User:Tomruen/List of finite Coxeter groups

Finite Coxeter groups are listed here up to rank 10 for connected groups, and disconnected groups up to rank 8.

1 rank

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# Coxeter group Coxeter-Dynkin diagram
1
1 A1 [ ]

2 rank

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Rank 2, one branch

Note: Above rank 2, only the general I2(p) group is enumerated in place of the others.

# Coxeter group Coxeter-Dynkin diagram
2
1 A2 [3]
2 BC2 [4]
2 H2 [5]
2 G2 [6]
3 I2(p) [p]
1+1
1 A1×A1 [ ]×[ ]

3 rank

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Rank 3, three branches
# Coxeter group Coxeter-Dynkin diagram
3
1 A3 [3,3]
2 BC3 [4,3]
3 H3 [5,3]
2+1
4 I2(p)×A1 [p]×[ ]
1+1+1
5 A1×A1×A1 [ ]×[ ]×[ ]

4 rank

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Rank 4, up to six branches
# Coxeter group Coxeter-Dynkin diagram
4
1 A4 [3,3,3]
2 BC4 [4,3,3]
3 D4 [31,1,1]
4 F4 [3,4,3]
5 H4 [5,3,3]
3+1
6 A3×A1 [3,3]×[ ]
7 BC3×A1 [4,3]×[ ]
8 H3×A1 [5,3]×[ ]
2+2
9 I2(p)×I2(q) [p]×[q]
2+1+1
10 I2(p)×A1×A1 [p]×[ ]×[ ]
1+1+1+1
11 A1×A1×A1×A1 [ ]×[ ]×[ ]×[ ]

5 rank

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Rank 5, up to 10 branches
# Coxeter group Coxeter-Dynkin diagram
5
1 A5 [34]
2 BC5 [4,33]
3 D5 [32,1,1]
4+1
1 A4×A1 [3,3,3]×[ ]
2 BC4×A1 [4,3,3]×[ ]
3 F4×A1 [3,4,3]×[ ]
4 H4×A1 [5,3,3]×[ ]
5 D4×A1 [31,1,1]×[ ]
3+2
1 A3×I2(p) [3,3]×[p]
2 BC3×I2(p) [4,3]×[p]
3. H3×I2(p) [5,3]×[p]
3+1+1
1 A3×A1×A1 [3,3]×[ ]×[ ]
2 BC3×A1×A1 [4,3]×[ ]×[ ]
3. H3×A1×A1 [5,3]×[ ]×[ ]
2+2+1
1 I2(p)×I2(q)×A1 [p]×[q]×[ ]
2+1+1+1
1 I2(p)×A1×A1×A1 [p]×[ ]×[ ]
1+1+1+1+1
1 A1×A1×A1×A1×A1 [ ]×[ ]×[ ]×[ ]×[ ]

6-rank

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Rank 6, up to 15 branches
# Coxeter group Coxeter-Dynkin diagram
6
1 A6 [35]
2 BC6 [4,34]
3 D6 [33,1,1]
4 E6 [32,2,1]
5+1
1 A5×A1 [3,3,3,3]×[ ]
2 BC5×A1 [4,3,3,3]×[ ]
3 D5×A1 [32,1,1]×[ ]
4+2
1 A4×I2(p) [3,3,3]×[p]
2 BC4×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]
4+1+1
1 A4×A1×A1 [3,3,3]×[ ]×[ ]
2 BC4×A1×A1 [4,3,3]×[ ]×[ ]
3 F4×A1×A1 [3,4,3]×[ ]×[ ]
4 H4×A1×A1 [5,3,3]×[ ]×[ ]
5 D4×A1×A1 [31,1,1]×[ ]×[ ]
3+3
6 A3×A3 [3,3]×[3,3]
7 A3×BC3 [3,3]×[4,3]
8 A3×H3 [3,3]×[5,3]
9 BC3×BC3 [4,3]×[4,3]
10 BC3×H3 [4,3]×[5,3]
11 H3×A3 [5,3]×[5,3]
3+2+1
4 A3×I2(p)×A1 [3,3]×[p]×[ ]
5 BC3×I2(p)×A1 [4,3]×[p]×[ ]
6 H3×I2(p)×A1 [5,3]×[p]×[ ]
3+1+1+1
4 A3×A1×A1×A1 [3,3]×[ ]×[ ]×[ ]
5 BC3×A1×A1×A1 [4,3]×[ ]×[ ]×[ ]
6 H3×A1×A1×A1 [5,3]×[ ]×[ ]×[ ]
2+2+2
1 I2(p)×I2(q)×I2(r) [p]×[q]×[r]
2+2+1+1
1 I2(p)×I2(q)×A1×A1 [p]×[q]×[ ]
2+1+1+1+1
1 I2(p)×A1×A1×A1×A1 [p]×[ ]×[ ]×[ ]×[ ]
1+1+1+1+1+1
1 A1×A1×A1×A1×A1×A1 [ ]×[ ]×[ ]×[ ]×[ ]×[ ]

7 rank

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Rank 7, up to 21 branches
# Coxeter group Coxeter-Dynkin diagram
7
1 A7 [36]
2 BC7 [4,35]
3 D7 [34,1,1]
4 E7 [33,2,1]
6+1
1 A6×A1 [35]×[ ]
2 BC6×A1 [4,34]×[ ]
3 D6×A1 [33,1,1]×[ ]
4 E6×A1 [32,2,1]×[ ]
5+2
1 A5×I2(p) [3,3,3]×[p]
2 BC5×I2(p) [4,3,3]×[p]
3 D5×I2(p) [32,1,1]×[p]
5+1+1
1 A5×A1×A1 [3,3,3]×[ ]×[ ]
2 BC5×A1×A1 [4,3,3]×[ ]×[ ]
3 D5×A1×A1 [32,1,1]×[ ]×[ ]
4+3
4 A4×A3 [3,3,3]×[3,3]
5 A4×BC3 [3,3,3]×[4,3]
6 A4×H3 [3,3,3]×[5,3]
7 BC4×A3 [4,3,3]×[3,3]
8 BC4×BC3 [4,3,3]×[4,3]
9 BC4×H3 [4,3,3]×[5,3]
10 H4×A3 [5,3,3]×[3,3]
11 H4×BC3 [5,3,3]×[4,3]
12 H4×H3 [5,3,3]×[5,3]
13 F4×A3 [3,4,3]×[3,3]
14 F4×BC3 [3,4,3]×[4,3]
15 F4×H3 [3,4,3]×[5,3]
16 D4×A3 [31,1,1]×[3,3]
17 D4×BC3 [31,1,1]×[4,3]
18 D4×H3 [31,1,1]×[5,3]
4+2+1
5 A4×I2(p)×A1 [3,3,3]×[p]×[ ]
6 BC4×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]×[ ]
4+1+1+1
5 A4×A1×A1×A1 [3,3,3]×[ ]×[ ]×[ ]
6 BC4×A1×A1×A1 [4,3,3]×[ ]×[ ]×[ ]
7 F4×A1×A1×A1 [3,4,3]×[ ]×[ ]×[ ]
8 H4×A1×A1×A1 [5,3,3]×[ ]×[ ]×[ ]
9 D4×A1×A1×A1 [31,1,1]×[ ]×[ ]×[ ]
3+3+1
10 A3×A3×A1 [3,3]×[3,3]×[ ]
11 A3×BC3×A1 [3,3]×[4,3]×[ ]
12 A3×H3×A1 [3,3]×[5,3]×[ ]
13 BC3×BC3×A1 [4,3]×[4,3]×[ ]
14 BC3×H3×A1 [4,3]×[5,3]×[ ]
15 H3×A3×A1 [5,3]×[5,3]×[ ]
3+2+2
1 A3×I2(p)×I2(q) [3,3]×[p]×[q]
2 BC3×I2(p)×I2(q) [4,3]×[p]×[q]
3 H3×I2(p)×I2(q) [5,3]×[p]×[q]
3+2+1+1
1 A3×I2(p)×A1×A1 [3,3]×[p]×[ ]×[ ]
2 BC3×I2(p)×A1×A1 [4,3]×[p]×[ ]×[ ]
3 H3×I2(p)×A1×A1 [5,3]×[p]×[ ]×[ ]
3+1+1+1+1
1 A3×A1×A1×A1×A1 [3,3]×[ ]×[ ]×[ ]×[ ]
2 BC3×A1×A1×A1×A1 [4,3]×[ ]×[ ]×[ ]×[ ]
3 H3×A1×A1×A1×A1 [5,3]×[ ]×[ ]×[ ]×[ ]
2+2+2+1
1 I2(p)×I2(q)×I2(r)×A1 [p]×[q]×[r]×[ ]
2+2+1+1+1
1 I2(p)×I2(q)×A1×A1×A1 [p]×[q]×[ ]×[ ]×[ ]
2+1+1+1+1+1
1 I2(p)×A1×A1×A1×A1×A1 [p]×[ ]×[ ]×[ ]×[ ]×[ ]
1+1+1+1+1+1+1
1 A1×A1×A1×A1×A1×A1×A1 [ ]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ]

8 rank

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Rank 8, up to 28 branches
# Coxeter group Coxeter-Dynkin diagram
8
1 A8 [37]
2 BC8 [4,36]
3 D8 [35,1,1]
4 E8 [34,2,1]
7+1
1 A7×A1 [3,3,3,3,3,3]×[ ]
2 BC7×A1 [4,3,3,3,3,3]×[ ]
3 D7×A1 [34,1,1]×[ ]
4 E7×A1 [33,2,1]×[ ]
6+2
1 A6×I2(p) [3,3,3,3,3]×[p]
2 BC6×I2(p) [4,3,3,3,3]×[p]
3 D6×I2(p) [33,1,1]×[p]
4 E6×I2(p) [3,3,3,3,3]×[p]
6+1+1
1 A6×A1×A1 [3,3,3,3,3]×[ ]x[ ]
2 BC6×A1×A1 [4,3,3,3,3]×[ ]x[ ]
3 D6×A1×A1 [33,1,1]×[ ]x[ ]
4 E6×A1×A1 [3,3,3,3,3]×[ ]x[ ]
5+3
1 A5×A3 [34]×[3,3]
2 BC5×A3 [4,33]×[3,3]
3 D5×A3 [32,1,1]×[3,3]
1 A5×BC3 [34]×[4,3]
2 BC5×BC3 [4,33]×[4,3]
3 D5×BC3 [32,1,1]×[4,3]
1 A5×H3 [34]×[5,3]
2 BC5×H3 [4,33]×[5,3]
3 D5×H3 [32,1,1]×[5,3]
5+2+1
5 A5×I2(p)×A1 [3,3,3]×[p]×[ ]
6 BC5×I2(p)×A1 [4,3,3]×[p]×[ ]
7 D5×I2(p)×A1 [32,1,1]×[p]×[ ]
5+1+1+1
5 A5×A1×A1×A1 [3,3,3]×[ ]×[ ]×[ ]
6 BC5×A1×A1×A1 [4,3,3]×[ ]×[ ]×[ ]
7 D5×A1×A1×A1 [32,1,1]×[ ]×[ ]×[ ]
4+4
1 A4×A4 [3,3,3]×[3,3,3]
2 BC4×A4 [4,3,3]×[3,3,3]
3 D4×A4 [31,1,1]×[3,3,3]
4 F4×A4 [3,4,3]×[3,3,3]
5 H4×A4 [5,3,3]×[3,3,3]
2 BC4×BC4 [4,3,3]×[4,3,3]
3 D4×BC4 [31,1,1]×[4,3,3]
4 F4×BC4 [3,4,3]×[4,3,3]
5 H4×BC4 [5,3,3]×[4,3,3]
3 D4×D4 [31,1,1]×[31,1,1]
4 F4×D4 [3,4,3]×[31,1,1]
5 H4×D4 [5,3,3]×[31,1,1]
4 F4×F4 [3,4,3]×[3,4,3]
5 H4×F4 [5,3,3]×[3,4,3]
5 H4×H4 [5,3,3]×[5,3,3]
4+3+1
8 A4×A3×A1 [3,3,3]×[3,3]×[ ]
9 A4×BC3×A1 [3,3,3]×[4,3]×[ ]
10 A4×H3×A1 [3,3,3]×[5,3]×[ ]
11 BC4×A3×A1 [4,3,3]×[3,3]×[ ]
12 BC4×BC3×A1 [4,3,3]×[4,3]×[ ]
13 BC4×H3×A1 [4,3,3]×[5,3]×[ ]
14 H4×A3×A1 [5,3,3]×[3,3]×[ ]
15 H4×BC3×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×BC3×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×BC3×A1 [31,1,1]×[4,3]×[ ]
22 D4×H3×A1 [31,1,1]×[5,3]×[ ]
4+2+2
1 A4×I2(p)×I2(q) [3,3,3]×[p]×[q]
2 BC4×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]
4+2+1+1
1 A4×I2(p)×A1 [3,3,3]×[p]×[ ]
2 BC4×I2(p)×A1 [4,3,3]×[p]×[ ]
3 F4×I2(p)×A1 [3,4,3]×[p]×[ ]
4 H4×I2(p)×A1 [5,3,3]×[p]×[ ]
5 D4×I2(p)×A1 [31,1,1]×[p]×[ ]
4+1+1+1+1
1 A4×A1×A1×A1×A1 [3,3,3]×[ ]×[ ]×[ ]×[ ]
2 BC4×A1×A1×A1×A1 [4,3,3]×[ ]×[ ]×[ ]×[ ]
3 F4×A1×A1×A1×A1 [3,4,3]×[ ]×[ ]×[ ]×[ ]
4 H4×A1×A1×A1×A1 [5,3,3]×[ ]×[ ]×[ ]×[ ]
5 D4×A1×A1×A1×A1 [31,1,1]×[ ]×[ ]×[ ]×[ ]
3+3+2
1 A3×A3×I2(p) [3,3]×[3,3]×[p]
2 BC3×A3×I2(p) [4,3]×[3,3]×[p]
3 H3×A3×I2(p) [5,3]×[3,3]×[p]
2 BC3×BC3×I2(p) [4,3]×[4,3]×[p]
3 H3×BC3×I2(p) [5,3]×[4,3]×[p]
3 H3×H3×I2(p) [5,3]×[5,3]×[p]
3+3+1+1
1 A3×A3×A1×A1 [3,3]×[3,3]×[ ]×[ ]
2 BC3×A3×A1×A1 [4,3]×[3,3]×[ ]×[ ]
3 H3×A3×A1×A1 [5,3]×[3,3]×[ ]×[ ]
2 BC3×BC3×A1×A1 [4,3]×[4,3]×[ ]×[ ]
3 H3×BC3×A1×A1 [5,3]×[4,3]×[ ]×[ ]
3 H3×H3×A1×A1 [5,3]×[5,3]×[ ]×[ ]
3+2+2+1
23 A3×I2(p)×I2(q)×A1 [3,3]×[p]×[q]×[ ]
24 BC3×I2(p)×I2(q)×A1 [4,3]×[p]×[q]×[ ]
25 H3×I2(p)×I2(q)×A1 [5,3]×[p]×[q]×[ ]
3+2+1+1+1
23 A3×I2(p)×A1×A1×A1 [3,3]×[p]×[ ]x[ ]×[ ]
24 BC3×I2(p)×A1×A1×A1 [4,3]×[p]×[ ]x[ ]×[ ]
25 H3×I2(p)×A1×A1×A1 [5,3]×[p]×[ ]x[ ]×[ ]
3+1+1+1+1+1
23 A3×A1×A1×A1×A1×A1 [3,3]×[ ]x[ ]×[ ]x[ ]×[ ]
24 BC3×A1×A1×A1×A1×A1 [4,3]×[ ]x[ ]×[ ]x[ ]×[ ]
25 H3×A1×A1×A1×A1×A1 [5,3]×[ ]x[ ]×[ ]x[ ]×[ ]
2+2+2+2
1 I2(p)×I2(q)×I2(r)×I2(s) [p]×[q]×[r]×[s]
2+2+2+1+1
1 I2(p)×I2(q)×I2(r)×A1×A1 [p]×[q]×[r]×[ ]×[ ]
2+2+1+1+1+1
1 I2(p)×I2(q)×A1×A1×A1×A1 [p]×[q]×[ ]×[ ]×[ ]×[ ]
2+1+1+1+1+1+1
1 I2(p)×A1×A1×A1×A1×A1×A1 [p]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ]
1+1+1+1+1+1+1+1
1 A1×A1×A1×A1×A1×A1×A1×A1 [ ]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ]

9 rank

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Rank 9, up to 36 branches

There are 3 fundamental groups for rank 9 and higher.

# Coxeter group Coxeter-Dynkin diagram
9
1 A9 [38]
2 BC9 [4,37]
3 D9 [36,1,1]

10 rank

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Rank 10, up to 45 branches

There are 3 fundamental groups for rank 9 or higher.

# Coxeter group Coxeter-Dynkin diagram
10
1 A10 [39]
2 BC10 [4,38]
3 D10 [37,1,1]

Special case prismatic reductions

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The product of n A1 groups can be associated with the higher symmetry BCn.

  • A1×A1 = BC2 : =
  • A1×A1×A1= BC3 : =
  • A1×A1×A1×A1 = BC4 : =
  • A1×A1×A1×A1×A1 = BC5 : =
  • ....