167.161.1.1.L
Spin space group detail page with configuration, notation, nontrivial spin-space point group, spin-space point group, symmetry operations, and spin Wyckoff-position data.
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Configuration: Collinear
Type: type-t
G0 / L0: 167 / 161
Identifier: 167.161.1.1.L
it / ik: 2 / 1
Nontrivial spin-space point group: $-1$
Spin-space point group: $∞/mm$
International Notation: $$R\ce{^{\text{-}1}{\text{-}3}}\ce{^{1}{c}}\ce{^{\infty m}{1}}$$
Conventional to Primitive Matrix P:
$$\begin{bmatrix}\frac{2}{3} & \frac{1}{3} & \frac{1}{3} \\ -\frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ -\frac{1}{3} & -\frac{2}{3} & \frac{1}{3}\end{bmatrix}$$
Note: (aC, bC, cC) P = (aP, bP, cP)
Transformation Matrix M:
$$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
Note: [a_G0, b_G0, c_G0] M = [a_L0, b_L0, c_L0]
Translational group generators:
$$\begin{aligned}a &= (1, 0, 0) \\ b &= (0, 1, 0) \\ c &= (0, 0, 1)\end{aligned}$$
Note: The listed operations below are understood modulo the translational group generated by a, b, and c.
E.g. for 167.161.1.1.L, if a=(1, 0, 0), then (x, y, z) and (x+1, y, z) represent the same translation part.
E_E_tau
op_index Seitz coordinate spin_expr space_expr
1 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
2 $\left\{1||1|\frac{2}{3} \frac{1}{3} \frac{1}{3}\right\}$ $(x + \frac{2}{3},y + \frac{1}{3},z + \frac{1}{3},+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & \frac{2}{3} \\ 0 & 1 & 0 & \frac{1}{3} \\ 0 & 0 & 1 & \frac{1}{3}\end{array}\right]$$
3 $\left\{1||1|\frac{1}{3} \frac{2}{3} \frac{2}{3}\right\}$ $(x + \frac{1}{3},y + \frac{2}{3},z + \frac{2}{3},+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & \frac{1}{3} \\ 0 & 1 & 0 & \frac{2}{3} \\ 0 & 0 & 1 & \frac{2}{3}\end{array}\right]$$
U_E_0
op_index Seitz coordinate spin_expr space_expr
1 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
2 $\left\{\ce{^{\infty}{1}}||1|0 0 0\right\}$ $(x,y,z,+1,\cos(\phi) u - \sin(\phi) v,\sin(\phi) u + \cos(\phi) v,w)$ $$\begin{bmatrix}\cos(\phi) & -\sin(\phi) & 0 \\ \sin(\phi) & \cos(\phi) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
3 $\left\{\ce{^{m_{100}}{1}}||1|0 0 0\right\}$ $(x,y,z,-1,-u,v,w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
U_E_tau
op_index Seitz coordinate spin_expr space_expr
1 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
U_R_tau
op_index Seitz coordinate spin_expr space_expr
1 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
2 $\left\{-1||-3^{1}_{001}|0 0 0\right\}$ $(y,-x + y,-z,-1,-u,-v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
3 $\left\{1||m_{100}|0 0 \frac{1}{2}\right\}$ $(-x + y,y,z + \frac{1}{2},+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
4 $\left\{1||3^{2}_{001}|0 0 0\right\}$ $(-x + y,-x,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
5 $\left\{-1||2_{110}|0 0 \frac{1}{2}\right\}$ $(y,x,\frac{1}{2} - z,-1,-u,-v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
6 $\left\{-1||-1|0 0 0\right\}$ $(-x,-y,-z,-1,-u,-v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
7 $\left\{1||m_{010}|0 0 \frac{1}{2}\right\}$ $(x,x - y,z + \frac{1}{2},+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
8 $\left\{1||3^{1}_{001}|0 0 0\right\}$ $(-y,x - y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
9 $\left\{-1||2_{100}|0 0 \frac{1}{2}\right\}$ $(x - y,-y,\frac{1}{2} - z,-1,-u,-v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & -1 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
10 $\left\{1||m_{110}|0 0 \frac{1}{2}\right\}$ $(-y,-x,z + \frac{1}{2},+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
11 $\left\{-1||-3^{2}_{001}|0 0 0\right\}$ $(x - y,x,-z,-1,-u,-v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
12 $\left\{-1||2_{010}|0 0 \frac{1}{2}\right\}$ $(-x,-x + y,\frac{1}{2} - z,-1,-u,-v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
No.
op_index Seitz coordinate spin_expr space_expr
1 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
2 $\left\{-1||-3^{1}_{001}|0 0 0\right\}$ $(y,-x + y,-z,-1,-u,-v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
3 $\left\{1||m_{100}|0 0 \frac{1}{2}\right\}$ $(-x + y,y,z + \frac{1}{2},+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
4 $\left\{1||1|\frac{2}{3} \frac{1}{3} \frac{1}{3}\right\}$ $(x + \frac{2}{3},y + \frac{1}{3},z + \frac{1}{3},+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & \frac{2}{3} \\ 0 & 1 & 0 & \frac{1}{3} \\ 0 & 0 & 1 & \frac{1}{3}\end{array}\right]$$
5 $\left\{1||1|\frac{1}{3} \frac{2}{3} \frac{2}{3}\right\}$ $(x + \frac{1}{3},y + \frac{2}{3},z + \frac{2}{3},+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & \frac{1}{3} \\ 0 & 1 & 0 & \frac{2}{3} \\ 0 & 0 & 1 & \frac{2}{3}\end{array}\right]$$
6 $\left\{1||3^{2}_{001}|0 0 0\right\}$ $(-x + y,-x,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
7 $\left\{-1||2_{110}|0 0 \frac{1}{2}\right\}$ $(y,x,\frac{1}{2} - z,-1,-u,-v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
8 $\left\{-1||-3^{1}_{001}|\frac{1}{3} \frac{2}{3} \frac{2}{3}\right\}$ $(y + \frac{1}{3},-x + y + \frac{2}{3},\frac{2}{3} - z,-1,-u,-v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & \frac{1}{3} \\ -1 & 1 & 0 & \frac{2}{3} \\ 0 & 0 & -1 & \frac{2}{3}\end{array}\right]$$
9 $\left\{-1||-3^{1}_{001}|\frac{2}{3} \frac{1}{3} \frac{1}{3}\right\}$ $(y + \frac{2}{3},-x + y + \frac{1}{3},\frac{1}{3} - z,-1,-u,-v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & \frac{2}{3} \\ -1 & 1 & 0 & \frac{1}{3} \\ 0 & 0 & -1 & \frac{1}{3}\end{array}\right]$$
10 $\left\{1||3^{2}_{001}|\frac{1}{3} \frac{2}{3} \frac{2}{3}\right\}$ $(-x + y + \frac{1}{3},\frac{2}{3} - x,z + \frac{2}{3},+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & \frac{1}{3} \\ -1 & 0 & 0 & \frac{2}{3} \\ 0 & 0 & 1 & \frac{2}{3}\end{array}\right]$$
11 $\left\{-1||-1|0 0 0\right\}$ $(-x,-y,-z,-1,-u,-v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
12 $\left\{1||m_{010}|0 0 \frac{1}{2}\right\}$ $(x,x - y,z + \frac{1}{2},+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
13 $\left\{1||3^{2}_{001}|\frac{2}{3} \frac{1}{3} \frac{1}{3}\right\}$ $(-x + y + \frac{2}{3},\frac{1}{3} - x,z + \frac{1}{3},+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & \frac{2}{3} \\ -1 & 0 & 0 & \frac{1}{3} \\ 0 & 0 & 1 & \frac{1}{3}\end{array}\right]$$
14 $\left\{-1||-1|\frac{2}{3} \frac{1}{3} \frac{1}{3}\right\}$ $(\frac{2}{3} - x,\frac{1}{3} - y,\frac{1}{3} - z,-1,-u,-v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & \frac{2}{3} \\ 0 & -1 & 0 & \frac{1}{3} \\ 0 & 0 & -1 & \frac{1}{3}\end{array}\right]$$
15 $\left\{1||3^{1}_{001}|0 0 0\right\}$ $(-y,x - y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
16 $\left\{-1||2_{100}|0 0 \frac{1}{2}\right\}$ $(x - y,-y,\frac{1}{2} - z,-1,-u,-v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & -1 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
17 $\left\{-1||-1|\frac{1}{3} \frac{2}{3} \frac{2}{3}\right\}$ $(\frac{1}{3} - x,\frac{2}{3} - y,\frac{2}{3} - z,-1,-u,-v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & \frac{1}{3} \\ 0 & -1 & 0 & \frac{2}{3} \\ 0 & 0 & -1 & \frac{2}{3}\end{array}\right]$$
18 $\left\{1||3^{1}_{001}|\frac{2}{3} \frac{1}{3} \frac{1}{3}\right\}$ $(\frac{2}{3} - y,x - y + \frac{1}{3},z + \frac{1}{3},+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & \frac{2}{3} \\ 1 & -1 & 0 & \frac{1}{3} \\ 0 & 0 & 1 & \frac{1}{3}\end{array}\right]$$
19 $\left\{1||3^{1}_{001}|\frac{1}{3} \frac{2}{3} \frac{2}{3}\right\}$ $(\frac{1}{3} - y,x - y + \frac{2}{3},z + \frac{2}{3},+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & \frac{1}{3} \\ 1 & -1 & 0 & \frac{2}{3} \\ 0 & 0 & 1 & \frac{2}{3}\end{array}\right]$$
20 $\left\{-1||-3^{2}_{001}|0 0 0\right\}$ $(x - y,x,-z,-1,-u,-v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
21 $\left\{1||m_{110}|0 0 \frac{1}{2}\right\}$ $(-y,-x,z + \frac{1}{2},+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
22 $\left\{-1||-3^{2}_{001}|\frac{1}{3} \frac{2}{3} \frac{2}{3}\right\}$ $(x - y + \frac{1}{3},x + \frac{2}{3},\frac{2}{3} - z,-1,-u,-v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & -1 & 0 & \frac{1}{3} \\ 1 & 0 & 0 & \frac{2}{3} \\ 0 & 0 & -1 & \frac{2}{3}\end{array}\right]$$
23 $\left\{-1||-3^{2}_{001}|\frac{2}{3} \frac{1}{3} \frac{1}{3}\right\}$ $(x - y + \frac{2}{3},x + \frac{1}{3},\frac{1}{3} - z,-1,-u,-v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & -1 & 0 & \frac{2}{3} \\ 1 & 0 & 0 & \frac{1}{3} \\ 0 & 0 & -1 & \frac{1}{3}\end{array}\right]$$
24 $\left\{-1||2_{010}|0 0 \frac{1}{2}\right\}$ $(-x,-x + y,\frac{1}{2} - z,-1,-u,-v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
25 $\left\{1||m_{100}|\frac{2}{3} \frac{1}{3} \frac{5}{6}\right\}$ $(-x + y + \frac{2}{3},y + \frac{1}{3},z + \frac{5}{6},+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & \frac{2}{3} \\ 0 & 1 & 0 & \frac{1}{3} \\ 0 & 0 & 1 & \frac{5}{6}\end{array}\right]$$
26 $\left\{1||m_{100}|\frac{1}{3} \frac{2}{3} \frac{1}{6}\right\}$ $(-x + y + \frac{1}{3},y + \frac{2}{3},z + \frac{1}{6},+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & \frac{1}{3} \\ 0 & 1 & 0 & \frac{2}{3} \\ 0 & 0 & 1 & \frac{1}{6}\end{array}\right]$$
27 $\left\{-1||2_{010}|\frac{1}{3} \frac{2}{3} \frac{1}{6}\right\}$ $(\frac{1}{3} - x,-x + y + \frac{2}{3},\frac{1}{6} - z,-1,-u,-v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & \frac{1}{3} \\ -1 & 1 & 0 & \frac{2}{3} \\ 0 & 0 & -1 & \frac{1}{6}\end{array}\right]$$
28 $\left\{-1||2_{010}|\frac{2}{3} \frac{1}{3} \frac{5}{6}\right\}$ $(\frac{2}{3} - x,-x + y + \frac{1}{3},\frac{5}{6} - z,-1,-u,-v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & \frac{2}{3} \\ -1 & 1 & 0 & \frac{1}{3} \\ 0 & 0 & -1 & \frac{5}{6}\end{array}\right]$$
29 $\left\{1||m_{110}|\frac{1}{3} \frac{2}{3} \frac{1}{6}\right\}$ $(\frac{1}{3} - y,\frac{2}{3} - x,z + \frac{1}{6},+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & \frac{1}{3} \\ -1 & 0 & 0 & \frac{2}{3} \\ 0 & 0 & 1 & \frac{1}{6}\end{array}\right]$$
30 $\left\{1||m_{110}|\frac{2}{3} \frac{1}{3} \frac{5}{6}\right\}$ $(\frac{2}{3} - y,\frac{1}{3} - x,z + \frac{5}{6},+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & \frac{2}{3} \\ -1 & 0 & 0 & \frac{1}{3} \\ 0 & 0 & 1 & \frac{5}{6}\end{array}\right]$$
31 $\left\{-1||2_{100}|\frac{2}{3} \frac{1}{3} \frac{5}{6}\right\}$ $(x - y + \frac{2}{3},\frac{1}{3} - y,\frac{5}{6} - z,-1,-u,-v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & -1 & 0 & \frac{2}{3} \\ 0 & -1 & 0 & \frac{1}{3} \\ 0 & 0 & -1 & \frac{5}{6}\end{array}\right]$$
32 $\left\{-1||2_{100}|\frac{1}{3} \frac{2}{3} \frac{1}{6}\right\}$ $(x - y + \frac{1}{3},\frac{2}{3} - y,\frac{1}{6} - z,-1,-u,-v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & -1 & 0 & \frac{1}{3} \\ 0 & -1 & 0 & \frac{2}{3} \\ 0 & 0 & -1 & \frac{1}{6}\end{array}\right]$$
33 $\left\{1||m_{010}|\frac{2}{3} \frac{1}{3} \frac{5}{6}\right\}$ $(x + \frac{2}{3},x - y + \frac{1}{3},z + \frac{5}{6},+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & \frac{2}{3} \\ 1 & -1 & 0 & \frac{1}{3} \\ 0 & 0 & 1 & \frac{5}{6}\end{array}\right]$$
34 $\left\{1||m_{010}|\frac{1}{3} \frac{2}{3} \frac{1}{6}\right\}$ $(x + \frac{1}{3},x - y + \frac{2}{3},z + \frac{1}{6},+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & \frac{1}{3} \\ 1 & -1 & 0 & \frac{2}{3} \\ 0 & 0 & 1 & \frac{1}{6}\end{array}\right]$$
35 $\left\{-1||2_{110}|\frac{1}{3} \frac{2}{3} \frac{1}{6}\right\}$ $(y + \frac{1}{3},x + \frac{2}{3},\frac{1}{6} - z,-1,-u,-v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & \frac{1}{3} \\ 1 & 0 & 0 & \frac{2}{3} \\ 0 & 0 & -1 & \frac{1}{6}\end{array}\right]$$
36 $\left\{-1||2_{110}|\frac{2}{3} \frac{1}{3} \frac{5}{6}\right\}$ $(y + \frac{2}{3},x + \frac{1}{3},\frac{5}{6} - z,-1,-u,-v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & \frac{2}{3} \\ 1 & 0 & 0 & \frac{1}{3} \\ 0 & 0 & -1 & \frac{5}{6}\end{array}\right]$$
extra__U_E_0(without_identity)
op_index Seitz coordinate spin_expr space_expr
1 $\left\{\ce{^{\infty}{1}}||1|0 0 0\right\}$ $(x,y,z,+1,\cos(\phi) u - \sin(\phi) v,\sin(\phi) u + \cos(\phi) v,w)$ $$\begin{bmatrix}\cos(\phi) & -\sin(\phi) & 0 \\ \sin(\phi) & \cos(\phi) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
2 $\left\{\ce{^{m_{100}}{1}}||1|0 0 0\right\}$ $(x,y,z,-1,-u,v,w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
wyckoff position site symmetry Coordinates
$(0,0,0\mid mx,my,mz)+(\frac{2}{3},\frac{1}{3},\frac{1}{3}\mid mx,my,mz)+(\frac{1}{3},\frac{2}{3},\frac{2}{3}\mid mx,my,mz)$
36f $(x,y,z\mid 0,0,mx),(-y,x - y,z\mid 0,0,mx),(-x + y,-x,z\mid 0,0,mx),(y,x,\frac{1}{2} - z\mid 0,0,-mx),(x - y,-y,\frac{1}{2} - z\mid 0,0,-mx),(-x,-x + y,\frac{1}{2} - z\mid 0,0,-mx),(-x,-y,-z\mid 0,0,-mx),(y,-x + y,-z\mid 0,0,-mx),(x - y,x,-z\mid 0,0,-mx),(-y,-x,z + \frac{1}{2}\mid 0,0,mx),(-x + y,y,z + \frac{1}{2}\mid 0,0,mx),(x,x - y,z + \frac{1}{2}\mid 0,0,mx)$
18e $(x,0,\frac{1}{4}\mid 0,0,0),(0,x,\frac{1}{4}\mid 0,0,0),(-x,-x,\frac{1}{4}\mid 0,0,0),(-x,0,\frac{3}{4}\mid 0,0,0),(0,-x,\frac{3}{4}\mid 0,0,0),(x,x,\frac{3}{4}\mid 0,0,0)$
18d $(\frac{1}{2},0,0\mid 0,0,0),(0,\frac{1}{2},0\mid 0,0,0),(\frac{1}{2},\frac{1}{2},0\mid 0,0,0),(0,\frac{1}{2},\frac{1}{2}\mid 0,0,0),(\frac{1}{2},0,\frac{1}{2}\mid 0,0,0),(\frac{1}{2},\frac{1}{2},\frac{1}{2}\mid 0,0,0)$
12c $(0,0,z\mid 0,0,mx),(0,0,\frac{1}{2} - z\mid 0,0,-mx),(0,0,-z\mid 0,0,-mx),(0,0,z + \frac{1}{2}\mid 0,0,mx)$
6b $(0,0,0\mid 0,0,0),(0,0,\frac{1}{2}\mid 0,0,0)$
6a $(0,0,\frac{1}{4}\mid 0,0,0),(0,0,\frac{3}{4}\mid 0,0,0)$
Site symmetry: $^{1}1^{\infty m}1$
op_index Seitz coordinate spin_expr space_expr
1 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
extra__U_E_0(without_identity)
op_index Seitz coordinate spin_expr space_expr
1 $\left\{\ce{^{\infty}{1}}||1|0 0 0\right\}$ $(x,y,z,+1,\cos(\phi) u - \sin(\phi) v,\sin(\phi) u + \cos(\phi) v,w)$ $$\begin{bmatrix}\cos(\phi) & -\sin(\phi) & 0 \\ \sin(\phi) & \cos(\phi) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
2 $\left\{\ce{^{m_{100}}{1}}||1|0 0 0\right\}$ $(x,y,z,-1,-u,v,w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
Site symmetry: $.^{\!-\!1}2^{\infty m}1$
op_index Seitz coordinate spin_expr space_expr
1 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
16 $\left\{-1||2_{100}|0 0 \frac{1}{2}\right\}$ $(x - y,-y,\frac{1}{2} - z,-1,-u,-v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & -1 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
extra__U_E_0(without_identity)
op_index Seitz coordinate spin_expr space_expr
1 $\left\{\ce{^{\infty}{1}}||1|0 0 0\right\}$ $(x,y,z,+1,\cos(\phi) u - \sin(\phi) v,\sin(\phi) u + \cos(\phi) v,w)$ $$\begin{bmatrix}\cos(\phi) & -\sin(\phi) & 0 \\ \sin(\phi) & \cos(\phi) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
2 $\left\{\ce{^{m_{100}}{1}}||1|0 0 0\right\}$ $(x,y,z,-1,-u,v,w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
Site symmetry: $^{\!-\!1}\!-\!1^{\infty m}1$
op_index Seitz coordinate spin_expr space_expr
1 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
11 $\left\{-1||-1|0 0 0\right\}$ $(-x,-y,-z,-1,-u,-v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
extra__U_E_0(without_identity)
op_index Seitz coordinate spin_expr space_expr
1 $\left\{\ce{^{\infty}{1}}||1|0 0 0\right\}$ $(x,y,z,+1,\cos(\phi) u - \sin(\phi) v,\sin(\phi) u + \cos(\phi) v,w)$ $$\begin{bmatrix}\cos(\phi) & -\sin(\phi) & 0 \\ \sin(\phi) & \cos(\phi) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
2 $\left\{\ce{^{m_{100}}{1}}||1|0 0 0\right\}$ $(x,y,z,-1,-u,v,w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
Site symmetry: $^{1}3.^{\infty m}1$
op_index Seitz coordinate spin_expr space_expr
1 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
6 $\left\{1||3^{2}_{001}|0 0 0\right\}$ $(-x + y,-x,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
15 $\left\{1||3^{1}_{001}|0 0 0\right\}$ $(-y,x - y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
extra__U_E_0(without_identity)
op_index Seitz coordinate spin_expr space_expr
1 $\left\{\ce{^{\infty}{1}}||1|0 0 0\right\}$ $(x,y,z,+1,\cos(\phi) u - \sin(\phi) v,\sin(\phi) u + \cos(\phi) v,w)$ $$\begin{bmatrix}\cos(\phi) & -\sin(\phi) & 0 \\ \sin(\phi) & \cos(\phi) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
2 $\left\{\ce{^{m_{100}}{1}}||1|0 0 0\right\}$ $(x,y,z,-1,-u,v,w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
Site symmetry: $^{\!-\!1}\!-\!3.^{\infty m}1$
op_index Seitz coordinate spin_expr space_expr
1 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
2 $\left\{-1||-3^{1}_{001}|0 0 0\right\}$ $(y,-x + y,-z,-1,-u,-v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
6 $\left\{1||3^{2}_{001}|0 0 0\right\}$ $(-x + y,-x,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
11 $\left\{-1||-1|0 0 0\right\}$ $(-x,-y,-z,-1,-u,-v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
15 $\left\{1||3^{1}_{001}|0 0 0\right\}$ $(-y,x - y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
20 $\left\{-1||-3^{2}_{001}|0 0 0\right\}$ $(x - y,x,-z,-1,-u,-v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
extra__U_E_0(without_identity)
op_index Seitz coordinate spin_expr space_expr
1 $\left\{\ce{^{\infty}{1}}||1|0 0 0\right\}$ $(x,y,z,+1,\cos(\phi) u - \sin(\phi) v,\sin(\phi) u + \cos(\phi) v,w)$ $$\begin{bmatrix}\cos(\phi) & -\sin(\phi) & 0 \\ \sin(\phi) & \cos(\phi) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
2 $\left\{\ce{^{m_{100}}{1}}||1|0 0 0\right\}$ $(x,y,z,-1,-u,v,w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
Site symmetry: $^{1}3^{\!-\!1}2^{\infty m}1$
op_index Seitz coordinate spin_expr space_expr
1 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
6 $\left\{1||3^{2}_{001}|0 0 0\right\}$ $(-x + y,-x,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
7 $\left\{-1||2_{110}|0 0 \frac{1}{2}\right\}$ $(y,x,\frac{1}{2} - z,-1,-u,-v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
15 $\left\{1||3^{1}_{001}|0 0 0\right\}$ $(-y,x - y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
16 $\left\{-1||2_{100}|0 0 \frac{1}{2}\right\}$ $(x - y,-y,\frac{1}{2} - z,-1,-u,-v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & -1 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
24 $\left\{-1||2_{010}|0 0 \frac{1}{2}\right\}$ $(-x,-x + y,\frac{1}{2} - z,-1,-u,-v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
extra__U_E_0(without_identity)
op_index Seitz coordinate spin_expr space_expr
1 $\left\{\ce{^{\infty}{1}}||1|0 0 0\right\}$ $(x,y,z,+1,\cos(\phi) u - \sin(\phi) v,\sin(\phi) u + \cos(\phi) v,w)$ $$\begin{bmatrix}\cos(\phi) & -\sin(\phi) & 0 \\ \sin(\phi) & \cos(\phi) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
2 $\left\{\ce{^{m_{100}}{1}}||1|0 0 0\right\}$ $(x,y,z,-1,-u,v,w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$