User:Tomruen/List of finite Coxeter groups

From WikiProjectMed
Jump to navigation Jump to search

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

1 rank

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

2 rank

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

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

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

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

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

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

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

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

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

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 : =
  • ....
Retrieved from "https://mdwiki.org/wiki/User:Tomruen/List_of_finite_Coxeter_groups"