194.11.1.1.P
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: Coplanar
Type: type-t
G0 / L0: 194 / 11
Identifier: 194.11.1.1.P
it / ik: 6 / 1
Nontrivial spin-space point group: $32$
Spin-space point group: $\!-\!62m$
International Notation: $$P\ce{^{3^{2}_{001}}{6_{3}/}}\ce{^{1}{m}}\ce{^{2_{\frac{2}{3}\pi}}{m}}\ce{^{2_{\frac{1}{3}\pi}}{c}}\ce{^{m_{001}}{1}}$$
Conventional to Primitive Matrix P:
$$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$
Note: (aC, bC, cC) P = (aP, bP, cP)
Transformation Matrix M:
$$\left[\begin{array}{ccc|c}0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\end{array}\right]$$
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 194.11.1.1.P, 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 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
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 $\left\{\ce{^{m_{001}}{1}}||1|0 0 0\right\}$ $(x,y,z,-1,u,v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
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 $\left\{3^{2}_{001}||6^{1}_{001}|0 0 \frac{1}{2}\right\}$ $(x - y,x,z + \frac{1}{2},+1,\cos(\frac{4\pi}{3}) u - \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u + \cos(\frac{4\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & -\sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & \cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
3 $\left\{1||m_{001}|0 0 \frac{1}{2}\right\}$ $(x,y,\frac{1}{2} - z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
4 $\left\{2_{\frac{2\pi}{3}}||m_{100}|0 0 0\right\}$ $(-x + y,y,z,+1,\cos(\frac{4\pi}{3}) u + \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u - \cos(\frac{4\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & \sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & -\cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
5 $\left\{2_{\frac{\pi}{3}}||m_{210}|0 0 \frac{1}{2}\right\}$ $(-x,-x + y,z + \frac{1}{2},+1,\cos(\frac{2\pi}{3}) u + \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u - \cos(\frac{2\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & \sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & -\cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
6 $\left\{3^{1}_{001}||3^{1}_{001}|0 0 0\right\}$ $(-y,x - y,z,+1,\cos(\frac{2\pi}{3}) u - \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u + \cos(\frac{2\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & -\sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & \cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
7 $\left\{3^{2}_{001}||-3^{2}_{001}|0 0 0\right\}$ $(x - y,x,-z,+1,\cos(\frac{4\pi}{3}) u - \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u + \cos(\frac{4\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & -\sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & \cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
8 $\left\{2_{100}||m_{110}|0 0 0\right\}$ $(-y,-x,z,+1,u,-v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
9 $\left\{1||2_{001}|0 0 \frac{1}{2}\right\}$ $(-x,-y,z + \frac{1}{2},+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
10 $\left\{3^{1}_{001}||-6^{5}_{001}|0 0 \frac{1}{2}\right\}$ $(-y,x - y,\frac{1}{2} - z,+1,\cos(\frac{2\pi}{3}) u - \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u + \cos(\frac{2\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & -\sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & \cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
11 $\left\{2_{\frac{2\pi}{3}}||m_{120}|0 0 \frac{1}{2}\right\}$ $(x - y,-y,z + \frac{1}{2},+1,\cos(\frac{4\pi}{3}) u + \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u - \cos(\frac{4\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & \sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & -\cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & -1 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
12 $\left\{3^{2}_{001}||3^{2}_{001}|0 0 0\right\}$ $(-x + y,-x,z,+1,\cos(\frac{4\pi}{3}) u - \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u + \cos(\frac{4\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & -\sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & \cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
13 $\left\{1||-1|0 0 0\right\}$ $(-x,-y,-z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
14 $\left\{2_{\frac{\pi}{3}}||m_{010}|0 0 0\right\}$ $(x,x - y,z,+1,\cos(\frac{2\pi}{3}) u + \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u - \cos(\frac{2\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & \sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & -\cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
15 $\left\{3^{1}_{001}||6^{5}_{001}|0 0 \frac{1}{2}\right\}$ $(y,-x + y,z + \frac{1}{2},+1,\cos(\frac{2\pi}{3}) u - \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u + \cos(\frac{2\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & -\sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & \cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
16 $\left\{3^{2}_{001}||-6^{1}_{001}|0 0 \frac{1}{2}\right\}$ $(-x + y,-x,\frac{1}{2} - z,+1,\cos(\frac{4\pi}{3}) u - \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u + \cos(\frac{4\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & -\sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & \cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
17 $\left\{2_{100}||m_{1-10}|0 0 \frac{1}{2}\right\}$ $(y,x,z + \frac{1}{2},+1,u,-v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
18 $\left\{3^{1}_{001}||-3^{1}_{001}|0 0 0\right\}$ $(y,-x + y,-z,+1,\cos(\frac{2\pi}{3}) u - \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u + \cos(\frac{2\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & -\sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & \cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
19 $\left\{2_{\frac{2\pi}{3}}||2_{120}|0 0 \frac{1}{2}\right\}$ $(-x + y,y,\frac{1}{2} - z,+1,\cos(\frac{4\pi}{3}) u + \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u - \cos(\frac{4\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & \sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & -\cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
20 $\left\{2_{\frac{\pi}{3}}||2_{010}|0 0 0\right\}$ $(-x,-x + y,-z,+1,\cos(\frac{2\pi}{3}) u + \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u - \cos(\frac{2\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & \sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & -\cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
21 $\left\{2_{100}||2_{1-10}|0 0 \frac{1}{2}\right\}$ $(-y,-x,\frac{1}{2} - z,+1,u,-v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
22 $\left\{2_{\frac{2\pi}{3}}||2_{100}|0 0 0\right\}$ $(x - y,-y,-z,+1,\cos(\frac{4\pi}{3}) u + \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u - \cos(\frac{4\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & \sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & -\cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & -1 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
23 $\left\{2_{\frac{\pi}{3}}||2_{210}|0 0 \frac{1}{2}\right\}$ $(x,x - y,\frac{1}{2} - z,+1,\cos(\frac{2\pi}{3}) u + \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u - \cos(\frac{2\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & \sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & -\cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
24 $\left\{2_{100}||2_{110}|0 0 0\right\}$ $(y,x,-z,+1,u,-v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
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 $\left\{3^{2}_{001}||6^{1}_{001}|0 0 \frac{1}{2}\right\}$ $(x - y,x,z + \frac{1}{2},+1,\cos(\frac{4\pi}{3}) u - \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u + \cos(\frac{4\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & -\sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & \cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
3 $\left\{1||m_{001}|0 0 \frac{1}{2}\right\}$ $(x,y,\frac{1}{2} - z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
4 $\left\{2_{\frac{2\pi}{3}}||m_{100}|0 0 0\right\}$ $(-x + y,y,z,+1,\cos(\frac{4\pi}{3}) u + \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u - \cos(\frac{4\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & \sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & -\cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
5 $\left\{2_{\frac{\pi}{3}}||m_{210}|0 0 \frac{1}{2}\right\}$ $(-x,-x + y,z + \frac{1}{2},+1,\cos(\frac{2\pi}{3}) u + \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u - \cos(\frac{2\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & \sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & -\cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
6 $\left\{3^{1}_{001}||3^{1}_{001}|0 0 0\right\}$ $(-y,x - y,z,+1,\cos(\frac{2\pi}{3}) u - \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u + \cos(\frac{2\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & -\sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & \cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
7 $\left\{3^{2}_{001}||-3^{2}_{001}|0 0 0\right\}$ $(x - y,x,-z,+1,\cos(\frac{4\pi}{3}) u - \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u + \cos(\frac{4\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & -\sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & \cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
8 $\left\{2_{100}||m_{110}|0 0 0\right\}$ $(-y,-x,z,+1,u,-v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
9 $\left\{1||2_{001}|0 0 \frac{1}{2}\right\}$ $(-x,-y,z + \frac{1}{2},+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
10 $\left\{3^{1}_{001}||-6^{5}_{001}|0 0 \frac{1}{2}\right\}$ $(-y,x - y,\frac{1}{2} - z,+1,\cos(\frac{2\pi}{3}) u - \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u + \cos(\frac{2\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & -\sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & \cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
11 $\left\{2_{\frac{2\pi}{3}}||m_{120}|0 0 \frac{1}{2}\right\}$ $(x - y,-y,z + \frac{1}{2},+1,\cos(\frac{4\pi}{3}) u + \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u - \cos(\frac{4\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & \sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & -\cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & -1 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
12 $\left\{3^{2}_{001}||3^{2}_{001}|0 0 0\right\}$ $(-x + y,-x,z,+1,\cos(\frac{4\pi}{3}) u - \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u + \cos(\frac{4\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & -\sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & \cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
13 $\left\{1||-1|0 0 0\right\}$ $(-x,-y,-z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
14 $\left\{2_{\frac{\pi}{3}}||m_{010}|0 0 0\right\}$ $(x,x - y,z,+1,\cos(\frac{2\pi}{3}) u + \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u - \cos(\frac{2\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & \sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & -\cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
15 $\left\{3^{1}_{001}||6^{5}_{001}|0 0 \frac{1}{2}\right\}$ $(y,-x + y,z + \frac{1}{2},+1,\cos(\frac{2\pi}{3}) u - \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u + \cos(\frac{2\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & -\sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & \cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
16 $\left\{3^{2}_{001}||-6^{1}_{001}|0 0 \frac{1}{2}\right\}$ $(-x + y,-x,\frac{1}{2} - z,+1,\cos(\frac{4\pi}{3}) u - \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u + \cos(\frac{4\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & -\sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & \cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
17 $\left\{2_{100}||m_{1-10}|0 0 \frac{1}{2}\right\}$ $(y,x,z + \frac{1}{2},+1,u,-v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
18 $\left\{3^{1}_{001}||-3^{1}_{001}|0 0 0\right\}$ $(y,-x + y,-z,+1,\cos(\frac{2\pi}{3}) u - \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u + \cos(\frac{2\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & -\sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & \cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
19 $\left\{2_{\frac{2\pi}{3}}||2_{120}|0 0 \frac{1}{2}\right\}$ $(-x + y,y,\frac{1}{2} - z,+1,\cos(\frac{4\pi}{3}) u + \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u - \cos(\frac{4\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & \sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & -\cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
20 $\left\{2_{\frac{\pi}{3}}||2_{010}|0 0 0\right\}$ $(-x,-x + y,-z,+1,\cos(\frac{2\pi}{3}) u + \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u - \cos(\frac{2\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & \sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & -\cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
21 $\left\{2_{100}||2_{1-10}|0 0 \frac{1}{2}\right\}$ $(-y,-x,\frac{1}{2} - z,+1,u,-v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
22 $\left\{2_{\frac{2\pi}{3}}||2_{100}|0 0 0\right\}$ $(x - y,-y,-z,+1,\cos(\frac{4\pi}{3}) u + \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u - \cos(\frac{4\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & \sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & -\cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & -1 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
23 $\left\{2_{\frac{\pi}{3}}||2_{210}|0 0 \frac{1}{2}\right\}$ $(x,x - y,\frac{1}{2} - z,+1,\cos(\frac{2\pi}{3}) u + \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u - \cos(\frac{2\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & \sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & -\cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
24 $\left\{2_{100}||2_{110}|0 0 0\right\}$ $(y,x,-z,+1,u,-v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
extra__U_E_0(without_identity)
op_index Seitz coordinate spin_expr space_expr
1 $\left\{\ce{^{m_{001}}{1}}||1|0 0 0\right\}$ $(x,y,z,-1,u,v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
wyckoff position site symmetry Coordinates
$(0,0,0\mid mx,my,mz)$
24l $(x,y,z\mid mx,my,0),(-y,x - y,z\mid -mx/2 - \sqrt{3} my/2,\sqrt{3} mx/2 - my/2,0),(-x + y,-x,z\mid -mx/2 + \sqrt{3} my/2,-\sqrt{3} mx/2 - my/2,0),(-x,-y,z + \frac{1}{2}\mid mx,my,0),(y,-x + y,z + \frac{1}{2}\mid -mx/2 - \sqrt{3} my/2,\sqrt{3} mx/2 - my/2,0),(x - y,x,z + \frac{1}{2}\mid -mx/2 + \sqrt{3} my/2,-\sqrt{3} mx/2 - my/2,0),(y,x,-z\mid mx,-my,0),(x - y,-y,-z\mid -mx/2 - \sqrt{3} my/2,-\sqrt{3} mx/2 + my/2,0),(-x,-x + y,-z\mid -mx/2 + \sqrt{3} my/2,\sqrt{3} mx/2 + my/2,0),(-y,-x,\frac{1}{2} - z\mid mx,-my,0),(-x + y,y,\frac{1}{2} - z\mid -mx/2 - \sqrt{3} my/2,-\sqrt{3} mx/2 + my/2,0),(x,x - y,\frac{1}{2} - z\mid -mx/2 + \sqrt{3} my/2,\sqrt{3} mx/2 + my/2,0),(-x,-y,-z\mid mx,my,0),(y,-x + y,-z\mid -mx/2 - \sqrt{3} my/2,\sqrt{3} mx/2 - my/2,0),(x - y,x,-z\mid -mx/2 + \sqrt{3} my/2,-\sqrt{3} mx/2 - my/2,0),(x,y,\frac{1}{2} - z\mid mx,my,0),(-y,x - y,\frac{1}{2} - z\mid -mx/2 - \sqrt{3} my/2,\sqrt{3} mx/2 - my/2,0),(-x + y,-x,\frac{1}{2} - z\mid -mx/2 + \sqrt{3} my/2,-\sqrt{3} mx/2 - my/2,0),(-y,-x,z\mid mx,-my,0),(-x + y,y,z\mid -mx/2 - \sqrt{3} my/2,-\sqrt{3} mx/2 + my/2,0),(x,x - y,z\mid -mx/2 + \sqrt{3} my/2,\sqrt{3} mx/2 + my/2,0),(y,x,z + \frac{1}{2}\mid mx,-my,0),(x - y,-y,z + \frac{1}{2}\mid -mx/2 - \sqrt{3} my/2,-\sqrt{3} mx/2 + my/2,0),(-x,-x + y,z + \frac{1}{2}\mid -mx/2 + \sqrt{3} my/2,\sqrt{3} mx/2 + my/2,0)$
12k $(x,2 x,z\mid -mx/2,\sqrt{3} mx/2,0),(-2 x,-x,z\mid -mx/2,-\sqrt{3} mx/2,0),(x,-x,z\mid mx,0,0),(-x,-2 x,z + \frac{1}{2}\mid -mx/2,\sqrt{3} mx/2,0),(2 x,x,z + \frac{1}{2}\mid -mx/2,-\sqrt{3} mx/2,0),(-x,x,z + \frac{1}{2}\mid mx,0,0),(2 x,x,-z\mid -mx/2,-\sqrt{3} mx/2,0),(-x,-2 x,-z\mid -mx/2,\sqrt{3} mx/2,0),(-x,x,-z\mid mx,0,0),(-2 x,-x,\frac{1}{2} - z\mid -mx/2,-\sqrt{3} mx/2,0),(x,2 x,\frac{1}{2} - z\mid -mx/2,\sqrt{3} mx/2,0),(x,-x,\frac{1}{2} - z\mid mx,0,0)$
12j $(x,y,\frac{1}{4}\mid mx,my,0),(-y,x - y,\frac{1}{4}\mid -mx/2 - \sqrt{3} my/2,\sqrt{3} mx/2 - my/2,0),(-x + y,-x,\frac{1}{4}\mid -mx/2 + \sqrt{3} my/2,-\sqrt{3} mx/2 - my/2,0),(-x,-y,\frac{3}{4}\mid mx,my,0),(y,-x + y,\frac{3}{4}\mid -mx/2 - \sqrt{3} my/2,\sqrt{3} mx/2 - my/2,0),(x - y,x,\frac{3}{4}\mid -mx/2 + \sqrt{3} my/2,-\sqrt{3} mx/2 - my/2,0),(y,x,\frac{3}{4}\mid mx,-my,0),(x - y,-y,\frac{3}{4}\mid -mx/2 - \sqrt{3} my/2,-\sqrt{3} mx/2 + my/2,0),(-x,-x + y,\frac{3}{4}\mid -mx/2 + \sqrt{3} my/2,\sqrt{3} mx/2 + my/2,0),(-y,-x,\frac{1}{4}\mid mx,-my,0),(-x + y,y,\frac{1}{4}\mid -mx/2 - \sqrt{3} my/2,-\sqrt{3} mx/2 + my/2,0),(x,x - y,\frac{1}{4}\mid -mx/2 + \sqrt{3} my/2,\sqrt{3} mx/2 + my/2,0)$
12i $(x,0,0\mid -mx/2,\sqrt{3} mx/2,0),(0,x,0\mid -mx/2,-\sqrt{3} mx/2,0),(-x,-x,0\mid mx,0,0),(-x,0,\frac{1}{2}\mid -mx/2,\sqrt{3} mx/2,0),(0,-x,\frac{1}{2}\mid -mx/2,-\sqrt{3} mx/2,0),(x,x,\frac{1}{2}\mid mx,0,0),(-x,0,0\mid -mx/2,\sqrt{3} mx/2,0),(0,-x,0\mid -mx/2,-\sqrt{3} mx/2,0),(x,x,0\mid mx,0,0),(x,0,\frac{1}{2}\mid -mx/2,\sqrt{3} mx/2,0),(0,x,\frac{1}{2}\mid -mx/2,-\sqrt{3} mx/2,0),(-x,-x,\frac{1}{2}\mid mx,0,0)$
6h $(x,2 x,\frac{1}{4}\mid -mx/2,\sqrt{3} mx/2,0),(-2 x,-x,\frac{1}{4}\mid -mx/2,-\sqrt{3} mx/2,0),(x,-x,\frac{1}{4}\mid mx,0,0),(-x,-2 x,\frac{3}{4}\mid -mx/2,\sqrt{3} mx/2,0),(2 x,x,\frac{3}{4}\mid -mx/2,-\sqrt{3} mx/2,0),(-x,x,\frac{3}{4}\mid mx,0,0)$
6g $(\frac{1}{2},0,0\mid -mx/2,\sqrt{3} mx/2,0),(0,\frac{1}{2},0\mid -mx/2,-\sqrt{3} mx/2,0),(\frac{1}{2},\frac{1}{2},0\mid mx,0,0),(\frac{1}{2},0,\frac{1}{2}\mid -mx/2,\sqrt{3} mx/2,0),(0,\frac{1}{2},\frac{1}{2}\mid -mx/2,-\sqrt{3} mx/2,0),(\frac{1}{2},\frac{1}{2},\frac{1}{2}\mid mx,0,0)$
4f $(\frac{1}{3},\frac{2}{3},z\mid 0,0,0),(\frac{2}{3},\frac{1}{3},z + \frac{1}{2}\mid 0,0,0),(\frac{2}{3},\frac{1}{3},-z\mid 0,0,0),(\frac{1}{3},\frac{2}{3},\frac{1}{2} - z\mid 0,0,0)$
4e $(0,0,z\mid 0,0,0),(0,0,z + \frac{1}{2}\mid 0,0,0),(0,0,-z\mid 0,0,0),(0,0,\frac{1}{2} - z\mid 0,0,0)$
2d $(\frac{1}{3},\frac{2}{3},\frac{3}{4}\mid 0,0,0),(\frac{2}{3},\frac{1}{3},\frac{1}{4}\mid 0,0,0)$
2c $(\frac{1}{3},\frac{2}{3},\frac{1}{4}\mid 0,0,0),(\frac{2}{3},\frac{1}{3},\frac{3}{4}\mid 0,0,0)$
2b $(0,0,\frac{1}{4}\mid 0,0,0),(0,0,\frac{3}{4}\mid 0,0,0)$
2a $(0,0,0\mid 0,0,0),(0,0,\frac{1}{2}\mid 0,0,0)$
Site symmetry: $^{1}1^{m}1$
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 $\left\{\ce{^{m_{001}}{1}}||1|0 0 0\right\}$ $(x,y,z,-1,u,v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
Site symmetry: $.^{2_{\frac{2\pi}{3}}}m.^{m}1$
op_index Seitz coordinate spin_expr space_expr
1 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
4 $\left\{2_{\frac{2\pi}{3}}||m_{100}|0 0 0\right\}$ $(-x + y,y,z,+1,\cos(\frac{4\pi}{3}) u + \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u - \cos(\frac{4\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & \sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & -\cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
extra__U_E_0(without_identity)
op_index Seitz coordinate spin_expr space_expr
1 $\left\{\ce{^{m_{001}}{1}}||1|0 0 0\right\}$ $(x,y,z,-1,u,v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
Site symmetry: $^{1}m..^{m}1$
op_index Seitz coordinate spin_expr space_expr
1 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
3 $\left\{1||m_{001}|0 0 \frac{1}{2}\right\}$ $(x,y,\frac{1}{2} - z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
extra__U_E_0(without_identity)
op_index Seitz coordinate spin_expr space_expr
1 $\left\{\ce{^{m_{001}}{1}}||1|0 0 0\right\}$ $(x,y,z,-1,u,v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
Site symmetry: $.^{2_{\frac{2\pi}{3}}}2.^{m}1$
op_index Seitz coordinate spin_expr space_expr
1 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
22 $\left\{2_{\frac{2\pi}{3}}||2_{100}|0 0 0\right\}$ $(x - y,-y,-z,+1,\cos(\frac{4\pi}{3}) u + \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u - \cos(\frac{4\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & \sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & -\cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & -1 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
extra__U_E_0(without_identity)
op_index Seitz coordinate spin_expr space_expr
1 $\left\{\ce{^{m_{001}}{1}}||1|0 0 0\right\}$ $(x,y,z,-1,u,v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
Site symmetry: $^{2_{\frac{2\pi}{3}}}m^{1}m^{2_{\frac{2\pi}{3}}}2^{m}1$
op_index Seitz coordinate spin_expr space_expr
1 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
3 $\left\{1||m_{001}|0 0 \frac{1}{2}\right\}$ $(x,y,\frac{1}{2} - z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
4 $\left\{2_{\frac{2\pi}{3}}||m_{100}|0 0 0\right\}$ $(-x + y,y,z,+1,\cos(\frac{4\pi}{3}) u + \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u - \cos(\frac{4\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & \sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & -\cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
19 $\left\{2_{\frac{2\pi}{3}}||2_{120}|0 0 \frac{1}{2}\right\}$ $(-x + y,y,\frac{1}{2} - z,+1,\cos(\frac{4\pi}{3}) u + \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u - \cos(\frac{4\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & \sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & -\cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
extra__U_E_0(without_identity)
op_index Seitz coordinate spin_expr space_expr
1 $\left\{\ce{^{m_{001}}{1}}||1|0 0 0\right\}$ $(x,y,z,-1,u,v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
Site symmetry: $.^{2_{\frac{2\pi}{3}}}2/^{2_{\frac{2\pi}{3}}}m.^{m}1$
op_index Seitz coordinate spin_expr space_expr
1 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
4 $\left\{2_{\frac{2\pi}{3}}||m_{100}|0 0 0\right\}$ $(-x + y,y,z,+1,\cos(\frac{4\pi}{3}) u + \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u - \cos(\frac{4\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & \sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & -\cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
13 $\left\{1||-1|0 0 0\right\}$ $(-x,-y,-z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
22 $\left\{2_{\frac{2\pi}{3}}||2_{100}|0 0 0\right\}$ $(x - y,-y,-z,+1,\cos(\frac{4\pi}{3}) u + \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u - \cos(\frac{4\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & \sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & -\cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & -1 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
extra__U_E_0(without_identity)
op_index Seitz coordinate spin_expr space_expr
1 $\left\{\ce{^{m_{001}}{1}}||1|0 0 0\right\}$ $(x,y,z,-1,u,v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
Site symmetry: $^{3^{1}_{001}}3^{2_{\frac{2\pi}{3}}}m.^{m}1$
op_index Seitz coordinate spin_expr space_expr
1 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
4 $\left\{2_{\frac{2\pi}{3}}||m_{100}|0 0 0\right\}$ $(-x + y,y,z,+1,\cos(\frac{4\pi}{3}) u + \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u - \cos(\frac{4\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & \sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & -\cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
6 $\left\{3^{1}_{001}||3^{1}_{001}|0 0 0\right\}$ $(-y,x - y,z,+1,\cos(\frac{2\pi}{3}) u - \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u + \cos(\frac{2\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & -\sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & \cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
8 $\left\{2_{100}||m_{110}|0 0 0\right\}$ $(-y,-x,z,+1,u,-v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
12 $\left\{3^{2}_{001}||3^{2}_{001}|0 0 0\right\}$ $(-x + y,-x,z,+1,\cos(\frac{4\pi}{3}) u - \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u + \cos(\frac{4\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & -\sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & \cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
14 $\left\{2_{\frac{\pi}{3}}||m_{010}|0 0 0\right\}$ $(x,x - y,z,+1,\cos(\frac{2\pi}{3}) u + \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u - \cos(\frac{2\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & \sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & -\cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
extra__U_E_0(without_identity)
op_index Seitz coordinate spin_expr space_expr
1 $\left\{\ce{^{m_{001}}{1}}||1|0 0 0\right\}$ $(x,y,z,-1,u,v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
Site symmetry: $^{3^{1}_{001}}3^{2_{\frac{2\pi}{3}}}m.^{m}1$
op_index Seitz coordinate spin_expr space_expr
1 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
4 $\left\{2_{\frac{2\pi}{3}}||m_{100}|0 0 0\right\}$ $(-x + y,y,z,+1,\cos(\frac{4\pi}{3}) u + \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u - \cos(\frac{4\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & \sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & -\cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
6 $\left\{3^{1}_{001}||3^{1}_{001}|0 0 0\right\}$ $(-y,x - y,z,+1,\cos(\frac{2\pi}{3}) u - \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u + \cos(\frac{2\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & -\sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & \cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
8 $\left\{2_{100}||m_{110}|0 0 0\right\}$ $(-y,-x,z,+1,u,-v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
12 $\left\{3^{2}_{001}||3^{2}_{001}|0 0 0\right\}$ $(-x + y,-x,z,+1,\cos(\frac{4\pi}{3}) u - \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u + \cos(\frac{4\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & -\sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & \cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
14 $\left\{2_{\frac{\pi}{3}}||m_{010}|0 0 0\right\}$ $(x,x - y,z,+1,\cos(\frac{2\pi}{3}) u + \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u - \cos(\frac{2\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & \sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & -\cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
extra__U_E_0(without_identity)
op_index Seitz coordinate spin_expr space_expr
1 $\left\{\ce{^{m_{001}}{1}}||1|0 0 0\right\}$ $(x,y,z,-1,u,v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
Site symmetry: $^{3^{2}_{001}}\!-\!6^{2_{100}}m^{2_{\frac{\pi}{3}}}2^{m}1$
op_index Seitz coordinate spin_expr space_expr
1 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
3 $\left\{1||m_{001}|0 0 \frac{1}{2}\right\}$ $(x,y,\frac{1}{2} - z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
4 $\left\{2_{\frac{2\pi}{3}}||m_{100}|0 0 0\right\}$ $(-x + y,y,z,+1,\cos(\frac{4\pi}{3}) u + \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u - \cos(\frac{4\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & \sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & -\cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
6 $\left\{3^{1}_{001}||3^{1}_{001}|0 0 0\right\}$ $(-y,x - y,z,+1,\cos(\frac{2\pi}{3}) u - \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u + \cos(\frac{2\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & -\sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & \cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
8 $\left\{2_{100}||m_{110}|0 0 0\right\}$ $(-y,-x,z,+1,u,-v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
10 $\left\{3^{1}_{001}||-6^{5}_{001}|0 0 \frac{1}{2}\right\}$ $(-y,x - y,\frac{1}{2} - z,+1,\cos(\frac{2\pi}{3}) u - \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u + \cos(\frac{2\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & -\sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & \cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
12 $\left\{3^{2}_{001}||3^{2}_{001}|0 0 0\right\}$ $(-x + y,-x,z,+1,\cos(\frac{4\pi}{3}) u - \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u + \cos(\frac{4\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & -\sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & \cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
14 $\left\{2_{\frac{\pi}{3}}||m_{010}|0 0 0\right\}$ $(x,x - y,z,+1,\cos(\frac{2\pi}{3}) u + \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u - \cos(\frac{2\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & \sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & -\cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
16 $\left\{3^{2}_{001}||-6^{1}_{001}|0 0 \frac{1}{2}\right\}$ $(-x + y,-x,\frac{1}{2} - z,+1,\cos(\frac{4\pi}{3}) u - \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u + \cos(\frac{4\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & -\sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & \cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
19 $\left\{2_{\frac{2\pi}{3}}||2_{120}|0 0 \frac{1}{2}\right\}$ $(-x + y,y,\frac{1}{2} - z,+1,\cos(\frac{4\pi}{3}) u + \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u - \cos(\frac{4\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & \sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & -\cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
21 $\left\{2_{100}||2_{1-10}|0 0 \frac{1}{2}\right\}$ $(-y,-x,\frac{1}{2} - z,+1,u,-v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
23 $\left\{2_{\frac{\pi}{3}}||2_{210}|0 0 \frac{1}{2}\right\}$ $(x,x - y,\frac{1}{2} - z,+1,\cos(\frac{2\pi}{3}) u + \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u - \cos(\frac{2\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & \sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & -\cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
extra__U_E_0(without_identity)
op_index Seitz coordinate spin_expr space_expr
1 $\left\{\ce{^{m_{001}}{1}}||1|0 0 0\right\}$ $(x,y,z,-1,u,v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
Site symmetry: $^{3^{2}_{001}}\!-\!6^{2_{100}}m^{2_{\frac{\pi}{3}}}2^{m}1$
op_index Seitz coordinate spin_expr space_expr
1 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
3 $\left\{1||m_{001}|0 0 \frac{1}{2}\right\}$ $(x,y,\frac{1}{2} - z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
4 $\left\{2_{\frac{2\pi}{3}}||m_{100}|0 0 0\right\}$ $(-x + y,y,z,+1,\cos(\frac{4\pi}{3}) u + \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u - \cos(\frac{4\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & \sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & -\cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
6 $\left\{3^{1}_{001}||3^{1}_{001}|0 0 0\right\}$ $(-y,x - y,z,+1,\cos(\frac{2\pi}{3}) u - \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u + \cos(\frac{2\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & -\sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & \cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
8 $\left\{2_{100}||m_{110}|0 0 0\right\}$ $(-y,-x,z,+1,u,-v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
10 $\left\{3^{1}_{001}||-6^{5}_{001}|0 0 \frac{1}{2}\right\}$ $(-y,x - y,\frac{1}{2} - z,+1,\cos(\frac{2\pi}{3}) u - \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u + \cos(\frac{2\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & -\sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & \cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
12 $\left\{3^{2}_{001}||3^{2}_{001}|0 0 0\right\}$ $(-x + y,-x,z,+1,\cos(\frac{4\pi}{3}) u - \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u + \cos(\frac{4\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & -\sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & \cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
14 $\left\{2_{\frac{\pi}{3}}||m_{010}|0 0 0\right\}$ $(x,x - y,z,+1,\cos(\frac{2\pi}{3}) u + \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u - \cos(\frac{2\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & \sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & -\cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
16 $\left\{3^{2}_{001}||-6^{1}_{001}|0 0 \frac{1}{2}\right\}$ $(-x + y,-x,\frac{1}{2} - z,+1,\cos(\frac{4\pi}{3}) u - \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u + \cos(\frac{4\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & -\sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & \cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
19 $\left\{2_{\frac{2\pi}{3}}||2_{120}|0 0 \frac{1}{2}\right\}$ $(-x + y,y,\frac{1}{2} - z,+1,\cos(\frac{4\pi}{3}) u + \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u - \cos(\frac{4\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & \sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & -\cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
21 $\left\{2_{100}||2_{1-10}|0 0 \frac{1}{2}\right\}$ $(-y,-x,\frac{1}{2} - z,+1,u,-v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
23 $\left\{2_{\frac{\pi}{3}}||2_{210}|0 0 \frac{1}{2}\right\}$ $(x,x - y,\frac{1}{2} - z,+1,\cos(\frac{2\pi}{3}) u + \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u - \cos(\frac{2\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & \sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & -\cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
extra__U_E_0(without_identity)
op_index Seitz coordinate spin_expr space_expr
1 $\left\{\ce{^{m_{001}}{1}}||1|0 0 0\right\}$ $(x,y,z,-1,u,v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
Site symmetry: $^{3^{2}_{001}}\!-\!6^{2_{100}}m^{2_{\frac{\pi}{3}}}2^{m}1$
op_index Seitz coordinate spin_expr space_expr
1 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
3 $\left\{1||m_{001}|0 0 \frac{1}{2}\right\}$ $(x,y,\frac{1}{2} - z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
4 $\left\{2_{\frac{2\pi}{3}}||m_{100}|0 0 0\right\}$ $(-x + y,y,z,+1,\cos(\frac{4\pi}{3}) u + \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u - \cos(\frac{4\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & \sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & -\cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
6 $\left\{3^{1}_{001}||3^{1}_{001}|0 0 0\right\}$ $(-y,x - y,z,+1,\cos(\frac{2\pi}{3}) u - \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u + \cos(\frac{2\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & -\sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & \cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
8 $\left\{2_{100}||m_{110}|0 0 0\right\}$ $(-y,-x,z,+1,u,-v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
10 $\left\{3^{1}_{001}||-6^{5}_{001}|0 0 \frac{1}{2}\right\}$ $(-y,x - y,\frac{1}{2} - z,+1,\cos(\frac{2\pi}{3}) u - \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u + \cos(\frac{2\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & -\sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & \cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
12 $\left\{3^{2}_{001}||3^{2}_{001}|0 0 0\right\}$ $(-x + y,-x,z,+1,\cos(\frac{4\pi}{3}) u - \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u + \cos(\frac{4\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & -\sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & \cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
14 $\left\{2_{\frac{\pi}{3}}||m_{010}|0 0 0\right\}$ $(x,x - y,z,+1,\cos(\frac{2\pi}{3}) u + \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u - \cos(\frac{2\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & \sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & -\cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
16 $\left\{3^{2}_{001}||-6^{1}_{001}|0 0 \frac{1}{2}\right\}$ $(-x + y,-x,\frac{1}{2} - z,+1,\cos(\frac{4\pi}{3}) u - \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u + \cos(\frac{4\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & -\sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & \cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
19 $\left\{2_{\frac{2\pi}{3}}||2_{120}|0 0 \frac{1}{2}\right\}$ $(-x + y,y,\frac{1}{2} - z,+1,\cos(\frac{4\pi}{3}) u + \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u - \cos(\frac{4\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & \sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & -\cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
21 $\left\{2_{100}||2_{1-10}|0 0 \frac{1}{2}\right\}$ $(-y,-x,\frac{1}{2} - z,+1,u,-v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
23 $\left\{2_{\frac{\pi}{3}}||2_{210}|0 0 \frac{1}{2}\right\}$ $(x,x - y,\frac{1}{2} - z,+1,\cos(\frac{2\pi}{3}) u + \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u - \cos(\frac{2\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & \sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & -\cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
extra__U_E_0(without_identity)
op_index Seitz coordinate spin_expr space_expr
1 $\left\{\ce{^{m_{001}}{1}}||1|0 0 0\right\}$ $(x,y,z,-1,u,v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
Site symmetry: $^{3^{1}_{001}}\!-\!3^{2_{\frac{2\pi}{3}}}m.^{m}1$
op_index Seitz coordinate spin_expr space_expr
1 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
4 $\left\{2_{\frac{2\pi}{3}}||m_{100}|0 0 0\right\}$ $(-x + y,y,z,+1,\cos(\frac{4\pi}{3}) u + \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u - \cos(\frac{4\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & \sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & -\cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
6 $\left\{3^{1}_{001}||3^{1}_{001}|0 0 0\right\}$ $(-y,x - y,z,+1,\cos(\frac{2\pi}{3}) u - \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u + \cos(\frac{2\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & -\sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & \cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
7 $\left\{3^{2}_{001}||-3^{2}_{001}|0 0 0\right\}$ $(x - y,x,-z,+1,\cos(\frac{4\pi}{3}) u - \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u + \cos(\frac{4\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & -\sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & \cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
8 $\left\{2_{100}||m_{110}|0 0 0\right\}$ $(-y,-x,z,+1,u,-v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
12 $\left\{3^{2}_{001}||3^{2}_{001}|0 0 0\right\}$ $(-x + y,-x,z,+1,\cos(\frac{4\pi}{3}) u - \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u + \cos(\frac{4\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & -\sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & \cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
13 $\left\{1||-1|0 0 0\right\}$ $(-x,-y,-z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
14 $\left\{2_{\frac{\pi}{3}}||m_{010}|0 0 0\right\}$ $(x,x - y,z,+1,\cos(\frac{2\pi}{3}) u + \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u - \cos(\frac{2\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & \sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & -\cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
18 $\left\{3^{1}_{001}||-3^{1}_{001}|0 0 0\right\}$ $(y,-x + y,-z,+1,\cos(\frac{2\pi}{3}) u - \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u + \cos(\frac{2\pi}{3}) v,w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & -\sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & \cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
20 $\left\{2_{\frac{\pi}{3}}||2_{010}|0 0 0\right\}$ $(-x,-x + y,-z,+1,\cos(\frac{2\pi}{3}) u + \sin(\frac{2\pi}{3}) v,\sin(\frac{2\pi}{3}) u - \cos(\frac{2\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{2\pi}{3}) & \sin(\frac{2\pi}{3}) & 0 \\ \sin(\frac{2\pi}{3}) & -\cos(\frac{2\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
22 $\left\{2_{\frac{2\pi}{3}}||2_{100}|0 0 0\right\}$ $(x - y,-y,-z,+1,\cos(\frac{4\pi}{3}) u + \sin(\frac{4\pi}{3}) v,\sin(\frac{4\pi}{3}) u - \cos(\frac{4\pi}{3}) v,-w)$ $$\begin{bmatrix}\cos(\frac{4\pi}{3}) & \sin(\frac{4\pi}{3}) & 0 \\ \sin(\frac{4\pi}{3}) & -\cos(\frac{4\pi}{3}) & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & -1 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
24 $\left\{2_{100}||2_{110}|0 0 0\right\}$ $(y,x,-z,+1,u,-v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
extra__U_E_0(without_identity)
op_index Seitz coordinate spin_expr space_expr
1 $\left\{\ce{^{m_{001}}{1}}||1|0 0 0\right\}$ $(x,y,z,-1,u,v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$