(3,3)Δ,4] are subgroups of [3,3,4], index 2, 3 and 6 respectively. The generators of [(3,3),4] [[4,2,4]] [8,2+,8], order 128, are {02,1,3} from [3,3,4] generators {0,1,2,3}. And [(3,3)Δ,4] [[4,2+,4]], order 64, has generators {02,1021,3}.

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

Central inversion[edit]

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 0..n-1. The order of the mirrors doesn't matter in the case of an inversion.

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 and CDel node.pngCDel 2.pngCDel node h2.pngCDel 2x.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, with the double-open CDel node h4.png marking a shared node in the two alternations.

Dimension Coxeter notation Order Coxeter diagram Operation Generator
2 [2]+ 2 CDel node h2.pngCDel 2x.pngCDel node h2.png rotation {01}
3 [2+,2+] 2 CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.png rotary reflection {012}
4 [2+,2+,2+] 2 CDel 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+] 2 CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.png double rotary reflection {01234}
6 [2+,2+,2+,2+,2+] 2 CDel 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.png triple rotation {012345}
7 [2+,2+,2+,2+,2+,2+] 2 CDel 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.png triple rotary reflection {0123456}

Rotations and rotary reflections[edit]

Rotations and rotary reflections are constructed by a single single-generator product of all the reflections of a prismatic group, [2p]×[2q]×... When 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+], which include a central group, and are expressed by Conway as ±[Cp×Cq], order 2pq/gcd(p,q).[1]

Dimension Coxeter notation Order Coxeter diagram Operation Generator Direct subgroup
2 [p]+ p CDel node h2.pngCDel p.pngCDel node h2.png Rotation {01} [p]+
3 [2p+,2+] 2p CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h4.pngCDel 2x.pngCDel node h2.png rotary reflection {012}
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}
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.png double rotary reflection {01234}
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.png triple rotation {012345}
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 h2.pngCDel 2x.pngCDel node h2.png triple rotary reflection {0123456}
4 [2p+,2+,2q+] 2pq CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel q.pngCDel node h2.png double rotation {0123} [p+,2,q+]
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.png double rotary reflection {01234}
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.png triple rotation {012345}
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.png triple rotary reflection {0123456}
6 [2p+,2+,2q+,2+,2r+] 2pqr 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 h2.png triple rotation {012345} [p+,2,q+,2,r+]
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.png triple rotary reflection {0123456}

Commutator subgroups[edit]

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.[2] 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[edit]

Rank 2 example subgroups[edit]

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.

Subgroups of [4]
Index 1 2 (half) 4 (Rank-reduction)
Diagram Dihedral symmetry domains 4.png Dihedral symmetry 4 half1.png Dihedral symmetry 4 half2.png Dihedral symmetry 4 quarter1.png Dihedral symmetry 4 quarter2.png
Coxeter
CDel node n0.pngCDel 4.pngCDel node n1.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] = [1+,4] = [2]
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+] = [4,1+] = [2]
CDel node c1.png
[1] = [ ]
CDel node c3.png
[1] = [ ]
Generators {0,1} {101,1} {0,010} {0} {1}
Direct subgroups
Index 2 4 8
Diagram Cyclic symmetry 4.png Cyclic symmetry 4 half.png Dihedral symmetry 4 eighth.png
Coxeter CDel node h2.pngCDel 4.pngCDel node h2.png
[4]+
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
[4]+2 = [1+,4,1+] = [2]+
CDel node h2.png
[ ]+
Generators {01} {(01)2} {02} = {12} = {(01)4} = { }

Rank 3 Euclidean example subgroups[edit]

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}.

Small index subgroups of [4,4]
Index 1 2 4
Diagram 442 symmetry 000.png 442 symmetry a00.png 442 symmetry 00a.png 442 symmetry 0a0.png 442 symmetry a0b.png 442 symmetry xxx.png
Coxeter
CDel node n0.pngCDel 4.pngCDel node n1.pngCDel 4.pngCDel node n2.png
[1,4,1,4,1] = [4,4]
CDel node c5.pngCDel 4.pngCDel node c1.pngCDel 4.pngCDel node c3.png
[1+,4,4]
CDel node h0.pngCDel 4.pngCDel node c1.pngCDel 4.pngCDel node c3.png = CDel nodeab c1.pngCDel split2-44.pngCDel node c3.png
[4,4,1+]
CDel node c5.pngCDel 4.pngCDel node c1.pngCDel 4.pngCDel node h0.png = CDel node c5.pngCDel split1-44.pngCDel nodeab c1.png
[4,1+,4]
CDel node c5.pngCDel 4.pngCDel node h0.pngCDel 4.pngCDel node c3.png = CDel nodeab c5.pngCDel 2a2b-cross.pngCDel nodeab c3.png
[1+,4,4,1+]
CDel node h0.pngCDel 4.pngCDel node c1.pngCDel 4.pngCDel node h0.png = CDel nodeab c1.pngCDel 2a2b-cross.pngCDel nodeab c1.png
[4+,4+]
CDel node h2.pngCDel 4.pngCDel node h4.pngCDel 4.pngCDel node h2.png
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 442 symmetry 0aa.png 442 symmetry aa0.png 442 symmetry a0a.png 442 symmetry 0ab.png 442 symmetry ab0.png
Coxeter [4,4+]
CDel node c5.pngCDel 4.pngCDel node h2.pngCDel 4.pngCDel node h2.png
[4+,4]
CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 4.pngCDel node c3.png
[(4,4,2+)]
CDel node c1.pngCDel split1-44.pngCDel branch h2h2.pngCDel label2.png = CDel nodeab c1.pngCDel 2a2b-cross.pngCDel branch h2h2.pngCDel label2.png
[4,1+,4,1+]
CDel node c5.pngCDel 4.pngCDel node h0.pngCDel 4.pngCDel node h0.png = CDel node c5.pngCDel split1-44.pngCDel branch h2h2.pngCDel label2.png = CDel nodeab c5.pngCDel 2a2b-cross.pngCDel branch h2h2.pngCDel label2.png
[1+,4,1+,4]
CDel node h0.pngCDel 4.pngCDel node h0.pngCDel 4.pngCDel node c3.png = CDel label2.pngCDel branch h2h2.pngCDel split2-44.pngCDel node c3.png = CDel label2.pngCDel branch h2h2.pngCDel 2a2b-cross.pngCDel nodeab c3.png
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 442 symmetry aaa.png 442 symmetry abb.png 442 symmetry aab.png 442 symmetry aba.png 442 symmetry abc.png
Coxeter [4,4]+
CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 4.pngCDel node h2.png = CDel node h2.pngCDel split1-44.pngCDel branch h2h2.pngCDel label2.png
[4,4+]+
CDel node h0.pngCDel 4.pngCDel node h2.pngCDel 4.pngCDel node h2.png = CDel node h2.pngCDel split1-44.pngCDel branch h2h2.pngCDel label2.png
[4+,4]+
CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 4.pngCDel node h0.png = CDel node h2.pngCDel split1-44.pngCDel branch h2h2.pngCDel label2.png
[(4,4,2+)]+
CDel labelh.pngCDel node.pngCDel split1-44.pngCDel branch h2h2.pngCDel label2.png = CDel label2.pngCDel branch h2h2.pngCDel 2xa2xb-cross.pngCDel branch h2h2.pngCDel label2.png
[4,4]+3 = [(4+,4+,2+)] = [1+,4,1+,4,1+] = [4+,4+]+
CDel node h4.pngCDel split1-44.pngCDel branch h4h4.pngCDel label2.png =CDel node h2.pngCDel 4.pngCDel node h0.pngCDel 4.pngCDel node h0.png = CDel node h0.pngCDel 4.pngCDel node h0.pngCDel 4.pngCDel node h2.png = CDel node h0.pngCDel 4.pngCDel node h2.pngCDel 4.pngCDel node h0.png = CDel label2.pngCDel branch h2h2.pngCDel 2xa2xb-cross.pngCDel branch h2h2.pngCDel label2.png
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 442 symmetry 0dd.png 442 symmetry dd0.png 442 symmetry add.png 442 symmetry dda.png
Coxeter [4,4*]
CDel node c5.pngCDel 4.pngCDel node g.pngCDel 4sg.pngCDel node g.png = CDel nodeab c5.pngCDel 2a2b-cross.pngCDel nodeab c5.png
[4*,4]
CDel node g.pngCDel 4sg.pngCDel node g.pngCDel 4.pngCDel node c3.png = CDel nodeab c3.pngCDel 2a2b-cross.pngCDel nodeab c3.png
[4,4*]+
CDel node h2.pngCDel 4.pngCDel node g.pngCDel 4sg.pngCDel node g.png = CDel label2.pngCDel branch h2h2.pngCDel 2xa2xb-cross.pngCDel branch h2h2.pngCDel label2.png
[4*,4]+
CDel node g.pngCDel 4sg.pngCDel node g.pngCDel 4.pngCDel node h2.png = CDel label2.pngCDel branch h2h2.pngCDel 2xa2xb-cross.pngCDel branch h2h2.pngCDel label2.png
Orbifold *2222 2222

Hyperbolic example subgroups[edit]

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 642 symmetry 000.png 642 symmetry a00.png 642 symmetry 00a.png 642 symmetry 0a0.png 642 symmetry a0b.png 642 symmetry xxx.png
Coxeter
CDel node n0.pngCDel 6.pngCDel node n1.pngCDel 4.pngCDel node n2.png
[1,6,1,4,1] = [6,4]
CDel node c3.pngCDel 6.pngCDel node c1.pngCDel 4.pngCDel node c2.png
[1+,6,4]
CDel node h0.pngCDel 6.pngCDel node c1.pngCDel 4.pngCDel node c2.png = CDel branch c1.pngCDel split2-44.pngCDel node c2.png
[6,4,1+]
CDel node c3.pngCDel 6.pngCDel node c1.pngCDel 4.pngCDel node h0.png = CDel node c3.pngCDel split1-66.pngCDel nodeab c1.png
[6,1+,4]
CDel node c3.pngCDel 6.pngCDel node h0.pngCDel 4.pngCDel node c2.png = CDel branch c3.pngCDel 2a2b-cross.pngCDel nodeab c2.png
[1+,6,4,1+]
CDel node h0.pngCDel 6.pngCDel node c1.pngCDel 4.pngCDel node h0.png = CDel branch c1.pngCDel 2a2b-cross.pngCDel branch c1.png
[6+,4+]
CDel node h2.pngCDel 6.pngCDel node h4.pngCDel 4.pngCDel node h2.png
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 642 symmetry 0aa.png 642 symmetry aa0.png 642 symmetry a0a.png 642 symmetry 0ab.png 642 symmetry ab0.png
Coxeter [6,4+]
CDel node c3.pngCDel 6.pngCDel node h2.pngCDel 4.pngCDel node h2.png
[6+,4]
CDel node h2.pngCDel 6.pngCDel node h2.pngCDel 4.pngCDel node c2.png
[(6,4,2+)]
CDel node c1.pngCDel split1-46.pngCDel branch h2h2.pngCDel label2.png
[6,1+,4,1+]
CDel node c3.pngCDel 6.pngCDel node h0.pngCDel 4.pngCDel node h0.png = CDel node c3.pngCDel 6.pngCDel node h2.pngCDel 4.pngCDel node h0.png = CDel node c3.pngCDel split1-66.pngCDel branch h2h2.pngCDel label2.png
= CDel node c3.pngCDel 6.pngCDel node h0.pngCDel 4.pngCDel node h2.png = CDel branch c3.pngCDel 2a2b-cross.pngCDel branch h2h2.pngCDel label2.png
[1+,6,1+,4]
CDel node h0.pngCDel 6.pngCDel node h0.pngCDel 4.pngCDel node c2.png = CDel node h0.pngCDel 6.pngCDel node h2.pngCDel 4.pngCDel node c2.png = CDel branch h2h2.pngCDel split2-44.pngCDel node c2.png
= CDel node h2.pngCDel 6.pngCDel node h0.pngCDel 4.pngCDel node c2.png = CDel branch h2h2.pngCDel 2a2b-cross.pngCDel nodeab c2.png
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 642 symmetry aaa.png 642 symmetry abb.png 642 symmetry aab.png 642 symmetry aba.png 642 symmetry abc.png
Coxeter [6,4]+
CDel node h2.pngCDel 6.pngCDel node h2.pngCDel 4.pngCDel node h2.png = CDel node h2.pngCDel split1-64.pngCDel branch h2h2.pngCDel label2.png
[6,4+]+
CDel node h0.pngCDel 6.pngCDel node h2.pngCDel 4.pngCDel node h2.png = CDel branch h2h2.pngCDel split2-44.pngCDel node h2.png
[6+,4]+
CDel node h2.pngCDel 6.pngCDel node h2.pngCDel 4.pngCDel node h0.png = CDel node h2.pngCDel split1-66.pngCDel branch h2h2.pngCDel label2.png
[(6,4,2+)]+
CDel labelh.pngCDel node.pngCDel split1-46.pngCDel branch h2h2.pngCDel label2.png = CDel branch h2h2.pngCDel 2xa2xb-cross.pngCDel branch h2h2.pngCDel label2.png
[6+,4+]+ = [1+,6,1+,4,1+]
CDel node h4.pngCDel split1-46.pngCDel branch h4h4.pngCDel label2.png = CDel node h0.pngCDel 6.pngCDel node h0.pngCDel 4.pngCDel node h0.png
= CDel node h0.pngCDel 6.pngCDel node h2.pngCDel 4.pngCDel node h0.png = CDel branch h2h2.pngCDel 2xa2xb-cross.pngCDel branch h2h2.png
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 642 symmetry 0zz.png 642 symmetry zz0.png 642 symmetry azz.png 642 symmetry zza.png
Coxeter
(orbifold)
[6,4*]
CDel node c3.pngCDel 6.pngCDel node g.pngCDel 4sg.pngCDel node g.png = CDel branch c3.pngCDel 3a3b-cross.pngCDel branch c3.png
(*3333)
[6*,4]
CDel node g.pngCDel 6.pngCDel 3sg.pngCDel node g.pngCDel 4.pngCDel node c2.png
(*222222)
[6,4*]+
CDel node h2.pngCDel 6.pngCDel node g.pngCDel 4sg.pngCDel node g.png = CDel branch h2h2.pngCDel 3a3b-cross.pngCDel branch h2h2.png
(3333)
[6*,4]+
CDel node g.pngCDel 6.pngCDel 3sg.pngCDel node g.pngCDel 4.pngCDel node h2.png
(222222)

Extended symmetry[edit]

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 can either represent symmetry of the Coxeter diagram of [X], or symmetry of the fundamental domain of [X].

For example in these equivalent rectangle and rhombic geometry diagrams of {\tilde{A}}_3: 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 {\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[31,1,1]] = [3,4,3] and [3[32,2,2]] respectively while {\tilde{D}}_4 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 {\bar{L}}_5 = [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 astrick * superscript is effectively an inverse operation, creating radical subgroups removing connected of odd-ordered mirrors.[3]

Examples:

Extended groups Radical subgroups Coxeter diagrams Index
[3[2,2]] = [4,3] [4,3*] = [2,2] CDel node c1.pngCDel 4.pngCDel node g.pngCDel 3sg.pngCDel node g.png = CDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c1.png 6
[(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 node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c1.pngCDel 2.pngCDel node c1.png 24
[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.png 2
[3[31,1,1]] = [3,4,3] [3*,4,3] = [31,1,1] CDel node g.pngCDel 3sg.pngCDel node g.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.png = CDel node c1.pngCDel branch3 c1.pngCDel splitsplit2.pngCDel node c2.png 6
[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.png 4
[3[3,31,1,1]] = [3,3,4,3] [3*,4,3,3] = [31,1,1,1] CDel node g.pngCDel 3sg.pngCDel node g.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.png 6
[(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.png 24
[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.png 4
[(2,3)[1,131,1,1]] = [4,3,3,4,3] [1+,4,3,3,4,3+] = [31,1,1,1,1] CDel node g.pngCDel 3sg.pngCDel node g.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.png 12
[(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.png 24
[(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.png 120
Extended groups Radical subgroups Coxeter diagrams Index
[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.png 2
[3[3[3]]] = [6,3] [6,3*] = [3[3]] CDel node c1.pngCDel 6.pngCDel node g.pngCDel 3sg.pngCDel node g.png = CDel node c1.pngCDel split1.pngCDel branch c1.png 6
[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.png 2
[(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.png 24
[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.png 2
[2[∞]2] = [4,4] [1+,4,4,1+] = [(4,4,2*)] = [∞]2 CDel 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.png 4
[4[∞]2] = [4,4] [4,4*] = [∞]2 CDel 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.png 8
[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.png 4
[3[∞]3] = [4,3,4] [4,3*,4] = [∞]3 = [∞,2,∞,2,∞] CDel node c1.pngCDel 4.pngCDel node g.pngCDel 3sg.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.png 6
[(3,3)[∞]3] = [4,31,1] [4,(31,1)*] = [∞]3 CDel 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.png 24
[(4,3)[∞]3] = [4,3,4] [4,(3,4)*] = [∞]3 CDel 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.png 48
[(3,3)[∞]4] = [4,3,3,4] [4,(3,3)*,4] = [∞]4 CDel 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.png 24
[(4,3,3)[∞]4] = [4,3,3,4] [4,(3,3,4)*] = [∞]4 CDel 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.png 384

Looking at generators, the double symmmetry 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).

Computation with reflection matrices as symmetry generators[edit]

A Coxeter group, represented by Coxeter diagram CDel node n0.pngCDel p.pngCDel node n1.pngCDel q.pngCDel node n2.png, is given Coxeter notation [p,q] for the branch orders. Each node in the Coxeter diagram represents a mirror, by convention called ρi (and matrix Ri). The generators of this group [p,q] are reflections: ρ0, ρ1, and ρ2. Rotational subsymmetry is given as products of reflections: By convention, σ0,1 (and matrix S0,1) = ρ0ρ1 represents a rotation of angle π/p, and σ1,2 = ρ1ρ2 is a rotation of angle π/q, and σ0,2 = ρ0ρ2 represents a rotation of angle π/2.

[p,q]+ is an index 2 subgroup represented by two rotation generators, each a products of two reflections: σ0,1, σ1,2, and representing rotations of π/p, and π/q angles respectively.

If q is even, [p+,q] is another subgroup of index 2, represented by rotation generator σ0,1, and reflectional ρ2.

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: ψ0,1,2 (and matrix U0,1,2) = σ0,1ρ2 = ρ0σ1,2 = ρ0ρ1ρ2, which is an rotary reflection, representing a reflection and rotation.

In the case of affine Coxeter groups like CDel node n0.pngCDel 4.pngCDel node n1.pngCDel 4.pngCDel node n2.png, or CDel node n0.pngCDel infin.pngCDel node n1.png, one mirror, usually the last, is translated off the origin. A translation generator τ0,1 (and matrix T0,1) is constructed as the product of two (or an even number of) reflections, including the affine reflection. A transreflection (reflection plus a translation) can be the product of an odd number of reflections φ0,1,2 (and matrix V0,1,2), like the index 4 subgroup CDel node n0.pngCDel 4.pngCDel node n1.pngCDel 4.pngCDel node n2.png: [4+,4+] = CDel node h2.pngCDel 4.pngCDel node h4.pngCDel 4.pngCDel node h2.png.

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] (CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png), this subgroup is a rotary reflection [2+,h+].

Example, in 2D, the Coxeter group [p] (CDel node.pngCDel p.pngCDel node.png) is represented by two reflection matrices R0 and R1, The cyclic symmetry [p]+ (CDel node h2.pngCDel p.pngCDel node h2.png) is represented by rotation generator of matrix S0,1.

R0 R1 S0,1=R0xR1

\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] (CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png) (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]+ (CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 4.pngCDel node h2.png) (p4) is generated by S0,1 S1,2, and S0,2. [4+,4+] (CDel node h2.pngCDel 4.pngCDel node h4.pngCDel 4.pngCDel node h2.png) (pgg) is generated by 2-fold rotation S0,2 and transreflection V0,1,2. [4+,4] (CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 4.pngCDel node.png) (p4g) is generated by S0,1 and R3. The group [(4,4,2+)] (CDel node.pngCDel split1-44.pngCDel branch h2h2.pngCDel label2.png) (cmm), is generated by 2-fold rotation S1,3 and reflection R2.

R0 R1 R2 S0,1 S1,2 S0,2 V0,1,2

\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-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 Dihn, and cyclic groups are represented by Zn, with Dih1=Z2.

Rank one groups[edit]

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.

Group Coxeter notation Coxeter diagram Order Description
C1 [ ]+ CDel node h2.png 1 Identity
D1 [ ] CDel node.png 2 Reflection group

Rank two groups[edit]

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

In two dimensions, the rectangular group [2], abstract D12 or D2, 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), 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 Dp, (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's a full pseudogonal group [πi/λ], and pseudogonal subgroup [πi/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.

Type Finite Affine Hyperbolic
Geometry Dihedral symmetry domains 1.png Dihedral symmetry domains 2.png Dihedral symmetry domains 3.png Dihedral symmetry domains 4.png ... Dihedral symmetry domains infinity.png Horocycle mirrors.png Dihedral symmetry ultra.png
Coxeter CDel node c1.png
[ ]
CDel node c1.pngCDel 2x.pngCDel node c3.png = CDel node c1.pngCDel 2.pngCDel node c3.png
[2]=[ ]×[ ]
CDel node c1.pngCDel 3.pngCDel node c1.png
[3]
CDel node c1.pngCDel 4.pngCDel node c3.png
[4]
CDel node.pngCDel p.pngCDel node.png
[p]
CDel node c1.pngCDel infin.pngCDel node c3.png
[∞]
CDel node c2.pngCDel infin.pngCDel node c3.png
[∞]
CDel node c2.pngCDel ultra.pngCDel node c3.png
[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)
Cyclic symmetry 1.png Cyclic symmetry 2.png Cyclic symmetry 3.png Cyclic symmetry 4.png ... Cyclic symmetry infin.png Cyclic symmetry ultra.png
Odd
images
(inverted)
Cyclic symmetry 1b.png Cyclic symmetry 2b.png Cyclic symmetry 3b.png Cyclic symmetry 4b.png Cyclic symmetry infinb.png Cyclic symmetry ultrab.png
Coxeter CDel node h2.png
[ ]+
CDel node h2.pngCDel 2x.pngCDel node h2.png
[2]+
CDel node h2.pngCDel 3.pngCDel node h2.png
[3]+
CDel node h2.pngCDel 4.pngCDel node h2.png
[4]+
CDel node h2.pngCDel p.pngCDel node h2.png
[p]+
CDel node h2.pngCDel infin.pngCDel node h2.png
[∞]+
CDel node h2.pngCDel infin.pngCDel node h2.png
[∞]+
CDel node h2.pngCDel ultra.pngCDel node h2.png
[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]+ CDel node h2.pngCDel n.pngCDel node h2.png n Cyclic: n-fold rotations. Abstract group Zn, the group of integers under addition modulo n.
Dn nm *n• [n] CDel node.pngCDel n.pngCDel node.png 2n Dihedral: cyclic with reflections. Abstract group Dihn, the dihedral group.
Affine
Z ∞• [∞]+ CDel node h2.pngCDel infin.pngCDel node h2.png Cyclic: apeirogonal group. Abstract group Z, the group of integers under addition.
Dih m *∞• [∞] CDel node.pngCDel infin.pngCDel node.png Dihedral: parallel reflections. Abstract infinite dihedral group Dih.
Hyperbolic
Z [πi/λ]+ CDel node h2.pngCDel ultra.pngCDel node h2.png pseudogonal group
Dih [πi/λ] CDel node.pngCDel ultra.pngCDel node.png full pseudogonal group

Rank three groups[edit]

Finite family correspondence
Affine isomorphism and correspondences

In three dimensions, the full orthorhombic group [2,2], abtractly Z2×D2, order 8, represents three orthogonal mirrors, (also represented by Coxeter diagram as three separate dots CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png). 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+] (CDel node.pngCDel 2.pngCDel node h2.pngCDel 2x.pngCDel node h2.png), abstractly D1×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 diagram, CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png) 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]+ (CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel node h2.png), also order 4, and finally the central group [2+,2+] (CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.png) of order 2.

Next there is the full ortho-p-gonal group, [2,p] (CDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png), abstractly D1×Dp=Z2×Dp, 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 CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.png.

The direct subgroup is called the para-p-gonal group, [2,p]+ (CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel p.pngCDel node h2.png), abstractly Dp, of order 2p, and another subgroup is [2,p+] (CDel node.pngCDel 2.pngCDel node h2.pngCDel p.pngCDel node h2.png) abstractly Z2×Zp, also of order 2p.

The full gyro-p-gonal group, [2+,2p] (CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel p.pngCDel node.png), abstractly D2p, of order 4p. The gyro-p-gonal group, [2+,2p+] (CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel p.pngCDel node h2.png), abstractly Z2p, 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 diagrams CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png, CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png, and CDel node.pngCDel split1.pngCDel branch.png, 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]].

[[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+] are semidirect subgroups.

Tetrahedral symmetry Octahedral symmetry
Tetrahedral subgroup tree.png Octahedral symmetry tree conway.png
Icosahedral symmetry
Icosahedral subgroup tree.png
Finite (point groups in three dimensions)
Intl* Geo
[4]
Orb. Schön. Conway Struct. Coxeter Ord.
1 1 1 C1 C1 Z1 [ ]+ CDel node h2.png 1
2 = m 1 * Cs ±C1 = CD2 D1 [ ] CDel node.png 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
Z2
Z3
Z4
Z5
Z6
Zn
[2]+
[3]+
[4]+
[5]+
[6]+
[n]+
CDel node h2.pngCDel 2x.pngCDel node h2.png
CDel node h2.pngCDel 3.pngCDel node h2.png
CDel node h2.pngCDel 4.pngCDel node h2.png
CDel node h2.pngCDel 5.pngCDel node h2.png
CDel node h2.pngCDel 6.pngCDel node h2.png
CDel node h2.pngCDel n.pngCDel node h2.png
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
D2
D3
D4
D5
D6
Dn
[2]
[3]
[4]
[5]
[6]
[n]
CDel node.pngCDel 2.pngCDel node.png
CDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 4.pngCDel node.png
CDel node.pngCDel 5.pngCDel node.png
CDel node.pngCDel 6.pngCDel node.png
CDel node.pngCDel n.pngCDel node.png
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
D1×Z2
D1×Z3
D1×Z4
D1×Z5
D1×Z6
D1×Zn
[2,2+]
[2,3+]
[2,4+]
[2,5+]
[2,6+]
[2,n+]
CDel node.pngCDel 2.pngCDel node h2.pngCDel 2x.pngCDel node h2.png
CDel node.pngCDel 2.pngCDel node h2.pngCDel 3.pngCDel node h2.png
CDel node.pngCDel 2.pngCDel node h2.pngCDel 4.pngCDel node h2.png
CDel node.pngCDel 2.pngCDel node h2.pngCDel 5.pngCDel node h2.png
CDel node.pngCDel 2.pngCDel node h2.pngCDel 6.pngCDel node h2.png
CDel node.pngCDel 2.pngCDel node h2.pngCDel n.pngCDel node h2.png
4
6
8
10
12
2n
4
3
8
5
12
2n
n
2 2
1
4 2
6 2
8 2
10 2
12 2
2n 2






Ci = S2
S4
S6
S8
S10
S12
S2n
CC2
CC4
±C3
CC8
±C5
CC12
CC2n / ±Cn
Z2
Z4
Z6=Z2×Z3
Z8
Z10=Z2×Z5
Z12
Z2n
[2+,2+]
[2+,4+]
[2+,6+]
[2+,8+]
[2+,10+]
[2+,12+]
[2+,2n+]
CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.png
CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 4.pngCDel node h2.png
CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 6.pngCDel node h2.png
CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 8.pngCDel node h2.png
CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 10.pngCDel node h2.png
CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 12.pngCDel node h2.png
CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel n.pngCDel node h2.png
2
4
6
8
10
12
2n
Intl Geo Orb. Schön. Conway Struct. Coxeter Ord.
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
D2
D3
D4
D5
D6
Dn
[2,2]+
[2,3]+
[2,4]+
[2,5]+
[2,6]+
[2,n]+
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel node h2.png
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 3.pngCDel node h2.png
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 4.pngCDel node h2.png
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 5.pngCDel node h2.png
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 6.pngCDel node h2.png
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel n.pngCDel node h2.png
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
D1×D2
D1×D3
D1×D4
D1×D5
D1×D6
D1×Dn
[2,2]
[2,3]
[2,4]
[2,5]
[2,6]
[2,n]
CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
CDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.png
CDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.png
CDel node.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.png
CDel node.pngCDel 2.pngCDel node.pngCDel n.pngCDel node.png
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
D2
D3
D4
D5
D6
Dn
[2+,4]
[2+,6]
[2+,8]
[2+,10]
[2+,12]
[2+,2n]
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 4.pngCDel node.png
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 6.pngCDel node.png
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 8.pngCDel node.png
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 10.pngCDel node.png
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 12.pngCDel node.png
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel n.pngCDel node.png
8
12
16
20
24
4n
23 3 3 332 T T A4 [3,3]+ CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png 12
m3 4 3 3*2 Th ±T A4×S2 [3+,4] CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 4.pngCDel node.png 24
43m 3 3 *332 Td TO S4 [3,3] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 24
432 4 3 432 O O S4 [3,4]+ CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 4.pngCDel node h2.png 24
m3m 4 3 *432 Oh ±O S4×S2 [3,4] CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 48
532 5 3 532 I I A5 [3,5]+ CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 5.pngCDel node h2.png 60
53m 5 3 *532 Ih ±I A4×S2 [3,5] CDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png 120
Semiaffine (frieze groups)
IUC Orb. Geo Sch. Coxeter
p1 ∞∞ p1 C [∞]+ CDel node h2.pngCDel infin.pngCDel node h2.png
p1m1 *∞∞ p1 C∞v [∞] CDel node.pngCDel infin.pngCDel node.png
p11g ∞× p.g1 S2∞ [∞+,2+] CDel node h2.pngCDel infin.pngCDel node h4.pngCDel 2x.pngCDel node h2.png
p11m ∞* p. 1 C∞h [∞+,2] CDel node h2.pngCDel infin.pngCDel node h2.pngCDel 2.pngCDel node.png
p2 22∞ p2 D [∞,2]+ CDel node h2.pngCDel infin.pngCDel node h2.pngCDel 2x.pngCDel node h2.png
p2mg 2*∞ p2g D∞d [∞,2+] CDel node.pngCDel infin.pngCDel node h2.pngCDel 2x.pngCDel node h2.png
p2mm *22∞ p2 D∞h [∞,2] CDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.png
Affine (Wallpaper groups)
IUC Orb. Geo. Coxeter
p2 2222 p2 [4,1+,4]+ CDel labelh.pngCDel node.pngCDel split1-44.pngCDel branch h2h2.pngCDel label2.png
p2gg 22× pg2g [4+,4+] CDel node h2.pngCDel 4.pngCDel node h4.pngCDel 4.pngCDel node h2.png
p2mm *2222 p2 [4,1+,4] CDel node.pngCDel 4.pngCDel node h2.pngCDel 4.pngCDel node.png
CDel node.pngCDel 4.pngCDel node h0.pngCDel 4.pngCDel node.png
c2mm 2*22 c2 [[4+,4+]] CDel node h4.pngCDel split1-44.pngCDel nodes h2h2.png
p4 442 p4 [4,4]+ CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 4.pngCDel node h2.png
p4gm 4*2 pg4 [4+,4] CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 4.pngCDel node.png
p4mm *442 p4 [4,4] CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
p3 333 p3 [1+,6,3+] = [3[3]]+ CDel node h0.pngCDel 6.pngCDel node h2.pngCDel 3.pngCDel node h2.png = CDel branch h2h2.pngCDel split2.pngCDel node h2.png
p3m1 *333 p3 [1+,6,3] = [3[3]] CDel node h0.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png = CDel branch.pngCDel split2.pngCDel node.png
p31m 3*3 h3 [6,3+] = [3[3[3]]+] CDel node.pngCDel 6.pngCDel node h2.pngCDel 3.pngCDel node h2.png
p6 632 p6 [6,3]+ = [3[3[3]]]+ CDel node h2.pngCDel 6.pngCDel node h2.pngCDel 3.pngCDel node h2.png
p6mm *632 p6 [6,3] = [3[3[3]]] CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png

Subgroups[edit]

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

Reflection Reflection
subgroups
Rotation subgroup Mixed Improper rotation Commutator
subgroup
C1v, [1]=[ ], CDel node.png, (*) C1, [1]+=[ ]+, CDel node h2.png, (11) S2, [2+,2+], CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.png, (×) [ ]+
C2v, [2], CDel node.pngCDel 2.pngCDel node.png, (*22) [1+,2]=[1]=[ ], CDel node h0.pngCDel 2.pngCDel node.png (*) C2, [2]+, CDel node h2.pngCDel 2x.pngCDel node h2.png, (22) C2h, [2+,2], CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2.pngCDel node.png, (2*) S4, [4+,2+], CDel node h2.pngCDel 4.pngCDel node h4.pngCDel 2x.pngCDel node h2.png, (2×)
Cnv, [n], CDel node.pngCDel n.pngCDel node.png, (*nn) [1+,2n]=[n], CDel node h0.pngCDel 2x.pngCDel n.pngCDel node.png (*nn) Cn, [n]+, CDel node h2.pngCDel n.pngCDel node h2.png, (nn) Cnh, [n+,2], CDel node h2.pngCDel n.pngCDel node h2.pngCDel 2.pngCDel node.png, (n*) S2n, [2n+,2+], CDel node h2.pngCDel n.pngCDel node h4.pngCDel 2x.pngCDel node h2.png, (n×) [n]+, n odd
[n/2]+, n even
Dnh, [2,n], CDel node.pngCDel 2.pngCDel node.pngCDel n.pngCDel node.png, (*22n) [1+,2,n]=[1,n]=[n], CDel node h0.pngCDel 2x.pngCDel n.pngCDel node.png (*nn) Dn, [2,n]+, CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel n.pngCDel node h2.png, (22n) Dnd, [2+,2n], CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel n.pngCDel node.png, (2*n)
Td, [3,3], CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png, (*332) T, [3,3]+, CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png, (332) [3,3]+, (332)
Oh, [4,3], CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png, (*432) [1+,4,3]=[3,3], CDel node h0.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png (*332) O, [4,3]+, CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png, (432) Th, [3+,4], CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 4.pngCDel node.png, (3*2)
Ih, [5,3], CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png, (*532) I, [5,3]+, CDel node h2.pngCDel 5.pngCDel node h2.pngCDel 3.pngCDel node h2.png, (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
[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[edit]

Polychoral group tree.png
Subgroup relations

Point groups[edit]

Rank four groups defined the 4-dimensional point groups:

Finite groups
[ ]: CDel node.png
Symbol Order
[1]+ 1.1
[1] = [ ] 2.1
[2]: CDel node.pngCDel 2.pngCDel node.png
Symbol Order
[1+,2]+ 1.1
[2]+ 2.1
[2] 4.1
[2,2]: CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
Symbol Order
[2+,2+]+
= [(2+,2+,2+)]
1.1
[2+,2+] 2.1
[2,2]+ 4.1
[2+,2] 4.1
[2,2] 8.1
[2,2,2]: CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
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]: CDel node.pngCDel p.pngCDel node.png
Symbol Order
[p]+ p
[p] 2p
[p,2]: CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.png
Symbol Order
[p,2]+ 2p
[p,2] 4p
[2p,2+]: CDel node.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 2x.pngCDel node.png
Symbol Order
[2p,2+] 4p
[2p+,2+] 2p
[p,2,2]: CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
Symbol Order
[p+,2,2+] 2p
[(p,2)+,2+] 2p
[p,2,2]+ 4p
[p,2,2+] 4p
[p+,2,2] 4p
[(p,2)+,2] 4p
[p,2,2] 8p
[2p,2+,2]: CDel node.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
Symbol Order
[2p+,2+,2+] 2p
[2p+,2+,2] 4p
[2p+,(2,2)+] 4p
[2p,(2,2)+] 8p
[2p,2+,2] 8p
[p,2,q]: CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.png
Symbol Order
[p+,2,q+] pq
[p,2,q]+ 2pq
[p+,2,q] 2pq
[p,2,q] 4pq
[(p,2)+,2q]: CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2x.pngCDel q.pngCDel node.png
Symbol Order
[(p,2)+,2q+] 2pq
[(p,2)+,2q] 4pq
[2p,2+,2q]: CDel node.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2x.pngCDel q.pngCDel node.png
Symbol Order
[2p+,2+,2q+] 2pq
[((2p,2)+,(2q,2)+)] 4pq
[2p,2+,2q+] 4pq
[2p,2+,2q] 8pq
[2p,2,2q] 16pq
[[p,2,p]]: CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png
Symbol Order
[[p+,2,p+]] 2p2
[[p,2,p]]+ 4p2
[[p,2,p]+] 4p2
[[p,2,p]] 8p2
[[2p,2+,2p]]: CDel node.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2x.pngCDel p.pngCDel node.png
Symbol Order
[[2p+,2+,2p+]] 4p2
[[((2p,2)+,(2p,2)+)]] 8p2
[[2p,2+,2p]] 16p2
[[2p,2,2p]] 32p2
[3,3,2]: CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png
Symbol Order
[(3,3)Δ,2,1+]
≅ [2,2]+
4
[(3,3)Δ,2]
≅ [2,(2,2)+]
8
[(3,3),2,1+]
≅ [4,2+]
8
[(3,3)+,2,1+]
= [3,3]+
12.5
[(3,3),2]
≅ [2,4,2+]
16
[3,3,2,1+]
= [3,3]
24
[(3,3)+,2] 24.10
[3,3,2]+ 24.10
[3,3,2] 48.36
[4,3,2]: CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png
Symbol Order
[1+,4,3+,2,1+]
= [3,3]+
12
[3+,4,2+] 24
[(3,4)+,2+] 24
[1+,4,3+,2]
= [(3,3)+,2]
24.10
[3+,4,2,1+]
= [3+,4]
24.10
[(4,3)+,2,1+]
= [4,3]+
24.15
[1+,4,3,2,1+]
= [3,3]
24
[1+,4,(3,2)+]
= [3,3,2]+
24
[3,4,2+] 48
[4,3+,2] 48.22
[4,(3,2)+] 48
[(4,3)+,2] 48.36
[1+,4,3,2]
= [3,3,2]
48.36
[4,3,2,1+]
= [4,3]
48.36
[4,3,2]+ 48.36
[4,3,2] 96.5
[5,3,2]: CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png
Symbol Order
[(5,3)+,2,1+]
= [5,3]+
60.13
[5,3,2,1+]
= [5,3]
120.2
[(5,3)+,2] 120.2
[5,3,2]+ 120.2
[5,3,2] 240 (nc)
[31,1,1]: CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
Symbol Order
[31,1,1]Δ
≅[[4,2+,4]]+
32
[31,1,1] 64
[31,1,1]+ 96.1
[31,1,1] 192.2
<[3,31,1]>
= [4,3,3]
384.1
[3[31,1,1]]
= [3,4,3]
1152.1
[3,3,3]: CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Symbol Order
[3,3,3]+ 60.13
[3,3,3] 120.1
[[3,3,3]]+ 120.2
[[3,3,3]+] 120.1
[[3,3,3]] 240.1
[4,3,3]: CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Symbol Order
[1+,4,(3,3)Δ]
= [31,1,1]Δ
≅[[4,2+,4]]+
32
[4,(3,3)Δ]
= [2+,4[2,2,2]+]
≅[[4,2+,4]]
64
[1+,4,(3,3)]
= [31,1,1]
64
[1+,4,(3,3)+]
= [31,1,1]+
96.1
[4,(3,3)]
≅ [[4,2,4]]
128
[1+,4,3,3]
= [31,1,1]
192.2
[4,(3,3)+] 192.1
[4,3,3]+ 192.3
[4,3,3] 384.1
[3,4,3]: CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Symbol Order
[3+,4,3+] 288.1
[3,4,3]+ 576.2
[3+,4,3] 576.1
[[3+,4,3+]] 576 (nc)
[3,4,3] 1152.1
[[3,4,3]]+ 1152 (nc)
[[3,4,3]+] 1152 (nc)
[[3,4,3]] 2304 (nc)
[5,3,3]: CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Symbol Order
[5,3,3]+ 7200 (nc)
[5,3,3] 14400 (nc)

Subgroups[edit]

1D-4D reflective point groups and subgroups
Order Reflection Semidirect
subgroups
Direct
subgroups
Commutator
subgroup
2 [ ] CDel node.png [ ]+ CDel node h2.png [ ]+1 [ ]+
4 [2] CDel node.pngCDel 2.pngCDel node.png [2]+ CDel node h2.pngCDel 2x.pngCDel node h2.png [2]+2
8 [2,2] CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png [2+,2] CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2.pngCDel node.png [2+,2+] CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.png [2,2]+ CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel node h2.png [2,2]+3
16 [2,2,2] CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png [2+,2,2]
[(2,2)+,2]
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2.pngCDel node.png
[2+,2+,2]
[(2,2)+,2+]
[2+,2+,2+]
CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.pngCDel 2.pngCDel node.png
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.png
CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.png
[2,2,2]+
[2+,2,2+]
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel node h2.png
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2.pngCDel node h2.pngCDel 2x.pngCDel node h2.png
[2,2,2]+4
[21,1,1] CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel nodes.png [(2+)1,1,1] CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel split1-22.pngCDel nodes h2h2.png
2n [n] CDel node.pngCDel n.pngCDel node.png [n]+ CDel node h2.pngCDel n.pngCDel node h2.png [n]+1 [n]+
4n [2n] CDel node.pngCDel 2x.pngCDel n.pngCDel node.png [2n]+ CDel node h2.pngCDel 2x.pngCDel n.pngCDel node h2.png [2n]+2
4n [2,n] CDel node.pngCDel 2.pngCDel node.pngCDel n.pngCDel node.png [2,n+] CDel node.pngCDel 2.pngCDel node h2.pngCDel n.pngCDel node h2.png [2,n]+ CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel n.pngCDel node h2.png [2,n]+2
8n [2,2n] CDel node.pngCDel 2.pngCDel node.pngCDel 2x.pngCDel n.pngCDel node.png [2+,2n] CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel n.pngCDel node.png [2+,2n+] CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel n.pngCDel node h2.png [2,2n]+ CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel n.pngCDel node h2.png [2,2n]+3
8n [2,2,n] CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel n.pngCDel node.png [2+,2,n]
[2,2,n+]
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2.pngCDel node.pngCDel n.pngCDel node.png
CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node h2.pngCDel n.pngCDel node h2.png
[2+,(2,n)+] CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.pngCDel n.pngCDel node h2.png [2,2,n]+
[2+,2,n+]
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel node h2.pngCDel n.pngCDel node h2.png
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2.pngCDel node h2.pngCDel n.pngCDel node h2.png
[2,2,n]+3
16n [2,2,2n] CDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2x.pngCDel n.pngCDel node.png [2,2+,2n] CDel node.pngCDel 2.pngCDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel n.pngCDel node.png [2+,2+,2n]
[2,2+,2n+]
[(2,2)+,2n+]
[2+,2+,2n+]
CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel n.pngCDel node.png
CDel node.pngCDel 2.pngCDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel n.pngCDel node h2.png
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel n.pngCDel node h2.png
CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel n.pngCDel node h2.png
[2,2,2n]+
[2+,2n,2+]
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel n.pngCDel node h2.png
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2.pngCDel 2x.pngCDel n.pngCDel 2.pngCDel node h2.pngCDel 2x.pngCDel node h2.png
[2,2,2n]+4
[2,2n,2] CDel node.pngCDel 2.pngCDel node.pngCDel 2x.pngCDel n.pngCDel node.pngCDel 2.pngCDel node.png [2+,2n+,2+] CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel n.pngCDel node h4.pngCDel 2x.pngCDel node h2.png
[2n,21,1] CDel node.pngCDel 2x.pngCDel n.pngCDel node.pngCDel 2.pngCDel nodes.png [2n+,(2+)1,1] CDel node h2.pngCDel 2x.pngCDel n.pngCDel node h4.pngCDel split1-22.pngCDel nodes h2h2.png
24 [3,3] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png [3,3]+ CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png [3,3]+1 [3,3]+
48 [3,3,2] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png [(3,3)+,2] CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 2.pngCDel node.png [3,3,2]+ CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 2x.pngCDel node h2.png [3,3,2]+2
48 [4,3] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png [4,3+] CDel node.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png [4,3]+ CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png [4,3]+2
96 [4,3,2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png [(4,3)+,2]
[4,(3,2)+]
CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 2.pngCDel node.png
CDel node.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 2x.pngCDel node h2.png
[4,3,2]+ CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 2x.pngCDel node h2.png [4,3,2]+3
[3,4,2] CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.png [3,4,2+]
[3+,4,2]
CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h2.pngCDel 2x.pngCDel node h2.png
CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.png
[(3,4)+,2+] CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 4.pngCDel node h4.pngCDel 2x.pngCDel node h2.png [3+,4,2+] CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 2.pngCDel 4.pngCDel 2.pngCDel node h2.pngCDel 2x.pngCDel node h2.png
120 [5,3] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png [5,3]+ CDel node h2.pngCDel 5.pngCDel node h2.pngCDel 3.pngCDel node h2.png [5,3]+1 [5,3]+
240 [5,3,2] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png [(5,3)+,2] CDel node h2.pngCDel 5.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 2.pngCDel node.png [5,3,2]+ CDel node h2.pngCDel 5.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 2x.pngCDel node h2.png [5,3,2]+2
4pq [p,2,q] CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.png [p+,2,q] CDel node h2.pngCDel p.pngCDel node h2.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.png [p,2,q]+
[p+,2,q+]
CDel node h2.pngCDel p.pngCDel node h2.pngCDel 2x.pngCDel node h2.pngCDel q.pngCDel node h2.png
CDel node h2.pngCDel p.pngCDel node h2.pngCDel 2.pngCDel node h2.pngCDel q.pngCDel node h2.png
[p,2,q]+2 [p+,2,q+]
8pq [2p,2,q] CDel node.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.png [2p,(2,q)+] CDel node.pngCDel 2x.pngCDel p.pngCDel node h2.pngCDel 2x.pngCDel node h2.pngCDel q.pngCDel node h2.png [2p+,(2,q)+] CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h4.pngCDel 2x.pngCDel node h2.pngCDel q.pngCDel node h2.png [2p,2,q]+ CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h2.pngCDel 2x.pngCDel node h2.pngCDel q.pngCDel node h2.png [2p,2,q]+3
16pq [2p,2,2q] CDel node.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2x.pngCDel q.pngCDel node.png [2p,2+,2q] CDel node.pngCDel 2x.pngCDel p.pngCDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel q.pngCDel node.png [2p+,2+,2q]
[2p+,2+,2q+]
[(2p,(2,2q)+,2+)]
CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h4.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel q.pngCDel node.png
CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel q.pngCDel node h2.png
-
[2p,2,2q]+ CDel node h2.pngCDel 2x.pngCDel p.pngCDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel q.pngCDel node h2.png [2p,2,2q]+4
120 [3,3,3] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png [3,3,3]+ CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png [3,3,3]+1 [3,3,3]+
192 [31,1,1] CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png [31,1,1]+ CDel node h2.pngCDel 3.pngCDel node h2.pngCDel split1.pngCDel nodes h2h2.png [31,1,1]+1 [31,1,1]+
384 [4,3,3] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png [4,(3,3)+] CDel node.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png [4,3,3]+ CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png [4,3,3]+2
1152 [3,4,3] CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png [3+,4,3] CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png [3,4,3]+
[3+,4,3+]
CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 4.pngCDel node h2.pngCDel 3.pngCDel node h2.png
CDel node h2.pngCDel 3.pngCDel node h2.pngCDel 2.pngCDel 4.pngCDel 2.pngCDel node h2.pngCDel 3.pngCDel node h2.png
[3,4,3]+2 [3+,4,3+]
14400 [5,3,3] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png [5,3,3]+ CDel node h2.pngCDel 5.pngCDel node h2.pngCDel 3.pngCDel node h2.pngCDel 3.pngCDel node h2.png [5,3,3]+1 [5,3,3]+

Space groups[edit]

Coxeter diagram affine rank4 correspondence.png
Affine isomorphism and correspondences
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
35 cubic fibrifold groups.png
35 cubic space groups in International, Fibrifold notation, and Coxeter notation

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

Triclinic (1-2)
Coxeter Space group
[∞+,2,∞+,2,∞+] (1) P1
Monoclinic (3-15)
Coxeter Space group
[(∞,2,∞)+,2,∞+] (3) P2
[∞+,2,∞+,2,∞] (6) Pm
[(∞,2,∞)+,2,∞] (10) P2/m
Orthorhombic (16-74)
CDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
Coxeter Space group
[∞,2,∞,2,∞]+ (16) P222
[[∞,2,∞,2,∞]]+ (23) I222
[∞+,2,∞,2,∞] (25) Pmm2
[∞,2,∞,2,∞] (47) Pmmm
[[∞,2,∞,2,∞]] (71) Immm
[∞+,2,∞+,2,∞+]
[∞,2,∞,2+,∞]
[∞,2+,∞,2+,∞]
Tetragonal (75-142)
CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
Coxeter Space group
[(4,4)+,2,∞+] (75) P4
[2+[(4,4)+,2,∞+]] (79) I4
[(4,4)+,2,∞] (83) P4/m
[2+[(4,4)+,2,∞]] (87) I4/m
[4,4,2,∞]+ (89) P422
[2+[4,4,2,∞]]+ (97) I422
[4,4,2,∞+] (99) P4mm
[4,4,2,∞] (123) P4/mmm
[2+[4,4,2,∞]] (139) I4/mmm
[4,(4,2)+,∞] (140) I4/mcm
[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,∞]
[4,4+,2,∞+]
[4,4+,2+,∞+]
[((4,2+,4)),2,∞+]
Trigonal (143-167), rhombohedral
Coxeter Space group
Hexagonal (168-194)
CDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
[(6,3)+,2,∞+] (168) P6
[(6,3)+,2,∞] (175) P6/m
[6,3,2,∞]+ (177) P622
[6,3,2,∞+] (183) P6mm
[6,3,2,∞] (191) P6/mmm
[(3[3])+,2,∞+]
[3[3],2,∞]
[6,3+,2,∞]
[6,3+,2,∞+]
[3[3],2,∞]+
[3[3],2,∞+]
[(3[3])+,2,∞]
Cubic (195-230)
Group Coxeter Space group Index
[[4,3,4]] [[4,3,4]] (229) Im3m 1
[[4,3,4]]+ (211) I432 2
[[4,3,4]+] (223) Pm3n 2
[[4,3+,4]] (204) I3 2
[[(4,3,4,2+)]] (217) I43m 2
[[4,3+,4]]+ (197) I23 4
[[4,3,4]+]+ (208) P4232 4
[[4,3+,4)]+] (201) Pn43 4
[[(4,3,4,2+)]+] (218) P43n 4
[4,3,4]
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
[4,3,4] (221) Pm3m 2
[4,3,4]+ (207) P432 4
[4,3+,4] (200) Pm3 4
[4,(3,4)+] (226) Fm3c 4
[(4,3,4,2+)] (215) P43m 4
[[{4,(3}+,4)+]] (228) Fd3c 4
[4,3+,4]+ (195) P23 8
[{4,(3}+,4)+] (219) F43c 8
[4,31,1]
CDel node.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.png
[4,31,1] (225) Fm3m 4
[4,(31,1)+] (202) Fm3 8
[4,31,1]+ (209) F432 8
[[3[4]]]
CDel branch.pngCDel 3ab.pngCDel branch.png
[(4+,2+)[3[4]]] (222) Pn3n 2
[[3[4]]] (227) Fd3m 4
[[3[4]]]+ (203) Fd3 8
[[3[4]]+] (210) F4132 8
[3[4]]
CDel branch.pngCDel 3ab.pngCDel branch.png
CDel node h0.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.png
[3[4]] (216) F43m 8
[3[4]]+ (196) F23 16

Line groups[edit]

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 Wallpaper group diagram p1.svg [∞+,2,n+] CDel node h2.pngCDel infin.pngCDel node h2.pngCDel 2.pngCDel node h2.pngCDel n.pngCDel node h2.png
2n n S2n P2n Pn None p11g, pg(h) ×× Wallpaper group diagram pg.svg [(∞,2)+,2n+] CDel node h2.pngCDel infin.pngCDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel n.pngCDel node h2.png
n/m 2n Cnh Pn/m P2n None p11m, pm(h) ** Wallpaper group diagram pm.svg [∞+,2,n] CDel node h2.pngCDel infin.pngCDel node h2.pngCDel 2.pngCDel node.pngCDel n.pngCDel node.png
2n/m C2nh P2nn/m Zigzag c11m, cm(h) Wallpaper group diagram cm.svg [∞+,2+,2n] CDel node h2.pngCDel infin.pngCDel node h4.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel n.pngCDel node.png
nmm nm Cnv Pnmm Pnm None p1m1, pm(v) ** Wallpaper group diagram pm rotated.svg [∞,2,n+] CDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node h2.pngCDel n.pngCDel node h2.png
Pncc Pnc Planar reflection p1g1, pg(v) ×× Wallpaper group diagram pg rotated.svg [∞+,(2,n)+] CDel node h2.pngCDel infin.pngCDel node h4.pngCDel 2x.pngCDel node h2.pngCDel n.pngCDel node h2.png
2nmm C2nv P2nnmc Zigzag c1m1, cm(v) Wallpaper group diagram cm rotated.svg [∞,2+,2n+] CDel node.pngCDel infin.pngCDel node h2.pngCDel 2x.pngCDel node h4.pngCDel 2x.pngCDel n.pngCDel node h2.png
n22 n2 Dn Pnq22 Pnq2 Helical: q p2 2222 Wallpaper group diagram p2.svg [∞,2,n]+ CDel node h2.pngCDel infin.pngCDel node h2.pngCDel 2x.pngCDel node h2.pngCDel n.pngCDel node h2.png
2n2m nm Dnd P2n2m Pnm None p2mg, pmg(h) 22* Wallpaper group diagram pmg.svg [(∞,2)+,2n] CDel node h2.pngCDel infin.pngCDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel n.pngCDel node.png
P2n2c Pnc Planar reflection p2gg, pgg 22× Wallpaper group diagram pgg rhombic.svg [+(∞,(2),2n)+]
n/mmm 2n2m Dnh Pn/mmm P2n2m None p2mm, pmm *2222 Wallpaper group diagram pmm.svg [∞,2,n] CDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel n.pngCDel node.png
Pn/mcc P2n2c Planar reflection p2mg, pmg(v) 22* Wallpaper group diagram pmg rotated.svg [∞,(2,n)+] CDel node.pngCDel infin.pngCDel node h2.pngCDel 2x.pngCDel node h2.pngCDel n.pngCDel node h2.png
2n/mmm D2nh P2nn/mcm Zigzag c2mm, cmm 2*22 Wallpaper group diagram cmm.svg [∞,2+,2n] CDel node.pngCDel infin.pngCDel node h2.pngCDel 2x.pngCDel node h2.pngCDel 2x.pngCDel n.pngCDel node.png

Duoprismatic group[edit]

Extended duoprismatic groups, [p]×[p] or [p,2,p] or CDel labelp.pngCDel branch.pngCDel 2.pngCDel branch.pngCDel labelp.png, 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]
CDel node c1.pngCDel p.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel q.pngCDel node c4.png
Coxeter
[[p,2,p]]
CDel node c1.pngCDel p.pngCDel node c2.pngCDel 2.pngCDel node c2.pngCDel p.pngCDel node c1.png
Wallpaper
IUC Orbifold Diagram IUC Orbifold Diagram
p1 o Wallpaper group diagram p1 rect.svg [p+,2,q+] [[p+,2,p+]] p1 o Wallpaper group diagram p1 rhombic.svg
pg ×× Wallpaper group diagram pg.svg [(p,2)+,2q+] -
pm ** Wallpaper group diagram pm.svg [p+,2,q] -
cm Wallpaper group diagram cm.svg [2p+,2+,2q] -
p2 2222 Wallpaper group diagram p2 rect.svg [p,2,q]+ [[p,2,p]]+ p4 442 Wallpaper group diagram p4 square.svg
pmg 22* Wallpaper group diagram pmg.svg [(p,2)+,2q] -
pgg 22× Wallpaper group diagram pgg rhombic.svg [+(2p,(2),2q)+] [[+(2p,(2),2p)+]] cmm 2*22 Wallpaper group diagram cmm square.svg
pmm *2222 Wallpaper group diagram pmm.svg [p,2,q] [[p,2,p]] p4m *442 Wallpaper group diagram p4m square.svg
cmm 2*22 Wallpaper group diagram cmm.svg [2p,2+,2q] [[2p,2+,2p]] p4g 4*2 Wallpaper group diagram p4g square.svg

Wallpaper groups[edit]

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 p1 [∞+,2,∞+] CDel labelinfin.pngCDel branch h2h2.pngCDel 2.pngCDel branch h2h2.pngCDel labelinfin.png Wallpaper group diagram p1 rect.svg
[∞+,2+,∞+] CDel node h2.pngCDel infin.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel infin.pngCDel node h2.png Wallpaper group diagram p1 rhombic.svg
[(∞+,2+,∞+,2+)] CDel labelinfin.pngCDel branch h4h4.pngCDel 2a2b.pngCDel branch h4h4.pngCDel labelinfin.png Wallpaper group diagram p1 rect.svg
p2 2222 p2 [∞,2,∞]+ CDel label2.pngCDel branch h2h2.pngCDel iaib.pngCDel branch h2h2.pngCDel label2.png Wallpaper group diagram p2 rect.svg
[(∞,2+,∞,2+)] CDel label2.pngCDel branch h2h2.pngCDel 2.pngCDel iaib.pngCDel 2.pngCDel branch h2h2.pngCDel label2.png Wallpaper group diagram p2 rhombic.svg
p11g ×× pg1 h: [∞+,(2,∞)+] CDel node h2.pngCDel infin.pngCDel node h4.pngCDel split1-22.pngCDel branch h2h2.pngCDel labelinfin.png Wallpaper group diagram pg.svg
p1g1 v: [(∞,2)+,∞+] CDel labelinfin.pngCDel branch h2h2.pngCDel split2-22.pngCDel node h4.pngCDel infin.pngCDel node h2.png Wallpaper group diagram pg rotated.svg
p2gm 22* pg2 h: [(∞,2)+,∞] CDel labelinfin.pngCDel branch h2h2.pngCDel split2-22.pngCDel node h2.pngCDel infin.pngCDel node.png Wallpaper group diagram pmg.svg
p2mg v: [∞,(2,∞)+] CDel node.pngCDel infin.pngCDel node h2.pngCDel split1-22.pngCDel branch h2h2.pngCDel labelinfin.png Wallpaper group diagram pmg rotated.svg
IUC Orb. Geo Coxeter Diagram
p11m ** p1 h: [∞+,2,∞] CDel node h2.pngCDel infin.pngCDel node h2.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png Wallpaper group diagram pm.svg
p1m1 v: [∞,2,∞+] CDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node h2.pngCDel infin.pngCDel node h2.png Wallpaper group diagram pm rotated.svg
p2mm *2222 p2 [∞,2,∞] CDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png Wallpaper group diagram pmm.svg
c11m c1 h: [∞+,2+,∞] CDel node h2.pngCDel infin.pngCDel node h4.pngCDel 2x.pngCDel node h2.pngCDel infin.pngCDel node.png Wallpaper group diagram cm.svg
c1m1 v: [∞,2+,∞+] CDel node.pngCDel infin.pngCDel node h2.pngCDel 2x.pngCDel node h4.pngCDel infin.pngCDel node h2.png Wallpaper group diagram cm rotated.svg
p2gg 22× pg2g [+(∞,(2),∞)+]
[((∞,2)+)[2]]
CDel node h2.pngCDel split1-i2.pngCDel nodes h4h4.pngCDel split2-2i.pngCDel node h2.png Wallpaper group diagram pgg rhombic.svg
c2mm 2*22 c2 [∞,2+,∞] CDel node.pngCDel infin.pngCDel node h2.pngCDel 2x.pngCDel node h2.pngCDel infin.pngCDel node.png Wallpaper group diagram cmm.svg

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

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),∞)+], (22×)
[(∞+,2+,∞+,2+)], (°)
[∞,2+,∞], (2*22)
[(∞,2)+,∞], (22*)

Notes[edit]

  1. ^ Conway, 2003, p.46, Table 4.2 Chiral groups II
  2. ^ Coxeter and Moser, 1980, Sec 9.5 Commutator subgroup, p. 124–126
  3. ^ Norman W. Johnson, Asia Ivić Weiss, Quaternionic modular groups, Linear Algebra and its Applications, Volume 295, Issues 1–3, 1 July 1999, Pages 159–189 [1]
  4. ^ The Crystallographic Space groups in Geometric algebra, D. Hestenes and J. Holt, Journal of Mathematical Physics. 48, 023514 (2007) (22 pages) PDF [2]

References[edit]

  • 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 [3]
      • (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, (2015) Chapter 11: Finite symmetry groups
    • Norman W. Johnson and Asia Ivic Weiss Quadratic Integers and Coxeter Groups PDF Canad. J. Math. Vol. 51 (6), 1999 pp. 1307–1336
  • Conway, John Horton; Delgado Friedrichs, Olaf; Huson, Daniel H.; Thurston, William P. (2001), "On three-dimensional space groups", Beiträge zur Algebra und Geometrie. Contributions to Algebra and Geometry 42 (2): 475–507, ISSN 0138-4821, MR 1865535 
  • John H. Conway and Derek A. Smith, On Quaternions and Octonions, 2003, ISBN 978-1-56881-134-5
  • The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 Ch.22 35 prime space groups, ch.25 184 composite space groups, ch.26 Higher still, 4D point groups