User:Tomruen/List of Coxeter groups

This test article is my attempt to construct Coxeter groups by rank, including products. I subdivide them by their finite order, infinite, or hyperbolic infinite.

I name them by their group names, Coxeter's bracket notation names, and their Coxeter-Dynkin diagram graphs. I also give an example polytope or tessellation that has this symmetry.

Finite Definite (spherical)

Rank 1

(1-space)

• A₁: [ ]

Rank 2

(2-space)

• A₂: [3] (triangle)
• B₂: [4] square)
• H₂: [5] (pentagon)
• I₂(p) [p] (p-gon)

(2-space prismatic)

• A₁xA₁: [ ]x[ ] = (rectangle)

Rank 3

(3-space)

• A₃: [3,3] (tetrahedron)
• B₃: [4,3] (cube/octahedron)
• H₃: [5,3] (dodecahedron/icosahedron)

(3-space prismatic)

• I₂(p)xA₁: [p]x[ ] (p-gonal prism)
• A₁xA₁xA₁: [ ]x[ ]x[ ] = (cube/cuboid)

Rank 4

(4-space)

• A₄: [3,3,3] (5-cell)
• B₄: [4,3,3] (tesseract/16-cell)
• D₄: [3^1,1,1] (24-cell)
• F₄: [3,4,3] (24-cell)
• H₄: [5,3,3] (120-cell/600-cell)

(4-space prismatic)

• A₃xA₁: [3,3]x[ ] (tetrahedral prism)
• B₃xA₁: [4,3]x[ ] (tesseract/orthoplex)
• H₃xA₁: [5,3]x[ ] (dodecahedral prism)
• I₂(p)xA₁xA₁: [p]x[ ]x[ ] = (p-4 duoprism)
• A₁xA₁xA₁xA₁: [ ]x[ ]x[ ]x[ ] = (tesseract/orthoplex)
• I₂(p)xI₂(q) [p]x[q] (p-q duoprism)

Rank 5

(5-space)

• A₅: [3,3,3,3] (hexateron)
• B₅: [4,3,3,3] (Penteract)
• D₅: [3^2,1,1] (Demipenteract)

(5-space prismatic)

• A₄xA₁: [3,3,3]x[ ]
• B₄xA₁: [4,3,3]x[ ] -
• F₄xA₁: [3,4,3]x[ ] -
• H₄xA₁: [5,3,3]x[ ] -
• D₄xA₁: [3^1,1,1]x[ ] -
• A₃xI₂^p: [3,3]x[p] -
• B₃xI₂^p: [4,3]x[p] -
• H₃xI₂^p: [5,3]x[p] -
• I₂^pxI₂^qxA₁: [p]x[q]x[ ] -

Rank 6

(6-space)

• A₆: [3,3,3,3,3] (6-simplex)
• B₆: [4,3,3,3,3] (6-orthoplex/6-hypercube)
• D₆: [3^3,1,1] (6-orthoplex)
• E₆: [3^2,2,1] (1_22 polytope)

Uniform prism

There are 6 categorical uniform prisms based the uniform 5-polytopes.

#	Coxeter group		Coxeter-Dynkin diagram
1	A5×A1	[3,3,3,3] × [ ]
2	B5×A1	[4,3,3,3] × [ ]
3	D5×A1	[32,1,1] × [ ]

#	Coxeter group		Coxeter-Dynkin diagram
4	A3×I2(p)×A1	[3,3] × [p] × [ ]
5	B3×I2(p)×A1	[4,3] × [p] × [ ]
6	H3×I2(p)×A1	[5,3] × [p] × [ ]

Uniform duoprism

There are 11 categorical uniform duoprismatic families of polytopes based on Cartesian products of lower dimensional uniform polytopes. Five are formed as the product of a uniform polychoron with a regular polygon, and six are formed by the product of two uniform polyhedra:

#	Coxeter group		Coxeter-Dynkin diagram
1	A4×I2(p)	[3,3,3] × [p]
2	B4×I2(p)	[4,3,3] × [p]
3	F4×I2(p)	[3,4,3] × [p]
4	H4×I2(p)	[5,3,3] × [p]
5	D4×I2(p)	[31,1,1] × [p]

#	Coxeter group		Coxeter-Dynkin diagram
6	A3×A3	[3,3] × [3,3]
7	A3×B3	[3,3] × [4,3]
8	A3×H3	[3,3] × [5,3]
9	B3×B3	[4,3] × [4,3]
10	B3×H3	[4,3] × [5,3]
11	H3×A3	[5,3] × [5,3]

Uniform triprisms

There is one infinite family of uniform triaprismatic families of polytopes constructed as a Cartesian products of three regular polygons. Each combination of at least one ring on every connected group produces a uniform prismatic 6-polytope.

#	Coxeter group		Coxeter-Dynkin diagram
1	I2(p)×I2(q)×I2(r)	[p] × [q] × [r]

Rank 7

(7-space)

• A₇: [3,3,3,3,3,3] (8-simplex)
• B₇: [4,3,3,3,3,3] (8-orthoplex/8-hypercube)
• D₇: [3^4,1,1] (8-orthoplex)
• E₇: [3^3,2,1] (3_21 polytope)

There are 16 uniform prismatic families based on the uniform 6-polytopes.

#	Coxeter group		Coxeter-Dynkin diagram
1	A6×A1	[35] × [ ]
2	B6×A1	[4,34] × [ ]
3	D6×A1	[33,1,1] × [ ]
4	E6×A1	[32,2,1] × [ ]
5	A4×I2(p)×A1	[3,3,3] × [p] × [ ]
6	B4×I2(p)×A1	[4,3,3] × [p] × [ ]
7	F4×I2(p)×A1	[3,4,3] × [p] × [ ]
8	H4×I2(p)×A1	[5,3,3] × [p] × [ ]
9	D4×I2(p)×A1	[31,1,1] × [p] × [ ]
10	A3×A3×A1	[3,3] × [3,3] × [ ]
11	A3×B3×A1	[3,3] × [4,3] × [ ]
12	A3×H3×A1	[3,3] × [5,3] × [ ]
13	B3×B3×A1	[4,3] × [4,3] × [ ]
14	B3×H3×A1	[4,3] × [5,3] × [ ]
15	H3×A3×A1	[5,3] × [5,3] × [ ]
16	I2(p)×I2(q)×I2(r)×A1	[p] × [q] × [r] × [ ]

There are 18 uniform duoprismatic forms based on Cartesian products of lower dimensional uniform polytopes.

#	Coxeter group		Coxeter-Dynkin diagram
1	A5×I2(p)	[3,3,3] × [p]
2	B5×I2(p)	[4,3,3] × [p]
3	D5×I2(p)	[32,1,1] × [p]
4	A4×A3	[3,3,3] × [3,3]
5	A4×B3	[3,3,3] × [4,3]
6	A4×H3	[3,3,3] × [5,3]
7	B4×A3	[4,3,3] × [3,3]
8	B4×B3	[4,3,3] × [4,3]
9	B4×H3	[4,3,3] × [5,3]
10	H4×A3	[5,3,3] × [3,3]
11	H4×B3	[5,3,3] × [4,3]
12	H4×H3	[5,3,3] × [5,3]
13	F4×A3	[3,4,3] × [3,3]
14	F4×B3	[3,4,3] × [4,3]
15	F4×H3	[3,4,3] × [5,3]
16	D4×A3	[31,1,1] × [3,3]
17	D4×B3	[31,1,1] × [4,3]
18	D4×H3	[31,1,1] × [5,3]

There are 3 uniform triaprismatic families based on Cartesian products of uniform polyhedrons and two regular polygons.

#	Coxeter group		Coxeter-Dynkin diagram
1	A3×I2(p)×I2(q)	[3,3] × [p] × [q]
2	B3×I2(p)×I2(q)	[4,3] × [p] × [q]
3	H3×I2(p)×I2(q)	[5,3] × [p] × [q]

Rank 8

(8-space)

• A₈: [3,3,3,3,3,3,3] (8-simplex)
• B₈: [4,3,3,3,3,3,3] (8-orthoplex/8-hypercube)
• D₈: [3^5,1,1] (8-orthoplex)
• E₈: [3^4,2,1] (4_21 polytope)

Four are based on the uniform 7-polytopes:

#	Coxeter group		Coxeter-Dynkin diagram
1	A7×A1	[3,3,3,3,3,3] × [ ]
2	B7×A1	[4,3,3,3,3,3] × [ ]
3	D7×A1	[34,1,1] × [ ]
4	E7×A1	[33,2,1] × [ ]

Three are based on the uniform 6-polytopes and regular polygons:

#	Coxeter group		Coxeter-Dynkin diagram
5	A5×I2(p)×A1	[3,3,3] × [p] × [ ]
6	B5×I2(p)×A1	[4,3,3] × [p] × [ ]
7	D5×I2(p)×A1	[32,1,1] × [p] × [ ]

Fifteen are based on the product of the uniform polychora and uniform polyhedra:

#	Coxeter group		Coxeter-Dynkin diagram
8	A4×A3×A1	[3,3,3] × [3,3] × [ ]
9	A4×B3×A1	[3,3,3] × [4,3] × [ ]
10	A4×H3×A1	[3,3,3] × [5,3] × [ ]
11	B4×A3×A1	[4,3,3] × [3,3] × [ ]
12	B4×B3×A1	[4,3,3] × [4,3] × [ ]
13	B4×H3×A1	[4,3,3] × [5,3] × [ ]
14	H4×A3×A1	[5,3,3] × [3,3] × [ ]
15	H4×B3×A1	[5,3,3] × [4,3] × [ ]
16	H4×H3×A1	[5,3,3] × [5,3] × [ ]
17	F4×A3×A1	[3,4,3] × [3,3] × [ ]
18	F4×B3×A1	[3,4,3] × [4,3] × [ ]
19	F4×H3×A1	[3,4,3] × [5,3] × [ ]
20	D4×A3×A1	[31,1,1] × [3,3] × [ ]
21	D4×B3×A1	[31,1,1] × [4,3] × [ ]
22	D4×H3×A1	[31,1,1] × [5,3] × [ ]

Three are based on the uniform polyhedra and uniform duoprism:

#	Coxeter group		Coxeter-Dynkin diagram
23	A3×I2(p)×I2(q)×A1	[3,3] × [p] × [q] × [ ]
24	B3×I2(p)×I2(q)×A1	[4,3] × [p] × [q] × [ ]
25	H3×I2(p)×I2(q)×A1	[5,3] × [p] × [q] × [ ]

There are 28 categorical uniform duoprismatic forms based on Cartesian products of lower dimensional uniform polytopes.

There are 4 based on the uniform 6-polytopes and regular polygons:

#	Coxeter group		Coxeter-Dynkin diagram
1	A6×I2(p)	[3,3,3,3,3] × [p]
2	B6×I2(p)	[4,3,3,3,3] × [p]
3	D6×I2(p)	[33,1,1] × [p]
4	E6×I2(p)	[32,2,1] × [p]

There are 9 based on the uniform 5-polytopes and uniform polyhedra:

#	Coxeter group		Coxeter-Dynkin diagram
5	A5×A2	[3,3,3,3] × [3,3]
6	A5×B2	[3,3,3,3] × [4,3]
7	A5×H2	[3,3,3,3] × [5,3]
8	B5×A2	[4,3,3,3] × [3,3]
9	B5×B2	[4,3,3,3] × [4,3]
10	B5×H2	[4,3,3,3] × [5,3]
11	D5×A2	[32,1,1] × [3,3]
12	D5×B2	[32,1,1] × [4,3]
13	D5×H2	[32,1,1] × [5,3]

Finally there are 20 based on two uniform 4-polytopes:

#	Coxeter group		Coxeter-Dynkin diagram
14	A4xA4	[3,3,3] × [3,3,3]
15	A4xB4	[3,3,3] × [4,3,3]
16	A4xD4	[3,3,3] × [31,1,1]
17	A4xF4	[3,3,3] × [3,4,3]
18	A4xH4	[3,3,3] × [5,3,3]
19	B4xB4	[4,3,3] × [4,3,3]
20	B4xD4	[4,3,3] × [31,1,1]
21	B4xF4	[4,3,3] × [3,4,3]
22	B4xH4	[4,3,3] × [5,3,3]
23	D4xD4	[31,1,1] × [31,1,1]
24	D4xF4	[31,1,1] × [3,4,3]
25	D4xH4	[31,1,1] × [5,3,3]
26	F4xF4	[3,4,3] × [3,4,3]
27	F4xH4	[3,4,3] × [5,3,3]
28	H4xH4	[5,3,3] × [5,3,3]

There are 11 categorical uniform triaprismatic forms based on Cartesian products of lower dimensional uniform polytopes, for example these regular products:

Six are based on products of the uniform 4-polytopes and uniform duoprisms:

#	Coxeter group		Coxeter-Dynkin diagram
1	A4×I2(p)×I2(q)	[3,3,3] × [p] × [q]
2	B4×I2(p)×I2(q)	[4,3,3] × [p] × [q]
3	F4×I2(p)×I2(q)	[3,4,3] × [p] × [q]
4	H4×I2(p)×I2(q)	[5,3,3] × [p] × [q]
5	D4×I2(p)×I2(q)	[31,1,1] × [p] × [q]
6	A3×A3×I2(p)	[3,3] × [3,3] × [p]

Five are based on triprism products of two uniform polyhedra and regular polygons:

#	Coxeter group		Coxeter-Dynkin diagram
7	A3×B3×I2(p)	[3,3] × [4,3] × [p]
8	A3×H3×I2(p)	[3,3] × [5,3] × [p]
9	B3×B3×I2(p)	[4,3] × [4,3] × [p]
10	B3×H3×I2(p)	[4,3] × [5,3] × [p]
11	H3×A3×I2(p)	[5,3] × [5,3] × [p]

There is one infinite family of uniform quadriprismatic figures based on Cartesian products of four regular polygons:

#	Coxeter group		Coxeter-Dynkin diagram
1	I2(p) x I2(q) x I2(r) x I2(s)	[p] x [q] x [r] x [s]

Rank 9

(9-space)

• A₉: [3,3,3,3,3,3,3,3] (9-simplex)
• B₉: [4,3,3,3,3,3,3,3] (9-orthoplex/9-hypercube)
• D₉: [3^6,1,1] (9-orthoplex)

Rank 10

(10-space)

• A₁₀: [3,3,3,3,3,3,3,3,3]
• B₁₀: [4,3,3,3,3,3,3,3,3]
• D₁₀: [3^7,1,1]

Euclidean compact

Rank 2

(1-space)

• I^~₁: [∞] (apeirogon)

Rank 3

(2-space)

• A^~₂: (3 3 3) (triangular tiling)
• B^~₂: [4,4] (square tiling)
• H^~₂: [6,3] (triangular tiling/hexagonal tiling)

Rank 4

(3-space)

• A^~₃: (3 3 3 3) (quarter cubic honeycomb)
• B^~₂: [4,3,4] (cubic honeycomb)
• D^~₃: [4,3^1,1] (Alternated cubic honeycomb)

(2-space prismatic)

• I^~₁xI^~₁: [∞]x[∞] = (square tiling)

Rank 5

(4-space)

• A^~₄: (3 3 3 3 3)
• B^~₄: [4,3,3,4] (Tesseractic honeycomb)
• C^~₄: [4,3,3^1,1] (Demitesseractic honeycomb)
• D^~₄: [3^1,1,1,1] (Demitesseractic honeycomb)
• F^~₄: [3,4,3,3] (Icositetrachoric honeycomb/Demitesseractic honeycomb)

(3-space prismatic)

• A₃xI^~₁: [3,3]x[∞] -
• B₃xI^~₁: [4,3]x[∞] -
• H₃xI^~₁: [5,3]x[∞] -

Rank 6

(5-space)

• A^~₅:
• B^~₅: [4,3,3,3,4]
• C^~₅: [4,3²,3^1,1]
• D^~₅: [3^1,1,3,3^1,1]

(4-space prismatic)

• I^~₁xI^~₁xI₂^r: [∞] x [∞] x [r] = [4,4]x[r] - =
• B^~₃xI^~₁: [4,3,4]x[∞]
• D^~₃xI^~₁: [4,3^1,1]x[∞]
• A^~₃xI^~₁:
• A^~₂xA^~₂: [Δ]x[Δ]
• A^~₂xB^~₂: [Δ]x[4,4]
• A^~₂xH^~₂: [Δ]x[6,3]
• B^~₂xB^~₂: [4,4]x[4,4]
• B^~₂xH^~₂: [4,4]x[6,3]
• H^~₂xH^~₂: [6,3]x[6,3]
• A₄xI^~₁: [3,3,3]x[∞] (5-cell column)
• B₄xI^~₁: [4,3,3]x[∞] (tesseract/16-cell column)
• H₄xI^~₁: [5,3,3]x[∞] (120-cell/600-cell column)
• F₄xI^~₁: [3,4,3]x[∞] (24-cell column)
• D₄xI^~₁: [3^1,1,1]x[∞] (24-cell column)

(3-space prismatic)

• I^~₁xI^~₁xI^~₁: [∞] x [∞] x [∞] -

Rank 7

(6-space)

• A^~₆: (3 3 3 3 3 3)
• B^~₆: [4,3,3,3,3,4]
• C^~₆: [4,3²,3^1,1]
• D^~₆: [3^1,1,3²,3^1,1]
• E^~₆: [3^2,2,2]

(6-space prismatic)

• A₅xI^~₁: [3,3,3] x [∞]
• B₅xI^~₁: [4,3,3] x [∞]
• D₅xI^~₁: [3^2,1,1] x [∞]
• A₄xI^~₁xA₁: [3,3,3] x [∞] x [ ]
• B₄xI^~₁xA₁: [4,3,3] x [∞] x [ ]
• F₄xI^~₁xA₁: [3,4,3] x [∞] x [ ]
• H₄xI^~₁xA₁: [5,3,3] x [∞] x [ ]
• D₄xI^~₁xA₁: [3^1,1,1] x [∞] x [ ]
• A₃xI^~₁xI₂^q: [3,3] x [∞] x [q]
• B₃xI^~₁xI₂^q: [4,3] x [∞] x [q]
• H₃xI^~₁xI₂^q: [5,3] x [∞] x [q]
• I^~₁xI₂^qxI₂^rA₁: [∞] x [q] x [r] x [ ]

(5-space prismatic)

• A₃xI^~₁xI^~₁: [3,3] x [∞] x [∞]
• B₃xI^~₁xI^~₁: [4,3] x [∞] x [∞]
• H₃xI^~₁xI^~₁: [5,3] x [∞] x [∞]
• I^~₁xI^~₁xI₂^rxA₁: [∞] x [∞] x [r] x [ ]

(4-space prismatic)

• I^~₁xI^~₁xI^~₁A₁: [∞] x [∞] x [∞] x [ ]

Rank 8

(7-space)

• A^~₇: (3 3 3 3 3 3 3)
• B^~₇: [4,3,3,3,3,3,4]
• C^~₇: [4,3³,3^1,1]
• D^~₇: [3^1,1,3³,3^1,1]
• E^~₇: [3^3,3,1]

Rank 9

(8-space)

• A^~₉: (3 3 3 3 3 3 3 3)
• B^~₈: [4,3,3,3,3,3,3,4]
• C^~₈: [4,3⁴,3^1,1]
• D^~₈: [3^1,1,3⁴,3^1,1]
• E^~₈: [3^5,2,1] (E8 lattice)

Rank 10

(9-space)

• A^~₉: (3 3 3 3 3 3 3 3 3)
• B^~₉: [4,3,3,3,3,3,3,3,4]
• C^~₉: [4,3⁵,3^1,1]
• D^~₉: [3^1,1,3⁵,3^1,1]

Euclidean noncompact

Rank 3

(2-space)

• I^~₁xA₁: [∞]x[ ] (apeirogonal prism)

Rank 4

(3-space)

• B^~₂xA₁: [4,4]x[ ] (cubic prismatic slab)
• H^~₂xA₁: [6,3]x[ ] (triangular/hexagonal prismatic slab)
• A^~₂xA₁: (3 3 3 3)x[ ] (triangular prismatic slab)
• I^~₁xA₁xA₁: [∞]x[ ]x[ ] = (4-∞ semi-infinite duoprism)
• I₂(p)xI^~₁: [p]x[∞] (p-∞ semiinfinite duoprism)

Rank 5

(4-space)

• I₂^pxI^~₁xA₁: [p]x[∞]x[ ] -
• D^~₃xA₁: [4,3^1,1]x[ ]
• A^~₃xA₁: (3 3 3 3)x[ ]
• A^~₂xI₂^p: (3 3 3)x[p]
• B^~₂xI₂^p: [4,4]x[p]
• H^~₂xI₂^p: [6,3]x[p]

(3-space)

• I^~₁xI^~₁xA₁: [∞]x[∞]x[ ] -
• B^~₂xI₂^p: [4,4]x[∞]
• H^~₂xI₂^p: [6,3]x[∞]
• A^~₂xI₂^pxA₁: [Δ]x[∞]x[ ]
• B^~₂xA₁: [4,3,4]x[ ]

Rank 6

(5-space)

• A₄xI^~₁: [3,3,3]x[∞] -
• B₄xI^~₁: [4,3,3]x[∞] -
• F₄xI^~₁: [3,4,3]x[∞] -
• H₄xI^~₁: [5,3,3]x[∞] -
• D₄xI^~₁: [3^1,1,1] x [∞] -
• A₃xI^~₁xA₁: [3,3] x [∞] x [ ] -
• B₃xI^~₁xA₁: [4,3] x [∞] x [ ] -
• H₃xI^~₁xA₁: [5,3] x [∞] x [ ] -
• I^~₁xI₂^qxI₂^r: [∞] x [q] x [r] -

• A^~₄xA₁: (3 3 3 3)x[ ]
• B^~₄xA₁: [4,3,3,4]x[ ]
• C^~₄xA₁: [4,3,3^1,1]x[ ]
• D^~₄xA₁: [3^1,1,1,1]x[ ]
• F^~₄xA₁: [3,4,3,3]x[ ]

Hyperbolic compact

Rank 3

(2-space)

• [p,q] , p,q>=3, p+q>9
• [p,q,r:] , p,q,r>=3, p+q+r>9

Rank 4

(3-space)

• [3,5,3] (Order-3 icosahedral honeycomb)
• [5,3,4] (Order-5 cubic honeycomb/Order-4 dodecahedral honeycomb)
• [5,3,5] (Order-5 dodecahedral honeycomb)
• [5,3^1,1]
• (4 3 3 3)
• (5 3 3 3)
• (4 3 3 3)
• (4 3 5 3)
• (5 3 5 3)

(2-space)

• (p q r s) , p,q,r,s>=2, p+q+r+s>8

Rank 5

(4-space)

• [5,3,3,3] (Order-5 pentachoronic tetracomb/Order-3 hecatonicosachoronic tetracomb)
• [5,3,3,4] (Order-5 tesseractic tetracomb/Order-4 hecatonicosachoronic tetracomb)
• [5,3,3,5] (Order-5 hecatonicosachoronic tetracomb)
• [5,3,3^1,1] (Order-5 demitesseractic tetracomb)
• (4 3 3 3 3)

Rank 6

Noncompact!

(5-space)

• [3,4,3,3,3]
• [3,3,4,3,3]
• [4,3,4,3,3]

Hyperbolic noncompact

Rank 3

(2-space)

• [p,∞] , p>=3
• (p q ∞) , p,q>=3, p+q>6
• (p ∞ ∞) , p>=3
• (∞ ∞ ∞)

Rank 4

(3-space)

• [6,3,3]
• [4,4,3]
• [3,6,3]
• [6,3,4]
• [4,4,4]
• [6,3,5]
• [6,3,6]
• ...

Rank 5

(4-space)

• [4,3,4,3]

Finite Coexter groups

Group symbol	Alternate symbol	Rank	Order	Related polytopes	Coxeter-Dynkin diagram
An	An	n	(n + 1)!	n-simplex	...
Bn = Cn	Cn	n	2n n!	n-hypercube / n-cross-polytope	...
Dn	Bn	n	2n−1 n!	demihypercube	...
I2(p)	D2p	2	2p	p-gon
H3	G3	3	120	icosahedron / dodecahedron
F4	F4	4	1152	24-cell
H4	G4	4	14400	120-cell / 600-cell
E6	E6	6	51840	122 polytope
E7	E7	7	2903040	321 polytope
E8	E8	8	696729600	E8 polytope

Finite Coxeter groups

Families of convex uniform polytopes are defined by Coxeter groups.

n	A1+ [3n-1]	D4+ [3n-3,1,1]	C2+ [4,3n-2]	I2p [p]	E6-8 [4,3n-3,2,1]	F4 [3,4,3]	H2-4 [5,3n-1]
1	A1=[]
2	A2=[3]		C2=[4]	I2p=[p]			H2=[5]
3	A3=[32]	D3=A3=[30,1,1]	C3=[4,3]				H3=[5,3]
4	A4=[33]	D4=[31,1,1]	C4=[4,32]		E4=A4=[30,2,1]	F4=[3,4,3]	H4=[5,3,3]
5	A5=[34]	D5=[32,1,1]	C5=[4,33]		E5=B5=[31,2,1]
6	A6=[35]	D6=[33,1,1]	C6=[4,34]		E6=[32,2,1]
7	A7=[36]	D7=[34,1,1]	C7=[4,35]		E7=[33,2,1]
8	A8=[37]	D8=[35,1,1]	C8=[4,36]		E8=[34,2,1]
9	A9=[38]	D9=[36,1,1]	C9=[4,37]
10+	..	..	..

Note: (Alternate names as Simple Lie groups also given)

1. A_n forms the simplex polytope family. (Same A_n)
2. B_n is the family of demihypercubes, beginning at n=4 with the 16-cell, and n=5 with the demipenteract. (Also named D_n)
3. C_n forms the hypercube polytope family. (Same C_n)
4. D₂ⁿ forms the regular polygons. (Also named I_1ⁿ)
5. E₆,E₇,E₈ are the generators of the Gosset Semiregular polytopes (Same E₆,E₇,E₈)
6. F₄ is the 24-cell polychoron family. (Same F₄)
7. G₃ is the dodecahedron/icosahedron polyhedron family. (Also named H₃₎
8. G₄ is the 120-cell/600-cell polychoron family. (Also named H₄₎

Affine Coxeter groups (Euclidean)

Group symbol	Alternate symbol	Related uniform tessellation(s)	Coxeter-Dynkin diagram
A~n-1	Pn	Simplex-rectified-simplex honeycomb A~2:Triangular tiling A~3:Tetrahedral-octahedral honeycomb	...
B~n-1	Rn	Hypercubic honeycomb	...
C~n-1	Sn	Demihypercubic honeycomb	...
D~n-1	Qn	Demihypercubic honeycomb	...
I~1	W2	apeirogon
H~2	G3	Hexagonal tiling and Triangular tiling
F~4	V5	Hexadecachoric tetracomb and Icositetrachoronic tetracomb or F4 lattice
E~6	T7	E6 lattice
E~7	T8	E7 lattice
E~8	T9	E8 lattice

n	A~2+ (3n)	D~4+ [31,1,3n-5,31,1]	B~2+ [4,3n-3,4]	C~3+ [4,3n-4,31,1]	E~6-8 [3a,b,c]	F~4 [3,4,3,3]	H~2 [6,3]	I~1
2								I~1=[∞]
3	A~2=h[6,3]		B~2=[4,4]				F~2=[6,3]
4	A~3=q[4,3,4]		B~3=[4,3,4]	C~3=h[4,3,4]
5	A~4	D~4=q[4,32,4]	B~4=[4,32,4]	C~4=h[4,32,4]		U5=[3,4,3,3]
6	A~5	D~5=q[4,33,4]	B~5=[4,33,4]	C~5=h[4,33,4]
7	A~6	D~6=q[4,34,4]	B~6=[4,34,4]	C~6=h[4,34,4]	E~6=[32,2,2]
8	A~7	D~7=q[4,35,4]	B~7=[4,35,4]	C~7=h[4,35,4]	E~7=[33,3,1]
9	A~8	D~8=q[4,36,4]	B~8=[4,36,4]	C~8=h[4,36,4]	E~8=[35,2,1]
10	A~9	D~9=q[4,37,4]	B~9=[4,37,4]	C~9=h[4,37,4]
11	...	...	...	...

See also

• List of regular polytopes

References

• THE GROWTH SERIES OF COMPACT HYPERBOLIC COXETER GROUPS WITH 4 AND 5 GENERATORS
• Spacelike Singularities and Hidden Symmetries of Gravity: 3 Hyperbolic Coxeter Groups
• Reflection Groups and Coxeter groups (Cambridge studies in advanced mathematics 29) , James E. Humphreys, Cambridge University press, 1990 ISBN 0521436133 (paperback), ISBN 052137510X (hardcopy)
  • 2.4 Some positive definite graphs, 2.5 Some positive semidefinite graphs, 6.8 Coxeter Hyperbolic groups