107.99.4.1.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-k
G0 / L0: 107 / 99
Identifier: 107.99.4.1.P
it / ik: 1 / 4
Nontrivial spin-space point group: $4$
Spin-space point group: $4/m$
International Notation: $$P\ce{^{1}{4}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{4^{1}_{001}}{(1/2 \ 1/2 \ 1/4)}}\ce{^{m_{001}}{1}}$$
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:
$$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 2 & 0\end{array}\right]$$
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 107.99.4.1.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 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
2 $\left\{4^{1}_{001}||1|\frac{1}{2} \frac{1}{2} \frac{1}{4}\right\}$ $(x + \frac{1}{2},y + \frac{1}{2},z + \frac{1}{4},+1,-v,u,w)$ $$\begin{bmatrix}0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & \frac{1}{2} \\ 0 & 1 & 0 & \frac{1}{2} \\ 0 & 0 & 1 & \frac{1}{4}\end{array}\right]$$
3 $\left\{2_{001}||1|0 0 \frac{1}{2}\right\}$ $(x,y,z + \frac{1}{2},+1,-u,-v,w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
4 $\left\{4^{3}_{001}||1|\frac{1}{2} \frac{1}{2} \frac{3}{4}\right\}$ $(x + \frac{1}{2},y + \frac{1}{2},z + \frac{3}{4},+1,v,-u,w)$ $$\begin{bmatrix}0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & \frac{1}{2} \\ 0 & 1 & 0 & \frac{1}{2} \\ 0 & 0 & 1 & \frac{3}{4}\end{array}\right]$$
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 $\left\{1||4^{1}_{001}|0 0 0\right\}$ $(-y,x,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
3 $\left\{1||m_{100}|0 0 0\right\}$ $(-x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
4 $\left\{1||m_{110}|0 0 0\right\}$ $(-y,-x,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
5 $\left\{1||2_{001}|0 0 0\right\}$ $(-x,-y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
6 $\left\{1||m_{010}|0 0 0\right\}$ $(x,-y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
7 $\left\{1||4^{3}_{001}|0 0 0\right\}$ $(y,-x,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
8 $\left\{1||m_{1-10}|0 0 0\right\}$ $(y,x,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
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 $\left\{1||4^{1}_{001}|0 0 0\right\}$ $(-y,x,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
3 $\left\{1||m_{100}|0 0 0\right\}$ $(-x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
4 $\left\{1||m_{110}|0 0 0\right\}$ $(-y,-x,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
5 $\left\{4^{1}_{001}||1|\frac{1}{2} \frac{1}{2} \frac{1}{4}\right\}$ $(x + \frac{1}{2},y + \frac{1}{2},z + \frac{1}{4},+1,-v,u,w)$ $$\begin{bmatrix}0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & \frac{1}{2} \\ 0 & 1 & 0 & \frac{1}{2} \\ 0 & 0 & 1 & \frac{1}{4}\end{array}\right]$$
6 $\left\{1||2_{001}|0 0 0\right\}$ $(-x,-y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
7 $\left\{1||m_{010}|0 0 0\right\}$ $(x,-y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
8 $\left\{4^{1}_{001}||4^{1}_{001}|\frac{1}{2} \frac{1}{2} \frac{1}{4}\right\}$ $(\frac{1}{2} - y,x + \frac{1}{2},z + \frac{1}{4},+1,-v,u,w)$ $$\begin{bmatrix}0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & \frac{1}{2} \\ 1 & 0 & 0 & \frac{1}{2} \\ 0 & 0 & 1 & \frac{1}{4}\end{array}\right]$$
9 $\left\{1||4^{3}_{001}|0 0 0\right\}$ $(y,-x,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
10 $\left\{1||m_{1-10}|0 0 0\right\}$ $(y,x,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
11 $\left\{4^{1}_{001}||2_{001}|\frac{1}{2} \frac{1}{2} \frac{1}{4}\right\}$ $(\frac{1}{2} - x,\frac{1}{2} - y,z + \frac{1}{4},+1,-v,u,w)$ $$\begin{bmatrix}0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & \frac{1}{2} \\ 0 & -1 & 0 & \frac{1}{2} \\ 0 & 0 & 1 & \frac{1}{4}\end{array}\right]$$
12 $\left\{4^{1}_{001}||4^{3}_{001}|\frac{1}{2} \frac{1}{2} \frac{1}{4}\right\}$ $(y + \frac{1}{2},\frac{1}{2} - x,z + \frac{1}{4},+1,-v,u,w)$ $$\begin{bmatrix}0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & \frac{1}{2} \\ -1 & 0 & 0 & \frac{1}{2} \\ 0 & 0 & 1 & \frac{1}{4}\end{array}\right]$$
13 $\left\{4^{1}_{001}||m_{100}|\frac{1}{2} \frac{1}{2} \frac{1}{4}\right\}$ $(\frac{1}{2} - x,y + \frac{1}{2},z + \frac{1}{4},+1,-v,u,w)$ $$\begin{bmatrix}0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & \frac{1}{2} \\ 0 & 1 & 0 & \frac{1}{2} \\ 0 & 0 & 1 & \frac{1}{4}\end{array}\right]$$
14 $\left\{4^{1}_{001}||m_{1-10}|\frac{1}{2} \frac{1}{2} \frac{1}{4}\right\}$ $(y + \frac{1}{2},x + \frac{1}{2},z + \frac{1}{4},+1,-v,u,w)$ $$\begin{bmatrix}0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & \frac{1}{2} \\ 1 & 0 & 0 & \frac{1}{2} \\ 0 & 0 & 1 & \frac{1}{4}\end{array}\right]$$
15 $\left\{4^{1}_{001}||m_{010}|\frac{1}{2} \frac{1}{2} \frac{1}{4}\right\}$ $(x + \frac{1}{2},\frac{1}{2} - y,z + \frac{1}{4},+1,-v,u,w)$ $$\begin{bmatrix}0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & \frac{1}{2} \\ 0 & -1 & 0 & \frac{1}{2} \\ 0 & 0 & 1 & \frac{1}{4}\end{array}\right]$$
16 $\left\{4^{1}_{001}||m_{110}|\frac{1}{2} \frac{1}{2} \frac{1}{4}\right\}$ $(\frac{1}{2} - y,\frac{1}{2} - x,z + \frac{1}{4},+1,-v,u,w)$ $$\begin{bmatrix}0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & \frac{1}{2} \\ -1 & 0 & 0 & \frac{1}{2} \\ 0 & 0 & 1 & \frac{1}{4}\end{array}\right]$$
17 $\left\{2_{001}||1|0 0 \frac{1}{2}\right\}$ $(x,y,z + \frac{1}{2},+1,-u,-v,w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
18 $\left\{2_{001}||4^{1}_{001}|0 0 \frac{1}{2}\right\}$ $(-y,x,z + \frac{1}{2},+1,-u,-v,w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
19 $\left\{2_{001}||2_{001}|0 0 \frac{1}{2}\right\}$ $(-x,-y,z + \frac{1}{2},+1,-u,-v,w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
20 $\left\{2_{001}||4^{3}_{001}|0 0 \frac{1}{2}\right\}$ $(y,-x,z + \frac{1}{2},+1,-u,-v,w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
21 $\left\{2_{001}||m_{100}|0 0 \frac{1}{2}\right\}$ $(-x,y,z + \frac{1}{2},+1,-u,-v,w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
22 $\left\{2_{001}||m_{1-10}|0 0 \frac{1}{2}\right\}$ $(y,x,z + \frac{1}{2},+1,-u,-v,w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
23 $\left\{2_{001}||m_{010}|0 0 \frac{1}{2}\right\}$ $(x,-y,z + \frac{1}{2},+1,-u,-v,w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
24 $\left\{2_{001}||m_{110}|0 0 \frac{1}{2}\right\}$ $(-y,-x,z + \frac{1}{2},+1,-u,-v,w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
25 $\left\{4^{3}_{001}||1|\frac{1}{2} \frac{1}{2} \frac{3}{4}\right\}$ $(x + \frac{1}{2},y + \frac{1}{2},z + \frac{3}{4},+1,v,-u,w)$ $$\begin{bmatrix}0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & \frac{1}{2} \\ 0 & 1 & 0 & \frac{1}{2} \\ 0 & 0 & 1 & \frac{3}{4}\end{array}\right]$$
26 $\left\{4^{3}_{001}||4^{1}_{001}|\frac{1}{2} \frac{1}{2} \frac{3}{4}\right\}$ $(\frac{1}{2} - y,x + \frac{1}{2},z + \frac{3}{4},+1,v,-u,w)$ $$\begin{bmatrix}0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & \frac{1}{2} \\ 1 & 0 & 0 & \frac{1}{2} \\ 0 & 0 & 1 & \frac{3}{4}\end{array}\right]$$
27 $\left\{4^{3}_{001}||2_{001}|\frac{1}{2} \frac{1}{2} \frac{3}{4}\right\}$ $(\frac{1}{2} - x,\frac{1}{2} - y,z + \frac{3}{4},+1,v,-u,w)$ $$\begin{bmatrix}0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & \frac{1}{2} \\ 0 & -1 & 0 & \frac{1}{2} \\ 0 & 0 & 1 & \frac{3}{4}\end{array}\right]$$
28 $\left\{4^{3}_{001}||4^{3}_{001}|\frac{1}{2} \frac{1}{2} \frac{3}{4}\right\}$ $(y + \frac{1}{2},\frac{1}{2} - x,z + \frac{3}{4},+1,v,-u,w)$ $$\begin{bmatrix}0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & \frac{1}{2} \\ -1 & 0 & 0 & \frac{1}{2} \\ 0 & 0 & 1 & \frac{3}{4}\end{array}\right]$$
29 $\left\{4^{3}_{001}||m_{100}|\frac{1}{2} \frac{1}{2} \frac{3}{4}\right\}$ $(\frac{1}{2} - x,y + \frac{1}{2},z + \frac{3}{4},+1,v,-u,w)$ $$\begin{bmatrix}0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & \frac{1}{2} \\ 0 & 1 & 0 & \frac{1}{2} \\ 0 & 0 & 1 & \frac{3}{4}\end{array}\right]$$
30 $\left\{4^{3}_{001}||m_{1-10}|\frac{1}{2} \frac{1}{2} \frac{3}{4}\right\}$ $(y + \frac{1}{2},x + \frac{1}{2},z + \frac{3}{4},+1,v,-u,w)$ $$\begin{bmatrix}0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & \frac{1}{2} \\ 1 & 0 & 0 & \frac{1}{2} \\ 0 & 0 & 1 & \frac{3}{4}\end{array}\right]$$
31 $\left\{4^{3}_{001}||m_{010}|\frac{1}{2} \frac{1}{2} \frac{3}{4}\right\}$ $(x + \frac{1}{2},\frac{1}{2} - y,z + \frac{3}{4},+1,v,-u,w)$ $$\begin{bmatrix}0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & \frac{1}{2} \\ 0 & -1 & 0 & \frac{1}{2} \\ 0 & 0 & 1 & \frac{3}{4}\end{array}\right]$$
32 $\left\{4^{3}_{001}||m_{110}|\frac{1}{2} \frac{1}{2} \frac{3}{4}\right\}$ $(\frac{1}{2} - y,\frac{1}{2} - x,z + \frac{3}{4},+1,v,-u,w)$ $$\begin{bmatrix}0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & \frac{1}{2} \\ -1 & 0 & 0 & \frac{1}{2} \\ 0 & 0 & 1 & \frac{3}{4}\end{array}\right]$$
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
$(0,0,0\mid mx,my,mz)+(\frac{1}{2},\frac{1}{2},\frac{1}{4}\mid -my,mx,mz)+(0,0,\frac{1}{2}\mid -mx,-my,mz)+(\frac{1}{2},\frac{1}{2},\frac{3}{4}\mid my,-mx,mz)$
32e $(x,y,z/2\mid mx,my,0),(-x,-y,z/2\mid mx,my,0),(-y,x,z/2\mid mx,my,0),(y,-x,z/2\mid mx,my,0),(x,-y,z/2\mid mx,my,0),(-x,y,z/2\mid mx,my,0),(-y,-x,z/2\mid mx,my,0),(y,x,z/2\mid mx,my,0)$
16d $(x,0,z/2\mid mx,my,0),(-x,0,z/2\mid mx,my,0),(0,x,z/2\mid mx,my,0),(0,-x,z/2\mid mx,my,0)$
16c $(x,x,z/2\mid mx,my,0),(-x,-x,z/2\mid mx,my,0),(-x,x,z/2\mid mx,my,0),(x,-x,z/2\mid mx,my,0)$
8b $(0,\frac{1}{2},z/2\mid mx,my,0),(\frac{1}{2},0,z/2\mid mx,my,0)$
4a $(0,0,z/2\mid mx,my,0)$
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: $.^{1}m.^{m}1$
op_index Seitz coordinate spin_expr space_expr
1 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
7 $\left\{1||m_{010}|0 0 0\right\}$ $(x,-y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
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: $..^{1}m^{m}1$
op_index Seitz coordinate spin_expr space_expr
1 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
10 $\left\{1||m_{1-10}|0 0 0\right\}$ $(y,x,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
extra__U_E_0(without_identity)
op_index Seitz coordinate spin_expr space_expr
1 $\left\{\ce{^{m_{001}}{1}}||1|0 0 0\right\}$ $(x,y,z,-1,u,v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
Site symmetry: $^{1}2^{1}m^{1}m .^{m}1$
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 $\left\{1||m_{100}|0 0 0\right\}$ $(-x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
6 $\left\{1||2_{001}|0 0 0\right\}$ $(-x,-y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
7 $\left\{1||m_{010}|0 0 0\right\}$ $(x,-y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
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: $^{1}4^{1}m^{1}m^{m}1$
op_index Seitz coordinate spin_expr space_expr
1 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
2 $\left\{1||4^{1}_{001}|0 0 0\right\}$ $(-y,x,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
3 $\left\{1||m_{100}|0 0 0\right\}$ $(-x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
4 $\left\{1||m_{110}|0 0 0\right\}$ $(-y,-x,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
6 $\left\{1||2_{001}|0 0 0\right\}$ $(-x,-y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
7 $\left\{1||m_{010}|0 0 0\right\}$ $(x,-y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
9 $\left\{1||4^{3}_{001}|0 0 0\right\}$ $(y,-x,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
10 $\left\{1||m_{1-10}|0 0 0\right\}$ $(y,x,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
extra__U_E_0(without_identity)
op_index Seitz coordinate spin_expr space_expr
1 $\left\{\ce{^{m_{001}}{1}}||1|0 0 0\right\}$ $(x,y,z,-1,u,v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$