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User:Tomruen/List of Coxeter groups

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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)

[edit]

Rank 1

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(1-space)

  • A1: [ ]

Rank 2

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(2-space)

(2-space prismatic)

Rank 3

[edit]

(3-space)

(3-space prismatic)

Rank 4

[edit]

(4-space)

(4-space prismatic)

Rank 5

[edit]

(5-space)

(5-space prismatic)

  • A4xA1: [3,3,3]x[ ]
  • B4xA1: [4,3,3]x[ ] -
  • F4xA1: [3,4,3]x[ ] -
  • H4xA1: [5,3,3]x[ ] -
  • D4xA1: [31,1,1]x[ ] -
  • A3xI2p: [3,3]x[p] -
  • B3xI2p: [4,3]x[p] -
  • H3xI2p: [5,3]x[p] -
  • I2pxI2qxA1: [p]x[q]x[ ] -

Rank 6

[edit]

(6-space)

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

[edit]

(7-space)

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

[edit]

(8-space)

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

[edit]

(9-space)

Rank 10

[edit]

(10-space)

  • A10: [3,3,3,3,3,3,3,3,3]
  • B10: [4,3,3,3,3,3,3,3,3]
  • D10: [37,1,1]

Euclidean compact

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Rank 2

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(1-space)

Rank 3

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(2-space)

Rank 4

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(3-space)

(2-space prismatic)

Rank 5

[edit]

(4-space)


(3-space prismatic)

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

Rank 6

[edit]

(5-space)

  • A~5:
  • B~5: [4,3,3,3,4]
  • C~5: [4,32,31,1]
  • D~5: [31,1,3,31,1]

(4-space prismatic)

  • I~1xI~1xI2r: [∞] x [∞] x [r] = [4,4]x[r] - =
  • B~3xI~1: [4,3,4]x[∞]
  • D~3xI~1: [4,31,1]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: [31,1,1]x[∞] (24-cell column)

(3-space prismatic)

  • I~1xI~1xI~1: [∞] x [∞] x [∞] -

Rank 7

[edit]

(6-space)

  • A~6: (3 3 3 3 3 3)
  • B~6: [4,3,3,3,3,4]
  • C~6: [4,32,31,1]
  • D~6: [31,1,32,31,1]
  • E~6: [32,2,2]

(6-space prismatic)

  • A5xI~1: [3,3,3] x [∞]
  • B5xI~1: [4,3,3] x [∞]
  • D5xI~1: [32,1,1] 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: [31,1,1] x [∞] x [ ]
  • A3xI~1xI2q: [3,3] x [∞] x [q]
  • B3xI~1xI2q: [4,3] x [∞] x [q]
  • H3xI~1xI2q: [5,3] x [∞] x [q]
  • I~1xI2qxI2rA1: [∞] 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~1xI2rxA1: [∞] x [∞] x [r] x [ ]

(4-space prismatic)

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

Rank 8

[edit]

(7-space)

  • A~7: (3 3 3 3 3 3 3)
  • B~7: [4,3,3,3,3,3,4]
  • C~7: [4,33,31,1]
  • D~7: [31,1,33,31,1]
  • E~7: [33,3,1]

Rank 9

[edit]

(8-space)

  • A~9: (3 3 3 3 3 3 3 3)
  • B~8: [4,3,3,3,3,3,3,4]
  • C~8: [4,34,31,1]
  • D~8: [31,1,34,31,1]
  • E~8: [35,2,1] (E8 lattice)

Rank 10

[edit]

(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: [4,35,31,1]
  • D~9: [31,1,35,31,1]

Euclidean noncompact

[edit]

Rank 3

[edit]

(2-space)

Rank 4

[edit]

(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

[edit]

(4-space)

  • I2pxI~1xA1: [p]x[∞]x[ ] -
  • D~3xA1: [4,31,1]x[ ]
  • A~3xA1: (3 3 3 3)x[ ]
  • A~2xI2p: (3 3 3)x[p]
  • B~2xI2p: [4,4]x[p]
  • H~2xI2p: [6,3]x[p]

(3-space)

  • I~1xI~1xA1: [∞]x[∞]x[ ] -
  • B~2xI2p: [4,4]x[∞]
  • H~2xI2p: [6,3]x[∞]
  • A~2xI2pxA1: [Δ]x[∞]x[ ]
  • B~2xA1: [4,3,4]x[ ]

Rank 6

[edit]

(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: [31,1,1] x [∞] -
  • A3xI~1xA1: [3,3] x [∞] x [ ] -
  • B3xI~1xA1: [4,3] x [∞] x [ ] -
  • H3xI~1xA1: [5,3] x [∞] x [ ] -
  • I~1xI2qxI2r: [∞] x [q] x [r] -
  • A~4xA1: (3 3 3 3)x[ ]
  • B~4xA1: [4,3,3,4]x[ ]
  • C~4xA1: [4,3,31,1]x[ ]
  • D~4xA1: [31,1,1,1]x[ ]
  • F~4xA1: [3,4,3,3]x[ ]

Rank 7

[edit]

Rank 8

[edit]

Rank 9

[edit]

Rank 10

[edit]

Hyperbolic compact

[edit]

Rank 3

[edit]

(2-space)

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

Rank 4

[edit]

(3-space)

(2-space)

  • (p q r s) , p,q,r,s>=2, p+q+r+s>8

Rank 5

[edit]

(4-space)

Rank 6

[edit]

Noncompact!

(5-space)

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

Rank 7

[edit]

Rank 8

[edit]

Rank 9

[edit]

Rank 10

[edit]

Hyperbolic noncompact

[edit]

Rank 3

[edit]

(2-space)

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

Rank 4

[edit]

(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

[edit]

(4-space)

  • [4,3,4,3]

Rank 6

[edit]

Rank 7

[edit]

Rank 8

[edit]

Rank 9

[edit]

Rank 10

[edit]

Finite Coexter groups

[edit]
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

[edit]

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. An forms the simplex polytope family. (Same An)
  2. Bn is the family of demihypercubes, beginning at n=4 with the 16-cell, and n=5 with the demipenteract. (Also named Dn)
  3. Cn forms the hypercube polytope family. (Same Cn)
  4. D2n forms the regular polygons. (Also named I1n)
  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)

[edit]
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

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References

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