Full list of elements in the A3 Coxeter group with generators G, R, B

Listed by word length.

0
 I = [[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]]
num 1
1
 G = [[0.0, 1.0, 0.0], [1.0, 0.0, 0.0], [0.0, 0.0, 1.0]]
 R = [[0.0, -1.0, 0.0], [-1.0, 0.0, 0.0], [0.0, 0.0, 1.0]]
 B = [[1.0, 0.0, 0.0], [0.0, 0.0, 1.0], [0.0, 1.0, 0.0]]
num 3
2
 G G == I
 G R = [[-1.0, 0.0, 0.0], [0.0, -1.0, 0.0], [0.0, 0.0, 1.0]]
 G B = [[0.0, 0.0, 1.0], [1.0, 0.0, 0.0], [0.0, 1.0, 0.0]]
 R G ==  G R
 R R == I
 R B = [[0.0, 0.0, -1.0], [-1.0, 0.0, 0.0], [0.0, 1.0, 0.0]]
 B G = [[0.0, 1.0, 0.0], [0.0, 0.0, 1.0], [1.0, 0.0, 0.0]]
 B R = [[0.0, -1.0, 0.0], [0.0, 0.0, 1.0], [-1.0, 0.0, 0.0]]
 B B == I
num 5
3
 G R G ==  R
 G R R ==  G
 G R B = [[-1.0, 0.0, 0.0], [0.0, 0.0, -1.0], [0.0, 1.0, 0.0]]
 G B G = [[0.0, 0.0, 1.0], [0.0, 1.0, 0.0], [1.0, 0.0, 0.0]]
 G B R = [[0.0, 0.0, 1.0], [0.0, -1.0, 0.0], [-1.0, 0.0, 0.0]]
 G B B ==  G
 R B G = [[0.0, 0.0, -1.0], [0.0, -1.0, 0.0], [1.0, 0.0, 0.0]]
 R B R = [[0.0, 0.0, -1.0], [0.0, 1.0, 0.0], [-1.0, 0.0, 0.0]]
 R B B ==  R
 B G G ==  B
 B G R = [[-1.0, 0.0, 0.0], [0.0, 0.0, 1.0], [0.0, -1.0, 0.0]]
 B G B ==  G B G
 B R G ==  B G R
 B R R ==  B
 B R B ==  R B R
num 6
4
 G R B G = [[0.0, -1.0, 0.0], [0.0, 0.0, -1.0], [1.0, 0.0, 0.0]]
 G R B R = [[0.0, 1.0, 0.0], [0.0, 0.0, -1.0], [-1.0, 0.0, 0.0]]
 G R B B ==  G R
 G B G G ==  G B
 G B G R = [[0.0, 0.0, 1.0], [-1.0, 0.0, 0.0], [0.0, -1.0, 0.0]]
 G B G B ==  B G
 G B R G ==  G B G R
 G B R R ==  G B
 G B R B ==  G R B R
 R B G G ==  R B
 R B G R = [[0.0, 0.0, -1.0], [1.0, 0.0, 0.0], [0.0, -1.0, 0.0]]
 R B G B ==  G R B G
 R B R G ==  R B G R
 R B R R ==  R B
 R B R B ==  B R
 B G R G ==  B R
 B G R R ==  B G
 B G R B = [[-1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, -1.0]]
num 5
5
 G R B G G ==  G R B
 G R B G R = [[1.0, 0.0, 0.0], [0.0, 0.0, -1.0], [0.0, -1.0, 0.0]]
 G R B G B ==  R B G
 G R B R G ==  G R B G R
 G R B R R ==  G R B
 G R B R B ==  G B R
 G B G R G ==  G B R
 G B G R R ==  G B G
 G B G R B = [[0.0, 1.0, 0.0], [-1.0, 0.0, 0.0], [0.0, 0.0, -1.0]]
 R B G R G ==  R B R
 R B G R R ==  R B G
 R B G R B = [[0.0, -1.0, 0.0], [1.0, 0.0, 0.0], [0.0, 0.0, -1.0]]
 B G R B G ==  R B G R B
 B G R B R ==  G B G R B
 B G R B B ==  B G R
num 3
6
 G R B G R G ==  G R B R
 G R B G R R ==  G R B G
 G R B G R B = [[1.0, 0.0, 0.0], [0.0, -1.0, 0.0], [0.0, 0.0, -1.0]]
 G B G R B G ==  G R B G R B
 G B G R B R ==  B G R B
 G B G R B B ==  G B G R
 R B G R B G ==  B G R B
 R B G R B R ==  G R B G R B
 R B G R B B ==  R B G R
num 1
7
 G R B G R B G ==  G B G R B
 G R B G R B R ==  R B G R B
 G R B G R B B ==  G R B G R
Total number of elements 24