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

#	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 BC_n.

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