An irreducible 2-dimensional finite reflective group is B2 =[4], order 8, . The reflection generators matrices are R0 , R1 . R0 2 =R1 2 =(R0 ×R1 )4 =Identity.
Chiral square symmetry, [4]+ , ( ) is generated by rotation: S0,1 .
Octahedral symmetry [ edit ]
Reflection lines for B3 =[4,3] =
Another irreducible 3-dimensional finite reflective group is octahedral symmetry , [4,3], order 48, . The reflection generators matrices are R0 , R1 , R2 . R0 2 =R1 2 =R2 2 =(R0 ×R1 )4 =(R1 ×R2 )3 =(R0 ×R2 )2 =Identity. Chiral octahedral symmetry, [4,3]+ , ( ) is generated by 2 of 3 rotations: S0,1 , S1,2 , and S0,2 . Pyritohedral symmetry [4,3+ ], ( ) is generated by reflection R0 and rotation S1,2 . A 6-fold rotoreflection is generated by V0,1,2 , the product of all 3 reflections.
[4,3],
Reflections
Rotations
Rotoreflection
Name
R0
R1
R2
S0,1
S1,2
S0,2
V0,1,2
Order
2
2
2
4
3
2
6
Matrix
[
1
0
0
0
1
0
0
0
−
1
]
{\displaystyle \left[{\begin{smallmatrix}1&0&0\\0&1&0\\0&0&-1\\\end{smallmatrix}}\right]}
[
1
0
0
0
0
1
0
1
0
]
{\displaystyle \left[{\begin{smallmatrix}1&0&0\\0&0&1\\0&1&0\\\end{smallmatrix}}\right]}
[
0
1
0
1
0
0
0
0
1
]
{\displaystyle \left[{\begin{smallmatrix}0&1&0\\1&0&0\\0&0&1\\\end{smallmatrix}}\right]}
[
1
0
0
0
0
1
0
−
1
0
]
{\displaystyle \left[{\begin{smallmatrix}1&0&0\\0&0&1\\0&-1&0\\\end{smallmatrix}}\right]}
[
0
1
0
0
0
1
1
0
0
]
{\displaystyle \left[{\begin{smallmatrix}0&1&0\\0&0&1\\1&0&0\\\end{smallmatrix}}\right]}
[
0
1
0
1
0
0
0
0
−
1
]
{\displaystyle \left[{\begin{smallmatrix}0&1&0\\1&0&0\\0&0&-1\\\end{smallmatrix}}\right]}
[
0
1
0
0
0
1
−
1
0
0
]
{\displaystyle \left[{\begin{smallmatrix}0&1&0\\0&0&1\\-1&0&0\\\end{smallmatrix}}\right]}
(0,0,1)n
(0,1,-1)n
(1,-1,0)n
(1,0,0)axis
(1,1,1)axis
(1,-1,0)axis
Hyperoctahedral symmetry [ edit ]
A irreducible 4-dimensional finite reflective group is hyperoctahedral group , B4 =[4,3,3], order 384, . The reflection generators matrices are R0 , R1 , R2 , R3 . R0 2 =R1 2 =R2 2 =R3 2 =(R0 ×R1 )4 =(R1 ×R2 )3 =(R2 ×R3 )3 =(R0 ×R2 )2 =(R1 ×R3 )2 =(R0 ×R3 )2 =Identity.
Chiral octahedral symmetry, [4,3,3]+ , ( ) is generated by 3 of 6 rotations: S0,1 , S1,2 , S2,3 , S0,2 , S1,3 , and S0,3 . Hyperpyritohedral symmetry [4,(3,3)+ ], ( ) is generated by reflection R0 and rotations S1,2 and S2,3 . An 8-fold double rotation is generated by W0,1,2,3 , the product of all 4 reflections.
[4,3,3],
Reflections
Rotations
Rotoreflection
Double rotation
Name
R0
R1
R2
R3
S0,1
S1,2
S2,3
S0,2
S1,3
S0,3
V0,1,2
V1,2,3
V0,1,3
V0,2,3
W0,1,2,3
Order
2
2
2
2
4 (B2 )
3 (A2 )
3 (A2 )
2
2
2
6 (B3 )
4 (A3 )
4
6
8 (B4 )
Matrix
[
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
−
1
]
{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&-1\\\end{smallmatrix}}\right]}
[
1
0
0
0
0
1
0
0
0
0
0
1
0
0
1
0
]
{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\\\end{smallmatrix}}\right]}
[
1
0
0
0
0
0
1
0
0
1
0
0
0
0
0
1
]
{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}
[
1
0
0
0
0
1
0
0
0
0
0
1
0
0
−
1
0
]
{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&-1&0\\\end{smallmatrix}}\right]}
[
1
0
0
0
0
1
0
0
0
0
0
1
0
0
−
1
0
]
{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&-1&0\\\end{smallmatrix}}\right]}
[
1
0
0
0
0
0
1
0
0
0
0
1
0
1
0
0
]
{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&0&0&1\\0&1&0&0\\\end{smallmatrix}}\right]}
[
0
1
0
0
0
0
1
0
1
0
0
1
0
0
0
1
]
{\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\0&0&1&0\\1&0&0&1\\0&0&0&1\\\end{smallmatrix}}\right]}
[
1
0
0
0
0
0
1
0
0
1
0
0
0
0
0
−
1
]
{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&-1\\\end{smallmatrix}}\right]}
[
0
1
0
0
1
0
0
0
0
0
0
1
0
0
1
0
]
{\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&0&1\\0&0&1&0\\\end{smallmatrix}}\right]}
[
0
1
0
0
1
0
0
0
0
0
1
0
0
0
0
−
1
]
{\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&1&0\\0&0&0&-1\\\end{smallmatrix}}\right]}
[
1
0
0
0
0
0
1
0
0
0
0
1
0
0
−
1
0
]
{\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&0&0&1\\0&0&-1&0\\\end{smallmatrix}}\right]}
[
0
1
0
0
0
0
1
0
0
0
0
1
1
0
0
0
]
{\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\0&0&1&0\\0&0&0&1\\1&0&0&0\\\end{smallmatrix}}\right]}
[
0
1
0
0
1
0
0
0
0
0
0
1
0
0
−
1
0
]
{\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&0&1\\0&0&-1&0\\\end{smallmatrix}}\right]}
[
0
1
0
0
0
0
1
0
1
0
0
0
0
0
0
−
1
]
{\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\0&0&1&0\\1&0&0&0\\0&0&0&-1\\\end{smallmatrix}}\right]}
[
0
1
0
0
0
0
1
0
0
0
0
1
−
1
0
0
0
]
{\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\0&0&1&0\\0&0&0&1\\-1&0&0&0\\\end{smallmatrix}}\right]}