182.4.4.2
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: Non-coplanar
Type: type-g
G0 / L0: 182 / 4
Identifier: 182.4.4.2
it / ik: 6 / 4
Nontrivial spin-space point group: $-43m$
Spin-space point group: $\!-\!43m$
International Notation: $$P\ce{^{3^{2}_{\text{-}11\text{-}1}}{6_{3}}}\ce{^{m_{110}}{2}}\ce{^{m_{011}}{2}} \vert (2_{001},2_{100},1)$$
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:
$$\left[\begin{array}{ccc|c}0 & 0 & -2 & 0 \\ -2 & 0 & 2 & 0 \\ 0 & 1 & 0 & \frac{1}{4}\end{array}\right]$$
Note: [a_G0, b_G0, c_G0] M = [a_L0, b_L0, c_L0]
Translational group generators:
$$\begin{aligned}a &= (2, 0, 0) \\ b &= (0, 2, 0) \\ c &= (0, 0, 1)\end{aligned}$$
Note: The listed operations below are understood modulo the translational group generated by a, b, and c.
E.g. for 182.4.4.2, if a=(2, 0, 0), then (x, y, z) and (x+2, 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 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
U_E_tau
op_index Seitz coordinate spin_expr space_expr
1 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
2 $\left\{2_{100}||1|0 1 0\right\}$ $(x,y + 1,z,+1,u,-v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0\end{array}\right]$$
3 $\left\{2_{001}||1|1 0 0\right\}$ $(x + 1,y,z,+1,-u,-v,w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
4 $\left\{2_{010}||1|1 1 0\right\}$ $(x + 1,y + 1,z,+1,-u,v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0\end{array}\right]$$
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 $\left\{3^{2}_{-11-1}||6^{1}_{001}|0 0 \frac{1}{2}\right\}$ $(x - y,x,z + \frac{1}{2},+1,-v,-w,u)$ $$\begin{bmatrix}0 & -1 & 0 \\ 0 & 0 & -1 \\ 1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
3 $\left\{m_{110}||2_{100}|0 0 0\right\}$ $(x - y,-y,-z,-1,-v,-u,w)$ $$\begin{bmatrix}0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & -1 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
4 $\left\{m_{011}||2_{210}|0 0 \frac{1}{2}\right\}$ $(x,x - y,\frac{1}{2} - z,-1,u,-w,-v)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & -1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
5 $\left\{3^{1}_{-11-1}||3^{1}_{001}|0 0 0\right\}$ $(-y,x - y,z,+1,w,-u,-v)$ $$\begin{bmatrix}0 & 0 & 1 \\ -1 & 0 & 0 \\ 0 & -1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
6 $\left\{m_{-101}||2_{110}|0 0 0\right\}$ $(y,x,-z,-1,w,v,u)$ $$\begin{bmatrix}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
7 $\left\{1||2_{001}|0 0 \frac{1}{2}\right\}$ $(-x,-y,z + \frac{1}{2},+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
8 $\left\{m_{110}||2_{120}|0 0 \frac{1}{2}\right\}$ $(-x + y,y,\frac{1}{2} - z,-1,-v,-u,w)$ $$\begin{bmatrix}0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
9 $\left\{3^{2}_{-11-1}||3^{2}_{001}|0 0 0\right\}$ $(-x + y,-x,z,+1,-v,-w,u)$ $$\begin{bmatrix}0 & -1 & 0 \\ 0 & 0 & -1 \\ 1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
10 $\left\{m_{011}||2_{010}|0 0 0\right\}$ $(-x,-x + y,-z,-1,u,-w,-v)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & -1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
11 $\left\{3^{1}_{-11-1}||6^{5}_{001}|0 0 \frac{1}{2}\right\}$ $(y,-x + y,z + \frac{1}{2},+1,w,-u,-v)$ $$\begin{bmatrix}0 & 0 & 1 \\ -1 & 0 & 0 \\ 0 & -1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
12 $\left\{m_{-101}||2_{1-10}|0 0 \frac{1}{2}\right\}$ $(-y,-x,\frac{1}{2} - z,-1,w,v,u)$ $$\begin{bmatrix}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
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 $\left\{m_{110}||2_{100}|0 0 0\right\}$ $(x - y,-y,-z,-1,-v,-u,w)$ $$\begin{bmatrix}0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & -1 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
3 $\left\{m_{011}||2_{210}|0 0 \frac{1}{2}\right\}$ $(x,x - y,\frac{1}{2} - z,-1,u,-w,-v)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & -1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
4 $\left\{2_{001}||1|1 0 0\right\}$ $(x + 1,y,z,+1,-u,-v,w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
5 $\left\{2_{100}||1|0 1 0\right\}$ $(x,y + 1,z,+1,u,-v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0\end{array}\right]$$
6 $\left\{3^{2}_{-11-1}||6^{1}_{001}|0 0 \frac{1}{2}\right\}$ $(x - y,x,z + \frac{1}{2},+1,-v,-w,u)$ $$\begin{bmatrix}0 & -1 & 0 \\ 0 & 0 & -1 \\ 1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
7 $\left\{3^{1}_{-11-1}||6^{5}_{001}|0 0 \frac{1}{2}\right\}$ $(y,-x + y,z + \frac{1}{2},+1,w,-u,-v)$ $$\begin{bmatrix}0 & 0 & 1 \\ -1 & 0 & 0 \\ 0 & -1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
8 $\left\{m_{1-10}||2_{100}|1 0 0\right\}$ $(x - y + 1,-y,-z,-1,v,u,w)$ $$\begin{bmatrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & -1 & 0 & 1 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
9 $\left\{-4^{1}_{001}||2_{100}|1 1 0\right\}$ $(x - y + 1,1 - y,-z,-1,v,-u,-w)$ $$\begin{bmatrix}0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & -1 & 0 & 1 \\ 0 & -1 & 0 & 1 \\ 0 & 0 & -1 & 0\end{array}\right]$$
10 $\left\{m_{-101}||2_{1-10}|0 0 \frac{1}{2}\right\}$ $(-y,-x,\frac{1}{2} - z,-1,w,v,u)$ $$\begin{bmatrix}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
11 $\left\{-4^{3}_{100}||2_{210}|1 1 \frac{1}{2}\right\}$ $(x + 1,x - y + 1,\frac{1}{2} - z,-1,-u,-w,v)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 1 \\ 1 & -1 & 0 & 1 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
12 $\left\{m_{01-1}||2_{210}|0 1 \frac{1}{2}\right\}$ $(x,x - y + 1,\frac{1}{2} - z,-1,u,w,v)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 1 & -1 & 0 & 1 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
13 $\left\{m_{-101}||2_{110}|0 0 0\right\}$ $(y,x,-z,-1,w,v,u)$ $$\begin{bmatrix}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
14 $\left\{3^{2}_{-1-11}||6^{1}_{001}|1 1 \frac{1}{2}\right\}$ $(x - y + 1,x + 1,z + \frac{1}{2},+1,v,-w,-u)$ $$\begin{bmatrix}0 & 1 & 0 \\ 0 & 0 & -1 \\ -1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & -1 & 0 & 1 \\ 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
15 $\left\{3^{2}_{111}||6^{1}_{001}|1 0 \frac{1}{2}\right\}$ $(x - y + 1,x,z + \frac{1}{2},+1,v,w,u)$ $$\begin{bmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & -1 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
16 $\left\{3^{1}_{-11-1}||3^{1}_{001}|0 0 0\right\}$ $(-y,x - y,z,+1,w,-u,-v)$ $$\begin{bmatrix}0 & 0 & 1 \\ -1 & 0 & 0 \\ 0 & -1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
17 $\left\{-4^{1}_{100}||2_{210}|1 0 \frac{1}{2}\right\}$ $(x + 1,x - y,\frac{1}{2} - z,-1,-u,w,-v)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 1 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
18 $\left\{2_{010}||1|1 1 0\right\}$ $(x + 1,y + 1,z,+1,-u,v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0\end{array}\right]$$
19 $\left\{3^{1}_{-1-11}||6^{5}_{001}|1 0 \frac{1}{2}\right\}$ $(y + 1,-x + y,z + \frac{1}{2},+1,-w,u,-v)$ $$\begin{bmatrix}0 & 0 & -1 \\ 1 & 0 & 0 \\ 0 & -1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 1 \\ -1 & 1 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
20 $\left\{-4^{3}_{001}||2_{100}|0 1 0\right\}$ $(x - y,1 - y,-z,-1,-v,u,-w)$ $$\begin{bmatrix}0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & -1 & 0 & 0 \\ 0 & -1 & 0 & 1 \\ 0 & 0 & -1 & 0\end{array}\right]$$
21 $\left\{-4^{1}_{010}||2_{1-10}|1 0 \frac{1}{2}\right\}$ $(1 - y,-x,\frac{1}{2} - z,-1,-w,-v,u)$ $$\begin{bmatrix}0 & 0 & -1 \\ 0 & -1 & 0 \\ 1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 1 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
22 $\left\{-4^{1}_{010}||2_{110}|1 0 0\right\}$ $(y + 1,x,-z,-1,-w,-v,u)$ $$\begin{bmatrix}0 & 0 & -1 \\ 0 & -1 & 0 \\ 1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
23 $\left\{3^{2}_{1-1-1}||6^{1}_{001}|0 1 \frac{1}{2}\right\}$ $(x - y,x + 1,z + \frac{1}{2},+1,-v,w,-u)$ $$\begin{bmatrix}0 & -1 & 0 \\ 0 & 0 & 1 \\ -1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & -1 & 0 & 0 \\ 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
24 $\left\{3^{1}_{-1-11}||3^{1}_{001}|1 0 0\right\}$ $(1 - y,x - y,z,+1,-w,u,-v)$ $$\begin{bmatrix}0 & 0 & -1 \\ 1 & 0 & 0 \\ 0 & -1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 1 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
25 $\left\{3^{1}_{111}||6^{5}_{001}|0 1 \frac{1}{2}\right\}$ $(y,-x + y + 1,z + \frac{1}{2},+1,w,u,v)$ $$\begin{bmatrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ -1 & 1 & 0 & 1 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
26 $\left\{-4^{3}_{010}||2_{1-10}|0 1 \frac{1}{2}\right\}$ $(-y,1 - x,\frac{1}{2} - z,-1,w,-v,-u)$ $$\begin{bmatrix}0 & 0 & 1 \\ 0 & -1 & 0 \\ -1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 1 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
27 $\left\{-4^{3}_{010}||2_{110}|0 1 0\right\}$ $(y,x + 1,-z,-1,w,-v,-u)$ $$\begin{bmatrix}0 & 0 & 1 \\ 0 & -1 & 0 \\ -1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 \\ 0 & 0 & -1 & 0\end{array}\right]$$
28 $\left\{3^{1}_{111}||3^{1}_{001}|0 1 0\right\}$ $(-y,x - y + 1,z,+1,w,u,v)$ $$\begin{bmatrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ 1 & -1 & 0 & 1 \\ 0 & 0 & 1 & 0\end{array}\right]$$
29 $\left\{3^{1}_{1-1-1}||6^{5}_{001}|1 1 \frac{1}{2}\right\}$ $(y + 1,-x + y + 1,z + \frac{1}{2},+1,-w,-u,v)$ $$\begin{bmatrix}0 & 0 & -1 \\ -1 & 0 & 0 \\ 0 & 1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 1 \\ -1 & 1 & 0 & 1 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
30 $\left\{m_{101}||2_{1-10}|1 1 \frac{1}{2}\right\}$ $(1 - y,1 - x,\frac{1}{2} - z,-1,-w,v,-u)$ $$\begin{bmatrix}0 & 0 & -1 \\ 0 & 1 & 0 \\ -1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 1 \\ -1 & 0 & 0 & 1 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
31 $\left\{m_{101}||2_{110}|1 1 0\right\}$ $(y + 1,x + 1,-z,-1,-w,v,-u)$ $$\begin{bmatrix}0 & 0 & -1 \\ 0 & 1 & 0 \\ -1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 1 \\ 1 & 0 & 0 & 1 \\ 0 & 0 & -1 & 0\end{array}\right]$$
32 $\left\{3^{1}_{1-1-1}||3^{1}_{001}|1 1 0\right\}$ $(1 - y,x - y + 1,z,+1,-w,-u,v)$ $$\begin{bmatrix}0 & 0 & -1 \\ -1 & 0 & 0 \\ 0 & 1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 1 \\ 1 & -1 & 0 & 1 \\ 0 & 0 & 1 & 0\end{array}\right]$$
33 $\left\{m_{110}||2_{120}|0 0 \frac{1}{2}\right\}$ $(-x + y,y,\frac{1}{2} - z,-1,-v,-u,w)$ $$\begin{bmatrix}0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
34 $\left\{1||2_{001}|0 0 \frac{1}{2}\right\}$ $(-x,-y,z + \frac{1}{2},+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
35 $\left\{-4^{1}_{001}||2_{120}|1 1 \frac{1}{2}\right\}$ $(-x + y + 1,y + 1,\frac{1}{2} - z,-1,v,-u,-w)$ $$\begin{bmatrix}0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 1 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
36 $\left\{2_{010}||2_{001}|1 1 \frac{1}{2}\right\}$ $(1 - x,1 - y,z + \frac{1}{2},+1,-u,v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 1 \\ 0 & -1 & 0 & 1 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
37 $\left\{m_{1-10}||2_{120}|1 0 \frac{1}{2}\right\}$ $(-x + y + 1,y,\frac{1}{2} - z,-1,v,u,w)$ $$\begin{bmatrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
38 $\left\{2_{001}||2_{001}|1 0 \frac{1}{2}\right\}$ $(1 - x,-y,z + \frac{1}{2},+1,-u,-v,w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 1 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
39 $\left\{-4^{3}_{001}||2_{120}|0 1 \frac{1}{2}\right\}$ $(-x + y,y + 1,\frac{1}{2} - z,-1,-v,u,-w)$ $$\begin{bmatrix}0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
40 $\left\{2_{100}||2_{001}|0 1 \frac{1}{2}\right\}$ $(-x,1 - y,z + \frac{1}{2},+1,u,-v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 1 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
41 $\left\{3^{2}_{-11-1}||3^{2}_{001}|0 0 0\right\}$ $(-x + y,-x,z,+1,-v,-w,u)$ $$\begin{bmatrix}0 & -1 & 0 \\ 0 & 0 & -1 \\ 1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
42 $\left\{m_{011}||2_{010}|0 0 0\right\}$ $(-x,-x + y,-z,-1,u,-w,-v)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & -1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
43 $\left\{3^{2}_{1-1-1}||3^{2}_{001}|0 1 0\right\}$ $(-x + y,1 - x,z,+1,-v,w,-u)$ $$\begin{bmatrix}0 & -1 & 0 \\ 0 & 0 & 1 \\ -1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ -1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0\end{array}\right]$$
44 $\left\{m_{01-1}||2_{010}|0 1 0\right\}$ $(-x,-x + y + 1,-z,-1,u,w,v)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ -1 & 1 & 0 & 1 \\ 0 & 0 & -1 & 0\end{array}\right]$$
45 $\left\{-4^{3}_{100}||2_{010}|1 1 0\right\}$ $(1 - x,-x + y + 1,-z,-1,-u,-w,v)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 1 \\ -1 & 1 & 0 & 1 \\ 0 & 0 & -1 & 0\end{array}\right]$$
46 $\left\{3^{2}_{-1-11}||3^{2}_{001}|1 1 0\right\}$ $(-x + y + 1,1 - x,z,+1,v,-w,-u)$ $$\begin{bmatrix}0 & 1 & 0 \\ 0 & 0 & -1 \\ -1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 1 \\ -1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0\end{array}\right]$$
47 $\left\{-4^{1}_{100}||2_{010}|1 0 0\right\}$ $(1 - x,-x + y,-z,-1,-u,w,-v)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 1 \\ -1 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
48 $\left\{3^{2}_{111}||3^{2}_{001}|1 0 0\right\}$ $(-x + y + 1,-x,z,+1,v,w,u)$ $$\begin{bmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 1 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
wyckoff position site symmetry Coordinates
$(0,0,0\mid mx,my,mz)+(1,0,0\mid -mx,-my,mz)+(0,1,0\mid mx,-my,-mz)+(1,1,0\mid -mx,my,-mz)$
48i $(x,y,z\mid mx,my,mz),(-y,x - y,z\mid mz,-mx,-my),(-x + y,-x,z\mid -my,-mz,mx),(-x,-y,z + \frac{1}{2}\mid mx,my,mz),(y,-x + y,z + \frac{1}{2}\mid mz,-mx,-my),(x - y,x,z + \frac{1}{2}\mid -my,-mz,mx),(y,x,-z\mid mz,my,mx),(x - y,-y,-z\mid -my,-mx,mz),(-x,-x + y,-z\mid mx,-mz,-my),(-y,-x,\frac{1}{2} - z\mid mz,my,mx),(-x + y,y,\frac{1}{2} - z\mid -my,-mx,mz),(x,x - y,\frac{1}{2} - z\mid mx,-mz,-my)$
24h $(x,2 x,\frac{1}{4}\mid -\sqrt{2} mx/2,\sqrt{2} mx/2,my),(-2 x,-x,\frac{1}{4}\mid my,\sqrt{2} mx/2,-\sqrt{2} mx/2),(x,-x,\frac{1}{4}\mid -\sqrt{2} mx/2,-my,-\sqrt{2} mx/2),(-x,-2 x,\frac{3}{4}\mid -\sqrt{2} mx/2,\sqrt{2} mx/2,my),(2 x,x,\frac{3}{4}\mid my,\sqrt{2} mx/2,-\sqrt{2} mx/2),(-x,x,\frac{3}{4}\mid -\sqrt{2} mx/2,-my,-\sqrt{2} mx/2)$
24g $(x,0,0\mid -\sqrt{2} mx/2,\sqrt{2} mx/2,my),(0,x,0\mid my,\sqrt{2} mx/2,-\sqrt{2} mx/2),(-x,-x,0\mid -\sqrt{2} mx/2,-my,-\sqrt{2} mx/2),(-x,0,\frac{1}{2}\mid -\sqrt{2} mx/2,\sqrt{2} mx/2,my),(0,-x,\frac{1}{2}\mid my,\sqrt{2} mx/2,-\sqrt{2} mx/2),(x,x,\frac{1}{2}\mid -\sqrt{2} mx/2,-my,-\sqrt{2} mx/2)$
16f $(\frac{1}{3},\frac{2}{3},z\mid -\sqrt{3} mx/3,\sqrt{3} mx/3,\sqrt{3} mx/3),(\frac{2}{3},\frac{1}{3},z + \frac{1}{2}\mid \sqrt{3} mx/3,\sqrt{3} mx/3,-\sqrt{3} mx/3),(\frac{2}{3},\frac{1}{3},-z\mid \sqrt{3} mx/3,\sqrt{3} mx/3,-\sqrt{3} mx/3),(\frac{1}{3},\frac{2}{3},\frac{1}{2} - z\mid -\sqrt{3} mx/3,\sqrt{3} mx/3,\sqrt{3} mx/3)$
16e $(0,0,z\mid \sqrt{3} mx/3,-\sqrt{3} mx/3,\sqrt{3} mx/3),(0,0,z + \frac{1}{2}\mid \sqrt{3} mx/3,-\sqrt{3} mx/3,\sqrt{3} mx/3),(0,0,-z\mid \sqrt{3} mx/3,-\sqrt{3} mx/3,\sqrt{3} mx/3),(0,0,\frac{1}{2} - z\mid \sqrt{3} mx/3,-\sqrt{3} mx/3,\sqrt{3} mx/3)$
8d $(\frac{1}{3},\frac{2}{3},\frac{3}{4}\mid -\sqrt{3} mx/3,\sqrt{3} mx/3,\sqrt{3} mx/3),(\frac{2}{3},\frac{1}{3},\frac{1}{4}\mid \sqrt{3} mx/3,\sqrt{3} mx/3,-\sqrt{3} mx/3)$
8c $(\frac{1}{3},\frac{2}{3},\frac{1}{4}\mid -\sqrt{3} mx/3,\sqrt{3} mx/3,\sqrt{3} mx/3),(\frac{2}{3},\frac{1}{3},\frac{3}{4}\mid \sqrt{3} mx/3,\sqrt{3} mx/3,-\sqrt{3} mx/3)$
8b $(0,0,\frac{1}{4}\mid \sqrt{3} mx/3,-\sqrt{3} mx/3,\sqrt{3} mx/3),(0,0,\frac{3}{4}\mid \sqrt{3} mx/3,-\sqrt{3} mx/3,\sqrt{3} mx/3)$
8a $(0,0,0\mid \sqrt{3} mx/3,-\sqrt{3} mx/3,\sqrt{3} mx/3),(0,0,\frac{1}{2}\mid \sqrt{3} mx/3,-\sqrt{3} mx/3,\sqrt{3} mx/3)$
Site symmetry: $^{1}1$
op_index Seitz coordinate spin_expr space_expr
1 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
Site symmetry: $..^{m_{110}}2$
op_index Seitz coordinate spin_expr space_expr
1 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
33 $\left\{m_{110}||2_{120}|0 0 \frac{1}{2}\right\}$ $(-x + y,y,\frac{1}{2} - z,-1,-v,-u,w)$ $$\begin{bmatrix}0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
Site symmetry: $.^{m_{110}}2.$
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 $\left\{m_{110}||2_{100}|0 0 0\right\}$ $(x - y,-y,-z,-1,-v,-u,w)$ $$\begin{bmatrix}0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & -1 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
Site symmetry: $^{3^{1}_{1\!-\!1\!-\!1}}3..$
op_index Seitz coordinate spin_expr space_expr
1 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
32 $\left\{3^{1}_{1-1-1}||3^{1}_{001}|1 1 0\right\}$ $(1 - y,x - y + 1,z,+1,-w,-u,v)$ $$\begin{bmatrix}0 & 0 & -1 \\ -1 & 0 & 0 \\ 0 & 1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 1 \\ 1 & -1 & 0 & 1 \\ 0 & 0 & 1 & 0\end{array}\right]$$
43 $\left\{3^{2}_{1-1-1}||3^{2}_{001}|0 1 0\right\}$ $(-x + y,1 - x,z,+1,-v,w,-u)$ $$\begin{bmatrix}0 & -1 & 0 \\ 0 & 0 & 1 \\ -1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ -1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0\end{array}\right]$$
Site symmetry: $^{3^{1}_{\!-\!11\!-\!1}}3..$
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 $\left\{3^{1}_{-11-1}||3^{1}_{001}|0 0 0\right\}$ $(-y,x - y,z,+1,w,-u,-v)$ $$\begin{bmatrix}0 & 0 & 1 \\ -1 & 0 & 0 \\ 0 & -1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
41 $\left\{3^{2}_{-11-1}||3^{2}_{001}|0 0 0\right\}$ $(-x + y,-x,z,+1,-v,-w,u)$ $$\begin{bmatrix}0 & -1 & 0 \\ 0 & 0 & -1 \\ 1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
Site symmetry: $^{3^{1}_{1\!-\!1\!-\!1}}3.^{m_{01\!-\!1}}2$
op_index Seitz coordinate spin_expr space_expr
1 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
12 $\left\{m_{01-1}||2_{210}|0 1 \frac{1}{2}\right\}$ $(x,x - y + 1,\frac{1}{2} - z,-1,u,w,v)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 1 & -1 & 0 & 1 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
30 $\left\{m_{101}||2_{1-10}|1 1 \frac{1}{2}\right\}$ $(1 - y,1 - x,\frac{1}{2} - z,-1,-w,v,-u)$ $$\begin{bmatrix}0 & 0 & -1 \\ 0 & 1 & 0 \\ -1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 1 \\ -1 & 0 & 0 & 1 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
32 $\left\{3^{1}_{1-1-1}||3^{1}_{001}|1 1 0\right\}$ $(1 - y,x - y + 1,z,+1,-w,-u,v)$ $$\begin{bmatrix}0 & 0 & -1 \\ -1 & 0 & 0 \\ 0 & 1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 1 \\ 1 & -1 & 0 & 1 \\ 0 & 0 & 1 & 0\end{array}\right]$$
33 $\left\{m_{110}||2_{120}|0 0 \frac{1}{2}\right\}$ $(-x + y,y,\frac{1}{2} - z,-1,-v,-u,w)$ $$\begin{bmatrix}0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
43 $\left\{3^{2}_{1-1-1}||3^{2}_{001}|0 1 0\right\}$ $(-x + y,1 - x,z,+1,-v,w,-u)$ $$\begin{bmatrix}0 & -1 & 0 \\ 0 & 0 & 1 \\ -1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ -1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0\end{array}\right]$$
Site symmetry: $^{3^{1}_{1\!-\!1\!-\!1}}3.^{m_{01\!-\!1}}2$
op_index Seitz coordinate spin_expr space_expr
1 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
12 $\left\{m_{01-1}||2_{210}|0 1 \frac{1}{2}\right\}$ $(x,x - y + 1,\frac{1}{2} - z,-1,u,w,v)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 1 & -1 & 0 & 1 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
30 $\left\{m_{101}||2_{1-10}|1 1 \frac{1}{2}\right\}$ $(1 - y,1 - x,\frac{1}{2} - z,-1,-w,v,-u)$ $$\begin{bmatrix}0 & 0 & -1 \\ 0 & 1 & 0 \\ -1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 1 \\ -1 & 0 & 0 & 1 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
32 $\left\{3^{1}_{1-1-1}||3^{1}_{001}|1 1 0\right\}$ $(1 - y,x - y + 1,z,+1,-w,-u,v)$ $$\begin{bmatrix}0 & 0 & -1 \\ -1 & 0 & 0 \\ 0 & 1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 1 \\ 1 & -1 & 0 & 1 \\ 0 & 0 & 1 & 0\end{array}\right]$$
33 $\left\{m_{110}||2_{120}|0 0 \frac{1}{2}\right\}$ $(-x + y,y,\frac{1}{2} - z,-1,-v,-u,w)$ $$\begin{bmatrix}0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
43 $\left\{3^{2}_{1-1-1}||3^{2}_{001}|0 1 0\right\}$ $(-x + y,1 - x,z,+1,-v,w,-u)$ $$\begin{bmatrix}0 & -1 & 0 \\ 0 & 0 & 1 \\ -1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ -1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0\end{array}\right]$$
Site symmetry: $^{3^{1}_{\!-\!11\!-\!1}}3.^{m_{011}}2$
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 $\left\{m_{011}||2_{210}|0 0 \frac{1}{2}\right\}$ $(x,x - y,\frac{1}{2} - z,-1,u,-w,-v)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & -1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
10 $\left\{m_{-101}||2_{1-10}|0 0 \frac{1}{2}\right\}$ $(-y,-x,\frac{1}{2} - z,-1,w,v,u)$ $$\begin{bmatrix}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
16 $\left\{3^{1}_{-11-1}||3^{1}_{001}|0 0 0\right\}$ $(-y,x - y,z,+1,w,-u,-v)$ $$\begin{bmatrix}0 & 0 & 1 \\ -1 & 0 & 0 \\ 0 & -1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
33 $\left\{m_{110}||2_{120}|0 0 \frac{1}{2}\right\}$ $(-x + y,y,\frac{1}{2} - z,-1,-v,-u,w)$ $$\begin{bmatrix}0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
41 $\left\{3^{2}_{-11-1}||3^{2}_{001}|0 0 0\right\}$ $(-x + y,-x,z,+1,-v,-w,u)$ $$\begin{bmatrix}0 & -1 & 0 \\ 0 & 0 & -1 \\ 1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
Site symmetry: $^{3^{1}_{\!-\!11\!-\!1}}3^{m_{110}}2.$
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 $\left\{m_{110}||2_{100}|0 0 0\right\}$ $(x - y,-y,-z,-1,-v,-u,w)$ $$\begin{bmatrix}0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & -1 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
13 $\left\{m_{-101}||2_{110}|0 0 0\right\}$ $(y,x,-z,-1,w,v,u)$ $$\begin{bmatrix}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
16 $\left\{3^{1}_{-11-1}||3^{1}_{001}|0 0 0\right\}$ $(-y,x - y,z,+1,w,-u,-v)$ $$\begin{bmatrix}0 & 0 & 1 \\ -1 & 0 & 0 \\ 0 & -1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
41 $\left\{3^{2}_{-11-1}||3^{2}_{001}|0 0 0\right\}$ $(-x + y,-x,z,+1,-v,-w,u)$ $$\begin{bmatrix}0 & -1 & 0 \\ 0 & 0 & -1 \\ 1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
42 $\left\{m_{011}||2_{010}|0 0 0\right\}$ $(-x,-x + y,-z,-1,u,-w,-v)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & -1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$