Coxeter notation
In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a CoxeterDynkin diagram, with modifiers to indicate certain subgroups. The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson.
C_{1v} 
C_{2v} 
C_{3v} 
C_{4v} 
C_{5v} 
C_{6v} 

Order 2 
Order 4 
Order 6 
Order 8 
Order 10 
Order 12 
[2]=[2,1] D_{1h} 
[2,2] D_{2h} 
[2,3] D_{3h} 
[2,4] D_{4h} 
[2,5] D_{5h} 
[2,6] D_{6h} 
Order 4 
Order 8 
Order 12 
Order 16 
Order 20 
Order 24 
Order 24 
Order 48 
Order 120  
Coxeter notation expresses Coxeter groups as a list of branch orders of a Coxeter diagram, like the polyhedral groups, 
Reflectional groups
For Coxeter groups, defined by pure reflections, there is a direct correspondence between the bracket notation and CoxeterDynkin diagram. The numbers in the bracket notation represent the mirror reflection orders in the branches of the Coxeter diagram. It uses the same simplification, suppressing 2s between orthogonal mirrors.
The Coxeter notation is simplified with exponents to represent the number of branches in a row for linear diagram. So the A_{n} group is represented by [3^{n1}], to imply n nodes connected by n1 order3 branches. Example A_{2} = [3,3] = [3^{2}] or [3^{1,1}] represents diagrams
Coxeter initially represented bifurcating diagrams with vertical positioning of numbers, but later abbreviated with an exponent notation, like [...,3^{p,q}] or [3^{p,q,r}], starting with [3^{1,1,1}] or [3,3^{1,1}] =
Coxeter groups formed by cyclic diagrams are represented by parentheseses inside of brackets, like [(p,q,r)] =
More complicated looping diagrams can also be expressed with care. The paracompact Coxeter group
The Coxeter diagram usually leaves order2 branches undrawn, but the bracket notation includes an explicit 2 to connect the subgraphs. So the Coxeter diagram



For the affine and hyperbolic groups, the subscript is one less than the number of nodes in each case, since each of these groups was obtained by adding a node to a finite group's diagram.
Subgroups
Coxeter's notation represents rotational/translational symmetry by adding a ^{+} superscript operator outside the brackets, [X]^{+} which cuts the order of the group [X] in half, thus an index 2 subgroup. This operator implies an even number of operators must be applied, replacing reflections with rotations (or translations). When applied to a Coxeter group, this is called a direct subgroup because what remains are only direct isometries without reflective symmetry.
The ^{+} operators can also be applied inside of the brackets, like [X,Y^{+}] or [X,(Y,Z)^{+}], and creates "semidirect" subgroups that may include both reflective and nonreflective generators. Semidirect subgroups can only apply to Coxeter group subgroups that have even order branches adjacent to it. Elements by parentheses inside of a Coxeter group can be give a ^{+} superscript operator, having the effect of dividing adjacent ordered branches into half order, thus is usually only applied with even numbers. For example, [4,3^{+}] and [4,(3,3)^{+}] (
If applied with adjacent odd branch, it doesn't create a subgroup of index 2, but instead creates overlapping fundamental domains, like [5,1^{+}] = [5/2], which can define doubly wrapped polygons like a pentagram, {5/2}, and [5,3^{+}] relates to Schwarz triangle [5/2,3], density 2.
Group  Order  Generators  Subgroup  Order  Generators  Notes  

[p]  2p  {0,1}  [p]^{+}  p  {01}  Direct subgroup  
[2p^{+}] = [2p]^{+}  2p  {01}  [2p^{+}]^{+} = [2p]^{+2} = [p]^{+}  p  {0101}  
[2p]  4p  {0,1}  [1^{+},2p] = [p]  2p  {101,1}  Half subgroups  
[2p,1^{+}] = [p]  {0,010}  
[1^{+},2p,1^{+}] = [2p]^{+2} = [p]^{+}  p  {0101}  Quarter group 
Groups without neighboring ^{+} elements can be seen in ringed nodes CoxeterDynkin 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,
Note: Pyritohedral symmetry
Halving subgroups and extended groups
[1,4,1] = [4] 
[1^{+},4,1]=[2]=[ ]×[ ]  
[1,4,1^{+}]=[2]=[ ]×[ ] 
[1^{+},4,1^{+}] = [2]^{+} 
Johnson extends the ^{+} operator to work with a placeholder 1^{+} nodes, which removes mirrors, doubling the size of the fundamental domain and cuts the group order in half.[1] In general this operation only applies to individual mirrors bounded by evenorder branches. The 1 represents a mirror so [2p] can be seen as [2p,1], [1,2p], or [1,2p,1], like diagram
Each of these mirrors can be removed so h[2p] = [1^{+},2p,1] = [1,2p,1^{+}] = [p], a reflective subgroup index 2. This can be shown in a Coxeter diagram by adding a ^{+} symbol above the node:
If both mirrors are removed, a quarter subgroup is generated, with the branch order becoming a gyration point of half the order:
 q[2p] = [1^{+},2p,1^{+}] = [p]^{+}, a rotational subgroup of index 4.
= = = = .
For example, (with p=2): [4,1^{+}] = [1^{+},4] = [2] = [ ]×[ ], order 4. [1^{+},4,1^{+}] = [2]^{+}, order 2.
The opposite to halving is doubling[2] which adds a mirror, bisecting a fundamental domain, and doubling the group order.
 [[p]] = [2p]
Halving operations apply for higher rank groups, like tetrahedral symmetry is a half group of octahedral group: h[4,3] = [1^{+},4,3] = [3,3], removing half the mirrors at the 4branch. The effect of a mirror removal is to duplicate all connecting nodes, which can be seen in the Coxeter diagrams:
If nodes are indexed, half subgroups can be labeled with new mirrors as composites. Like
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 

T_{d}, [3,3] = [1^{+},4,3] (Order 24) 
O_{h}, [4,3] = [[3,3]] (Order 48) 
Radical subgroups
Johnson also added an asterisk or star * operator for "radical" subgroups,[3] that acts similar to the ^{+} operator, but removes rotational symmetry. The index of the radical subgroup is the order of the removed element. For example, [4,3*] ≅ [2,2]. The removed [3] subgroup is order 6 so [2,2] is an index 6 subgroup of [4,3].
The radical subgroups represent the inverse operation to an extended symmetry operation. For example, [4,3*] ≅ [2,2], and in reverse [2,2] can be extended as [3[2,2]] ≅ [4,3]. The subgroups can be expressed as a Coxeter diagram:
If [4,3] has generators {0,1,2}, [4,3^{+}], index 2, has generators {0,12}; [1^{+},4,3] ≅ [3,3], index 2 has generators {010,1,2}; while radical subgroup [4,3*] ≅ [2,2], index 6, has generators {01210, 2, (012)^{3}}; and finally [1^{+},4,3*], index 12 has generators {0(12)^{2}0, (012)^{2}01}.
Trionic subgroups
A trionic subgroup is an index 3 subgroups. There are many
Johnson defines a trionic subgroup with operator ⅄, index 3. For rank 2 Coxeter groups, [3], the trionic subgroup, [3^{⅄}] is [ ], a single mirror. And for [3p], the trionic subgroup is [3p]^{⅄} ≅ [p]. Given
Trionic subgroups of tetrahedral symmetry: [3,3]^{⅄} ≅ [2^{+},4], relating the symmetry of the regular tetrahedron and tetragonal disphenoid.
For rank 3 Coxeter groups, [p,3], there is a trionic subgroup [p,3^{⅄}] ≅ [p/2,p], or
An oddorder adjacent branch, p, will not lower the group order, but create overlapping fundamental domains. The group order stays the same, while the density increases. For example, the icosahedral symmetry, [5,3], of the regular polyhedra icosahedron becomes [5/2,5], the symmetry of 2 regular star polyhedra. It also relates the hyperbolic tilings {p,3}, and star hyperbolic tilings {p/2,p}
For rank 4, [q,2p,3^{⅄}] = [2p,((p,q,q))],
For example, [3,4,3^{⅄}] = [4,3,3], or
Trionic subgroups of tetrahedral symmetry
Johnson identified two specific trionic subgroups[4] of [3,3], first an index 3 subgroup [3,3]^{⅄} ≅ [2^{+},4], with [3,3] (
Secondly he identifies a related index 6 subgroup [3,3]^{Δ} or [(3,3,2^{⅄})]^{+} (
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<sup>+</sup>,4]], order 64, has generators {02,1021,3}. As well, [3^{⅄},4,3^{⅄}] ≅ [(3,3)^{⅄},4].
Also related [3^{1,1,1}] = [3,3,4,1^{+}] has trionic subgroups: [3^{1,1,1}]^{⅄} = [(3,3)^{⅄},4,1^{+}], order 64, and 1=[3^{1,1,1}]^{Δ} = [(3,3)^{Δ},4,1^{+}] ≅ [[4,2<sup>+</sup>,4]]^{+}, order 32.
Central inversion
A central inversion, order 2, is operationally differently by dimension. The group [ ]^{n} = [2^{n1}] represents n orthogonal mirrors in ndimensional space, or an nflat subspace of a higher dimensional space. The mirrors of the group [2^{n1}] are numbered . The order of the mirrors doesn't matter in the case of an inversion. The matrix of a central inversion is , the Identity matrix with negative one on the diagonal.
From that basis, the central inversion has a generator as the product of all the orthogonal mirrors. In Coxeter notation this inversion group is expressed by adding an alternation ^{+} to each 2 branch. The alternation symmetry is marked on Coxeter diagram nodes as open nodes.
A CoxeterDynkin diagram can be marked up with explicit 2 branches defining a linear sequence of mirrors, opennodes, and shared doubleopen nodes to show the chaining of the reflection generators.
For example, [2^{+},2] and [2,2^{+}] are subgroups index 2 of [2,2],
Dimension  Coxeter notation  Order  Coxeter diagram  Operation  Generator 

2  [2]^{+}  2  180° rotation, C_{2}  {01}  
3  [2^{+},2^{+}]  2  rotoreflection, C_{i} or S_{2}  {012}  
4  [2^{+},2^{+},2^{+}]  2  double rotation  {0123}  
5  [2^{+},2^{+},2^{+},2^{+}]  2  double rotary reflection  {01234}  
6  [2^{+},2^{+},2^{+},2^{+},2^{+}]  2  triple rotation  {012345}  
7  [2^{+},2^{+},2^{+},2^{+},2^{+},2^{+}]  2  triple rotary reflection  {0123456} 
Rotations and rotary reflections
Rotations and rotary reflections are constructed by a single singlegenerator product of all the reflections of a prismatic group, [2p]×[2q]×... where gcd(p,q,...)=1, they are isomorphic to the abstract cyclic group Z_{n}, of order n=2pq.
The 4dimensional double rotations, [2p^{+},2^{+},2q^{+}] (with gcd(p,q)=1), which include a central group, and are expressed by Conway as ±[C_{p}×C_{q}],[5] order 2pq. From Coxeter diagram
If there is a common factor f, the double rotation can be written as ^{1}⁄_{f}[2pf^{+},2^{+},2qf^{+}] (with gcd(p,q)=1), generator {0123}, order 2pqf. For example, p=q=1, f=2, ^{1}⁄_{2}[4^{+},2^{+},4^{+}] is order 4. And ^{1}⁄_{f}[2pf^{+},2^{+},2qf^{+}]^{+}, generator {01230123}, is order pqf. For example, ^{1}⁄_{2}[4^{+},2^{+},4^{+}]^{+} is order 2, a central inversion.
Dimension  Coxeter notation  Order  Coxeter diagram  Operation  Generator  Direct subgroup  

2  [2p]^{+}  2p  Rotation  {01}  [2p]^{+2}  Simple rotation: [2p]^{+2} = [p]^{+} order p  
3  [2p^{+},2^{+}]  rotary reflection  {012}  [2p^{+},2^{+}]^{+}  
4  [2p^{+},2^{+},2^{+}]  double rotation  {0123}  [2p^{+},2^{+},2^{+}]^{+}  
5  [2p^{+},2^{+},2^{+},2^{+}]  double rotary reflection  {01234}  [2p^{+},2^{+},2^{+},2^{+}]^{+}  
6  [2p^{+},2^{+},2^{+},2^{+},2^{+}]  triple rotation  {012345}  [2p^{+},2^{+},2^{+},2^{+},2^{+}]^{+}  
7  [2p^{+},2^{+},2^{+},2^{+},2^{+},2^{+}]  triple rotary reflection  {0123456}  [2p^{+},2^{+},2^{+},2^{+},2^{+},2^{+}]^{+}  
4  [2p^{+},2^{+},2q^{+}]  2pq  double rotation  {0123}  [2p^{+},2^{+},2q^{+}]^{+}  Double rotation: [2p^{+},2^{+},2q^{+}]^{+} order pq gcd(p,q)=1  
5  [2p^{+},2^{+},2q^{+},2^{+}]  double rotary reflection  {01234}  [2p^{+},2^{+},2q^{+},2^{+}]^{+}  
6  [2p^{+},2^{+},2q^{+},2^{+},2^{+}]  triple rotation  {012345}  [2p^{+},2^{+},2q^{+},2^{+},2^{+}]  
7  [2p^{+},2^{+},2q^{+},2^{+},2^{+},2^{+}]  triple rotary reflection  {0123456}  [2p^{+},2^{+},2q^{+},2^{+},2^{+},2^{+}]^{+}  
6  [2p^{+},2^{+},2q^{+},2^{+},2r^{+}]  2pqr  triple rotation  {012345}  [2p^{+},2^{+},2q^{+},2^{+},2r^{+}]^{+}  Triple rotation: [2p^{+},2^{+},2q^{+},2^{+},2r^{+}]^{+} order pqr gcd(p,q,r)=1  
7  [2p^{+},2^{+},2q^{+},2^{+},2r^{+},2^{+}]  triple rotary reflection  {0123456}  [2p^{+},2^{+},2q^{+},2^{+},2r^{+},2^{+}]^{+} 
Commutator subgroups
Simple groups with only oddorder 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 evenorder branches, the commutator subgroup has index 2^{c}, where c is the number of disconnected subgraphs when all the evenorder branches are removed.[6] For example, [4,4] has three independent nodes in the Coxeter diagram when the 4s are removed, so its commutator subgroup is index 2^{3}, and can have different representations, all with three ^{+} operators: [4^{+},4^{+}]^{+}, [1^{+},4,1^{+},4,1^{+}], [1^{+},4,4,1^{+}]^{+}, or [(4^{+},4^{+},2^{+})]. A general notation can be used with +c as a group exponent, like [4,4]^{+3}.
Example subgroups
Rank 2 example subgroups
Dihedral symmetry groups with evenorders have a number of subgroups. This example shows two generator mirrors of [4] in red and green, and looks at all subgroups by halfing, rankreduction, and their direct subgroups. The group [4],
Subgroups of [4]  

Index  1  2 (half)  4 (Rankreduction)  
Diagram  
Coxeter 
[1,4,1] = [4] 
[1^{+},4,1] = [1^{+},4] = [2] 
[1,4,1^{+}] = [4,1^{+}] = [2] 
[1] = [ ] 
[1] = [ ]  
Generators  {0,1}  {101,1}  {0,010}  {0}  {1}  
Direct subgroups  
Index  2  4  8  
Diagram  
Coxeter  [4]^{+} 
[4]^{+2} = [1^{+},4,1^{+}] = [2]^{+} 
[ ]^{+}  
Generators  {01}  {(01)^{2}}  {0^{2}} = {1^{2}} = {(01)^{4}} = { } 
Rank 3 Euclidean example subgroups
The [4,4] group has 15 small index subgroups. This table shows them all, with a yellow fundamental domain for pure reflective groups, and alternating white and blue domains which are paired up to make rotational domains. Cyan, red, and green mirror lines correspond to the same colored nodes in the Coxeter diagram. Subgroup generators can be expressed as products of the original 3 mirrors of the fundamental domain, {0,1,2}, corresponding to the 3 nodes of the Coxeter diagram,
Small index subgroups of [4,4]  

Index  1  2  4  
Diagram  
Coxeter 
[1,4,1,4,1] = [4,4] 
[1^{+},4,4] 
[4,4,1^{+}] 
[4,1^{+},4] 
[1^{+},4,4,1^{+}] 
[4^{+},4^{+}]  
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  
Coxeter  [4,4^{+}] 
[4^{+},4] 
[(4,4,2^{+})] 
[4,1^{+},4,1^{+}] 
[1^{+},4,1^{+},4]  
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  
Coxeter  [4,4]^{+} 
[4,4^{+}]^{+} 
[4^{+},4]^{+} 
[(4,4,2^{+})]^{+} 
[4,4]^{+3} = [(4^{+},4^{+},2^{+})] = [1^{+},4,1^{+},4,1^{+}] = [4^{+},4^{+}]^{+}  
Generators  {01,12}  {(01)^{2},12}  {01,(12)^{2}}  {02,(01)^{2},(12)^{2}}  {(01)^{2},(12)^{2},2(01)^{2}2}  
Orbifold  442  2222  
Radical subgroups  
Index  8  16  
Diagram  
Coxeter  [4,4*] 
[4*,4] 
[4,4*]^{+} 
[4*,4]^{+}  
Orbifold  *2222  2222 
Hyperbolic example subgroups
The same set of 15 small subgroups exists on all triangle groups with even order elements, like [6,4] in the hyperbolic plane:
Small index subgroups of [6,4]  

Index  1  2  4  
Diagram  
Coxeter 
[1,6,1,4,1] = [6,4] 
[1^{+},6,4] 
[6,4,1^{+}] 
[6,1^{+},4] 
[1^{+},6,4,1^{+}] 
[6^{+},4^{+}]  
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  
Coxeter  [6,4^{+}] 
[6^{+},4] 
[(6,4,2^{+})] 
[6,1^{+},4,1^{+}] = 
[1^{+},6,1^{+},4] =  
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  
Coxeter  [6,4]^{+} 
[6,4^{+}]^{+} 
[6^{+},4]^{+} 
[(6,4,2^{+})]^{+} 
[6^{+},4^{+}]^{+} = [1^{+},6,1^{+},4,1^{+}] =  
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  
Coxeter (orbifold) 
[6,4*] (*3333) 
[6*,4] (*222222) 
[6,4*]^{+} (3333) 
[6*,4]^{+} (222222) 
Extended symmetry
 
In the Euclidean plane, the , [3^{[3]}] Coxeter group can be extended in two ways into the , [6,3] Coxeter group and relates uniform tilings as ringed diagrams. 
Coxeter's notation includes double square bracket notation, [[X]] to express automorphic symmetry within a Coxeter diagram. Johnson added alternative of angledbracket <[X]> or ⟨[X]⟩ option as equivalent to square brackets for doubling to distinguish diagram symmetry through the nodes versus through the branches. Johnson also added a prefix symmetry modifier [Y[X]], where Y can either represent symmetry of the Coxeter diagram of [X], or symmetry of the fundamental domain of [X].
For example, in 3D these equivalent rectangle and rhombic geometry diagrams of :
Further symmetry exists in the cyclic and branching , , and diagrams. has order 2n symmetry of a regular ngon, {n}, and is represented by [n[3^{[n]}]]. and are represented by [3[3^{1,1,1}]] = [3,4,3] and [3[3^{2,2,2}]] respectively while by [(3,3)[3^{1,1,1,1}]] = [3,3,4,3], with the diagram containing the order 24 symmetry of the regular tetrahedron, {3,3}. The paracompact hyperbolic group = [3^{1,1,1,1,1}],
An asterisk * superscript is effectively an inverse operation, creating radical subgroups removing connected of oddordered mirrors.[7]
Examples:
Example Extended groups and radical subgroups  



Looking at generators, the double symmetry is seen as adding a new operator that maps symmetric positions in the Coxeter diagram, making some original generators redundant. For 3D space groups, and 4D point groups, Coxeter defines an index two subgroup of [[X]], [[X]^{+}], which he defines as the product of the original generators of [X] by the doubling generator. This looks similar to [[X]]^{+}, which is the chiral subgroup of [[X]]. So for example the 3D space groups [[4,3,4]]^{+} (I432, 211) and [[4,3,4]^{+}] (Pm3n, 223) are distinct subgroups of [[4,3,4]] (Im3m, 229).
Computation with reflection matrices as symmetry generators
A Coxeter group, represented by Coxeter diagram
[p,q]^{+},
With one even branch, [p^{+},2q],
With even branches, [2p^{+},2q^{+}],
In the case of affine Coxeter groups like
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 = (ρ_{0}ρ_{1}ρ_{2})^{h/2}, where h is 6 and 10 respectively, the Coxeter number for each family. For 3D Coxeter group [p,q] (
Coxeter groups are categorized by their rank, being the number of nodes in its CoxeterDynkin diagram. The structure of the groups are also given with their abstract group types: In this article, the abstract dihedral groups are represented as Dih_{n}, and cyclic groups are represented by Z_{n}, with Dih_{1}=Z_{2}.
Rank 2
Example, in 2D, the Coxeter group [p] (

 


Rank 3
The finite rank 3 Coxeter groups are [1,p], [2,p], [3,3], [3,4], and [3,5].
To reflect a point through a plane (which goes through the origin), one can use , where is the 3x3 identity matrix and is the threedimensional unit vector for the vector normal of the plane. If the L2 norm of and is unity, the transformation matrix can be expressed as:
Dihedral symmetry
The reducible 3dimensional finite reflective group is dihedral symmetry, [p,2], order 4p,
Reflections  Rotation  Rotoreflection  

Name  R_{0}  R_{1}  R_{2}  S_{0,1}  S_{1,2}  S_{0,2}  V_{0,1,2} 
Group  
Order  2  2  2  p  2  2p  
Matrix 







Tetrahedral symmetry
The simplest irreducible 3dimensional finite reflective group is tetrahedral symmetry, [3,3], order 24,
Reflections  Rotations  Rotoreflection  

Name  R_{0}  R_{1}  R_{2}  S_{0,1}  S_{1,2}  S_{0,2}  V_{0,1,2} 
Name  
Order  2  2  2  3  2  4  
Matrix 







(0,1,1)_{n}  (1,1,0)_{n}  (0,1,1)_{n}  (1,1,1)_{axis}  (1,1,1)_{axis}  (1,0,0)_{axis} 
Octahedral symmetry
Another irreducible 3dimensional finite reflective group is octahedral symmetry, [4,3], order 48,
Reflections  Rotations  Rotoreflection  

Name  R_{0}  R_{1}  R_{2}  S_{0,1}  S_{1,2}  S_{0,2}  V_{0,1,2} 
Group  
Order  2  2  2  4  3  2  6 
Matrix 







(0,0,1)_{n}  (0,1,1)_{n}  (1,1,0)_{n}  (1,0,0)_{axis}  (1,1,1)_{axis}  (1,1,0)_{axis} 
Icosahedral symmetry
A final irreducible 3dimensional finite reflective group is icosahedral symmetry, [5,3], order 120,
Reflections  Rotations  Rotoreflection  

Name  R_{0}  R_{1}  R_{2}  S_{0,1}  S_{1,2}  S_{0,2}  V_{0,1,2} 
Group  
Order  2  2  2  5  3  2  10 
Matrix  
(1,0,0)_{n}  _{n}  (0,1,0)_{n}  (φ,1,0)_{axis}  (1,1,1)_{axis}  (1,0,0)_{axis} 
Affine rank 3
A simple example affine group is [4,4] (
Reflections  Rotations  Rotoreflection  

Name  R_{0}  R_{1}  R_{2}  S_{0,1}  S_{1,2}  S_{0,2}  V_{0,1,2} 
Group  
Order  2  2  2  4  2  ∞  
Matrix 







Rank 4
Hyperoctahedral or hexadecachoric symmetry
A irreducible 4dimensional finite reflective group is hyperoctahedral group (or hexadecachoric group (for 16cell), B_{4}=[4,3,3], order 384,
Chiral hyperoctahedral symmetry, [4,3,3]^{+}, (
Reflections  Rotations  Rotoreflection  Double rotation  

Name  R_{0}  R_{1}  R_{2}  R_{3}  S_{0,1}  S_{1,2}  S_{2,3}  S_{0,2}  S_{1,3}  S_{0,3}  V_{1,2,3}  V_{0,1,3}  V_{0,1,2}  V_{0,2,3}  W_{0,1,2,3} 
Group  
Order  2  2  2  2  4  3  2  4  6  8  
Matrix 















(0,0,0,1)_{n}  (0,0,1,1)_{n}  (0,1,1,0)_{n}  (1,1,0,0)_{n} 
Icositetrachoric symmetry
A irreducible 4dimensional finite reflective group is Icositetrachoric group (for 24cell), F_{4}=[3,4,3], order 1152,
Chiral icositetrachoric symmetry, [3,4,3]^{+}, (
Reflections  Rotations  Rotoreflection  Double rotation  

Name  R_{0}  R_{1}  R_{2}  R_{3}  S_{0,1}  S_{1,2}  S_{2,3}  S_{0,2}  S_{1,3}  S_{0,3}  V_{1,2,3}  V_{0,1,3}  V_{0,1,2}  V_{0,2,3}  W_{0,1,2,3} 
Group  
Order  2  2  2  2  3  4  3  2  6  12  
Matrix 















(1,1,1,1)_{n}  (0,0,1,0)_{n}  (0,1,1,0)_{n}  (1,1,0,0)_{n} 
Hypericosahedral symmetry
The hypericosahedral symmetry, [5,3,3], order 14400,
Reflections  

Name  R_{0}  R_{1}  R_{2}  R_{3} 
Group  
Order  2  2  2  2 
Matrix  
(1,0,0,0)_{n}  _{n}  (0,1,0,0)_{n}  _{n} 
Rank one groups
In one dimension, the bilateral group [ ] represents a single mirror symmetry, abstract Dih_{1} or Z_{2}, symmetry order 2. It is represented as a Coxeter–Dynkin diagram with a single node,
Group  Coxeter notation  Coxeter diagram  Order  Description 

C_{1}  [ ]^{+}  1  Identity  
D_{1}  [ ]  2  Reflection group 
Rank two groups
In two dimensions, the rectangular group [2], abstract D_{1}^{2} or D_{2}, also can be represented as a direct product [ ]×[ ], being the product of two bilateral groups, represents two orthogonal mirrors, with Coxeter diagram,
Coxeter notation to allow a 1 placeholder for lower rank groups, so [1] is the same as [ ], and [1^{+}] or [1]^{+} is the same as [ ]^{+} and Coxeter diagram
The full pgonal group [p], abstract dihedral group D_{p}, (nonabelian for p>2), of order 2p, is generated by two mirrors at angle π/p, represented by Coxeter diagram
Coxeter notation uses doublebracking 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
In the hyperbolic plane, there is a full pseudogonal group [iπ/λ], and pseudogonal subgroup [iπ/λ]^{+},
Example rank 2 finite and hyperbolic symmetries  

Type  Finite  Affine  Hyperbolic  
Geometry  ...  
Coxeter  [ ] 
[2]=[ ]×[ ] 
[3] 
[4] 
[p] 
[∞] 
[∞] 
[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) 
...  
Odd images (inverted) 

Coxeter  [ ]^{+} 
[2]^{+} 
[3]^{+} 
[4]^{+} 
[p]^{+} 
[∞]^{+} 
[∞]^{+} 
[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  
Z_{n}  n  n•  [n]^{+}  n  Cyclic: nfold rotations. Abstract group Z_{n}, the group of integers under addition modulo n.  
D_{n}  nm  *n•  [n]  2n  Dihedral: cyclic with reflections. Abstract group Dih_{n}, the dihedral group.  
Affine  
Z_{∞}  ∞  ∞•  [∞]^{+}  ∞  Cyclic: apeirogonal group. Abstract group Z_{∞}, the group of integers under addition.  
Dih_{∞}  ∞m  *∞•  [∞]  ∞  Dihedral: parallel reflections. Abstract infinite dihedral group Dih_{∞}.  
Hyperbolic  
Z_{∞}  [πi/λ]^{+}  ∞  pseudogonal group  
Dih_{∞}  [πi/λ]  ∞  full pseudogonal group 
Rank three groups
Point groups in 3 dimensions can be expressed in bracket notation related to the rank 3 Coxeter groups:
Finite groups of isometries in 3space[2]  

Rotation groups  Extended groups  
Name  Bracket  Orb  Sch  Abstract  Order  Name  Bracket  Orb  Sch  Abstract  Order 
Identity  [ ]^{+}  11  C_{1}  Z_{1}  1  Bilateral  [1,1] = [ ]  *  D_{1}  D_{1}  2 
Central  [2^{+},2^{+}]  ×  C_{i}  2×Z_{1}  2  
Acrorhombic  [1,2]^{+} = [2]^{+}  22  C_{2}  Z_{2}  2  Acrorectangular  [1,2] = [2]  *22  C_{2v}  D_{2}  4 
Gyrorhombic  [2^{+},4^{+}]  2×  S_{4}  Z_{4}  4  
Orthorhombic  [2,2^{+}]  2*  D_{1d}  D_{1}×Z_{2}  4  
Pararhombic  [2,2]^{+}  222  D_{2}  D_{2}  4  Gyrorectangular  [2^{+},4]  2*2  D_{2d}  D_{4}  8 
Orthorectangular  [2,2]  *222  D_{2h}  D_{1}×D_{2}  8  
Acropgonal  [1,p]^{+} = [p]^{+}  pp  C_{p}  Z_{p}  p  Full acropgonal  [1,p] = [p]  *pp  C_{pv}  D_{p}  2p 
Gyropgonal  [2^{+},2p^{+}]  p×  S_{2p}  Z_{2p}  2p  
Orthopgonal  [2,p^{+}]  p*  C_{ph}  D_{1}×Z_{p}  2p  
Parapgonal  [2,p]^{+}  p22  D_{p}  D_{p}  2p  Full gyropgonal  [2^{+},2p]  2*p  D_{pd}  D_{2p}  4p 
Full orthopgonal  [2,p]  *p22  D_{ph}  D_{1}×D_{p}  4p  
Tetrahedral  [3,3]+  332  T  A_{4}  12  Full tetrahedral  [3,3]  *332  T_{d}  S_{4}  24 
Pyritohedral  [3^{+},4]  3*2  T_{h}  2×A_{4}  24  
Octahedral  [3,4]^{+}  432  O  S_{4}  24  Full octahedral  [3,4]  *432  O_{h}  2×S_{4}  48 
Icosahedral  [3,5]^{+}  532  I  A_{5}  60  Full icosahedral  [3,5]  *532  I_{h}  2×A_{5}  120 
In three dimensions, the full orthorhombic group or orthorectangular [2,2], abstractly D_{2}×D_{2}, order 8, represents three orthogonal mirrors, (also represented by Coxeter diagram as three separate dots
First there is a "semidirect" subgroup, the orthorhombic group, [2,2^{+}] (
Next there is the full orthopgonal group, [2,p] (
The direct subgroup is called the parapgonal group, [2,p]^{+} (
The full gyropgonal group, [2^{+},2p] (
The polyhedral groups are based on the symmetry of platonic solids: the tetrahedron, octahedron, cube, icosahedron, and dodecahedron, with Schläfli symbols {3,3}, {3,4}, {4,3}, {3,5}, and {5,3} respectively. The Coxeter groups for these are: [3,3] (
In all these symmetries, alternate reflections can be removed producing the rotational tetrahedral [3,3]^{+}(
The tetrahedral group, [3,3] (
Example rank 3 finite Coxeter groups subgroup trees  

Tetrahedral symmetry  Octahedral symmetry 
Icosahedral symmetry  
Finite (point groups in three dimensions)  



Affine
In the Euclidean plane there's 3 fundamental reflective groups generated by 3 mirrors, represented by Coxeter diagrams
[[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.


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
Subgroup relations 
Point groups
Rank four groups defined the 4dimensional point groups:
Finite groups  






Subgroups
1D4D reflective point groups and subgroups  

Order  Reflection  Semidirect subgroups 
Direct subgroups 
Commutator subgroup  
2  [ ]  [ ]^{+}  [ ]^{+1}  [ ]^{+}  
4  [2]  [2]^{+}  [2]^{+2}  
8  [2,2]  [2^{+},2]  [2^{+},2^{+}]  [2,2]^{+}  [2,2]^{+3}  
16  [2,2,2]  [2^{+},2,2] [(2,2)^{+},2]  [2^{+},2^{+},2] [(2,2)^{+},2^{+}] [2^{+},2^{+},2^{+}]  [2,2,2]^{+} [2^{+},2,2^{+}]  [2,2,2]^{+4}  
[2^{1,1,1}]  [(2^{+})^{1,1,1}]  
2n  [n]  [n]^{+}  [n]^{+1}  [n]^{+}  
4n  [2n]  [2n]^{+}  [2n]^{+2}  
4n  [2,n]  [2,n^{+}]  [2,n]^{+}  [2,n]^{+2}  
8n  [2,2n]  [2^{+},2n]  [2^{+},2n^{+}]  [2,2n]^{+}  [2,2n]^{+3}  
8n  [2,2,n]  [2^{+},2,n] [2,2,n^{+}]  [2^{+},(2,n)^{+}]  [2,2,n]^{+} [2^{+},2,n^{+}]  [2,2,n]^{+3}  
16n  [2,2,2n]  [2,2^{+},2n]  [2^{+},2^{+},2n] [2,2^{+},2n^{+}] [(2,2)^{+},2n^{+}] [2^{+},2^{+},2n^{+}]  [2,2,2n]^{+} [2^{+},2n,2^{+}]  [2,2,2n]^{+4}  
[2,2n,2]  [2^{+},2n^{+},2^{+}]  
[2n,2^{1,1}]  [2n^{+},(2^{+})^{1,1}]  
24  [3,3]  [3,3]^{+}  [3,3]^{+1}  [3,3]^{+}  
48  [3,3,2]  [(3,3)^{+},2]  [3,3,2]^{+}  [3,3,2]^{+2}  
48  [4,3]  [4,3^{+}]  [4,3]^{+}  [4,3]^{+2}  
96  [4,3,2]  [(4,3)^{+},2] [4,(3,2)^{+}]  [4,3,2]^{+}  [4,3,2]^{+3}  
[3,4,2]  [3,4,2^{+}] [3^{+},4,2]  [(3,4)^{+},2^{+}]  [3^{+},4,2^{+}]  
120  [5,3]  [5,3]^{+}  [5,3]^{+1}  [5,3]^{+}  
240  [5,3,2]  [(5,3)^{+},2]  [5,3,2]^{+}  [5,3,2]^{+2}  
4pq  [p,2,q]  [p^{+},2,q]  [p,2,q]^{+} [p^{+},2,q^{+}]  [p,2,q]^{+2}  [p^{+},2,q^{+}]  
8pq  [2p,2,q]  [2p,(2,q)^{+}]  [2p^{+},(2,q)^{+}]  [2p,2,q]^{+}  [2p,2,q]^{+3}  
16pq  [2p,2,2q]  [2p,2^{+},2q]  [2p^{+},2^{+},2q] [2p^{+},2^{+},2q^{+}] [(2p,(2,2q)^{+},2^{+})]   
[2p,2,2q]^{+}  [2p,2,2q]^{+4}  
120  [3,3,3]  [3,3,3]^{+}  [3,3,3]^{+1}  [3,3,3]^{+}  
192  [3^{1,1,1}]  [3^{1,1,1}]^{+}  [3^{1,1,1}]^{+1}  [3^{1,1,1}]^{+}  
384  [4,3,3]  [4,(3,3)^{+}]  [4,3,3]^{+}  [4,3,3]^{+2}  
1152  [3,4,3]  [3^{+},4,3]  [3,4,3]^{+} [3^{+},4,3^{+}]  [3,4,3]^{+2}  [3^{+},4,3^{+}]  
14400  [5,3,3]  [5,3,3]^{+}  [5,3,3]^{+1}  [5,3,3]^{+} 
Space groups
Space groups  

Affine isomorphism and correspondences 
8 cubic space groups as extended symmetry from [3^{[4]}], with square Coxeter diagrams and reflective fundamental domains 
35 cubic space groups in International, Fibrifold notation, and Coxeter notation 
Rank four groups as 3dimensional space groups  





Line groups
Rank four groups also defined the 3dimensional line groups:
Semiaffine (3D) groups  

Point group  Line group  
HermannMauguin  Schönflies  HermannMauguin  Offset type  Wallpaper  Coxeter [∞_{h},2,p_{v}]  
Even n  Odd n  Even n  Odd n  IUC  Orbifold  Diagram  
n  C_{n}  Pn_{q}  Helical: q  p1  o  [∞^{+},2,n^{+}]  
2n  n  S_{2n}  P2n  Pn  None  p11g, pg(h)  ××  [(∞,2)^{+},2n^{+}]  
n/m  2n  C_{nh}  Pn/m  P2n  None  p11m, pm(h)  **  [∞^{+},2,n]  
2n/m  C_{2nh}  P2n_{n}/m  Zigzag  c11m, cm(h)  *×  [∞^{+},2^{+},2n]  
nmm  nm  C_{nv}  Pnmm  Pnm  None  p1m1, pm(v)  **  [∞,2,n^{+}]  
Pncc  Pnc  Planar reflection  p1g1, pg(v)  ××  [∞^{+},(2,n)^{+}]  
2nmm  C_{2nv}  P2n_{n}mc  Zigzag  c1m1, cm(v)  *×  [∞,2^{+},2n^{+}]  
n22  n2  D_{n}  Pn_{q}22  Pn_{q}2  Helical: q  p2  2222  [∞,2,n]^{+}  
2n2m  nm  D_{nd}  P2n2m  Pnm  None  p2mg, pmg(h)  22*  [(∞,2)^{+},2n]  
P2n2c  Pnc  Planar reflection  p2gg, pgg  22×  [^{+}(∞,(2),2n)^{+}]  
n/mmm  2n2m  D_{nh}  Pn/mmm  P2n2m  None  p2mm, pmm  *2222  [∞,2,n]  
Pn/mcc  P2n2c  Planar reflection  p2mg, pmg(v)  22*  [∞,(2,n)^{+}]  
2n/mmm  D_{2nh}  P2n_{n}/mcm  Zigzag  c2mm, cmm  2*22  [∞,2^{+},2n] 
Duoprismatic group
Extended duoprismatic symmetry 

Extended duoprismatic groups, [p]×[p] or [p,2,p] or 
Rank four groups defined the 4dimensional 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] 
Coxeter [[p,2,p]] 
Wallpaper  
IUC  Orbifold  Diagram  IUC  Orbifold  Diagram  
p1  o  [p^{+},2,q^{+}]  [[p<sup>+</sup>,2,p<sup>+</sup>]]  p1  o  
pg  ××  [(p,2)^{+},2q^{+}]    
pm  **  [p^{+},2,q]    
cm  *×  [2p^{+},2^{+},2q]    
p2  2222  [p,2,q]^{+}  [[p,2,p]]^{+}  p4  442  
pmg  22*  [(p,2)^{+},2q]    
pgg  22×  [^{+}(2p,(2),2q)^{+}]  [[<sup>+</sup>(2p,(2),2p)<sup>+</sup>]]  cmm  2*22  
pmm  *2222  [p,2,q]  [[p,2,p]]  p4m  *442  
cmm  2*22  [2p,2^{+},2q]  [[2p,2<sup>+</sup>,2p]]  p4g  4*2 
Wallpaper groups
Rank four groups also defined some of the 2dimensional wallpaper groups, as limiting cases of the fourdimensional duoprism groups:
Affine (2D plane)  



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

Reflective group 
Reflective subgroup 
Mixed subgroup 
Rotation subgroup 
Improper rotation/ translation 
Commutator subgroup  
[∞,2,∞], (*2222)  [1^{+},∞,2,∞], (*2222)  [∞^{+},2,∞], (**)  [∞,2,∞]^{+}, (2222)  [∞,2^{+},∞]^{+}, (°) [∞^{+},2^{+},∞^{+}], (°) [∞^{+},2,∞^{+}], (°) [∞^{+},2^{+},∞], (*×) [(∞,2)^{+},∞^{+}], (××) [^{+}(∞,(2),∞)^{+}], (22×) 
[(∞^{+},2^{+},∞^{+},2^{+})], (°)  
[∞,2^{+},∞], (2*22) [(∞,2)^{+},∞], (22*) 
Complex reflections
Coxeter notation has been extended to Complex space, C^{n} where nodes are unitary reflections of period greater than 2. Nodes are labeled by an index, assumed to be 2 for ordinary real reflection if suppressed. Complex reflection groups are called Shephard groups rather than Coxeter groups, and can be used to construct complex polytopes.
In , a rank 1 shephard group
Coxeter writes the rank 2 complex group, _{p}[q]_{r} represents Coxeter diagram
The rank 2 solutions that generate complex polygons are: _{p}[4]_{2} (p is 2,3,4,...), _{3}[3]_{3}, _{3}[6]_{2}, _{3}[4]_{3}, _{4}[3]_{4}, _{3}[8]_{2}, _{4}[6]_{2}, _{4}[4]_{3}, _{3}[5]_{3}, _{5}[3]_{5}, _{3}[10]_{2}, _{5}[6]_{2}, and _{5}[4]_{3} with Coxeter diagrams
Infinite groups are _{3}[12]_{2}, _{4}[8]_{2}, _{6}[6]_{2}, _{3}[6]_{3}, _{6}[4]_{3}, _{4}[4]_{4}, and _{6}[3]_{6} or
Index 2 subgroups exists by removing a real reflection: _{p}[2q]_{2} → _{p}[q]_{p}. Also index r subgroups exist for 4 branches: _{p}[4]_{r} → _{p}[r]_{p}.
For the infinite family _{p}[4]_{2}, for any p = 2, 3, 4,..., there are two subgroups: _{p}[4]_{2} → [p], index p, while and _{p}[4]_{2} → _{p}[]×_{p}[], index 2.
Notes
 Johnson (2018), 11.6 Subgroups and extensions, p 255, halving subgroups
 Johnson (2018), pp.231236, and p 245 Table 11.4 Finite groups of isometries in 3space
 Johnson (2018), 11.6 Subgroups and extensions, p 259, radical subgroup
 Johnson (2018), 11.6 Subgroups and extensions, p 258, trionic subgroups
 Conway, 2003, p.46, Table 4.2 Chiral groups II
 Coxeter and Moser, 1980, Sec 9.5 Commutator subgroup, p. 124–126
 Johnson, Norman W.; Weiss, Asia Ivić (1999). "Quaternionic modular groups". Linear Algebra and Its Applications. 295 (1–3): 159–189. doi:10.1016/S00243795(99)00107X.
 The Crystallographic Space groups in Geometric algebra, D. Hestenes and J. Holt, Journal of Mathematical Physics. 48, 023514 (2007) (22 pages) PDF
 Coxeter, Regular Complex Polytopes, 9.7 Twogenerator subgroups reflections. pp. 178–179
References
 H.S.M. Coxeter:
 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, WileyInterscience Publication, 1995, ISBN 9780471010036
 (Paper 22) Coxeter, H.S.M. (1940), "Regular and Semi Regular Polytopes I", Math. Z., 46: 380–407, doi:10.1007/bf01181449
 (Paper 23) Coxeter, H.S.M. (1985), "Regular and SemiRegular Polytopes II", Math. Z., 188 (4): 559–591, doi:10.1007/bf01161657
 (Paper 24) Coxeter, H.S.M. (1988), "Regular and SemiRegular Polytopes III", Math. Z., 200: 3–45, doi:10.1007/bf01161745
 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, WileyInterscience Publication, 1995, ISBN 9780471010036
 Coxeter, H. S. M.; Moser, W. O. J. (1980). Generators and Relations for Discrete Groups. New York: SpringerVerlag. ISBN 0387092129.
 Norman Johnson Uniform Polytopes, Manuscript (1991)
 N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
 Norman W. Johnson and Asia Ivic Weiss Quadratic Integers and Coxeter Groups PDF Can. J. Math. Vol. 51 (6), 1999 pp. 1307–1336
 N. W. Johnson: Geometries and Transformations, (2018) ISBN 9781107103405 Chapter 11: Finite symmetry groups
 Conway, John Horton; Delgado Friedrichs, Olaf; Huson, Daniel H.; Thurston, William P. (2001), "On threedimensional space groups", Beiträge zur Algebra und Geometrie. Contributions to Algebra and Geometry, 42 (2): 475–507, ISSN 01384821, MR 1865535
 John H. Conway and Derek A. Smith, On Quaternions and Octonions, 2003, ISBN 9781568811345
 The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim GoodmanStrass, ISBN 9781568812205 Ch.22 35 prime space groups, ch.25 184 composite space groups, ch.26 Higher still, 4D point groups