205.2.1.1
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-t
G0 / L0: 205 / 2
Identifier: 205.2.1.1
it / ik: 12 / 1
Nontrivial spin-space point group: $23$
Spin-space point group: $23$
International Notation: $$P\ce{^{2_{100}}{a}}\ce{^{3^{1}_{111}}{\text{-}3}}$$
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 205.2.1.1, 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 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
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 $\left\{2_{100}||m_{100}|\frac{1}{2} \frac{1}{2} 0\right\}$ $(\frac{1}{2} - x,y + \frac{1}{2},z,+1,u,-v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & \frac{1}{2} \\ 0 & 1 & 0 & \frac{1}{2} \\ 0 & 0 & 1 & 0\end{array}\right]$$
3 $\left\{3^{1}_{111}||-3^{1}_{111}|0 0 0\right\}$ $(-z,-x,-y,+1,w,u,v)$ $$\begin{bmatrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 0 & -1 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0\end{array}\right]$$
4 $\left\{3^{1}_{-11-1}||3^{1}_{-11-1}|\frac{1}{2} \frac{1}{2} 0\right\}$ $(z + \frac{1}{2},\frac{1}{2} - x,-y,+1,w,-u,-v)$ $$\begin{bmatrix}0 & 0 & 1 \\ -1 & 0 & 0 \\ 0 & -1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 0 & 1 & \frac{1}{2} \\ -1 & 0 & 0 & \frac{1}{2} \\ 0 & -1 & 0 & 0\end{array}\right]$$
5 $\left\{3^{1}_{-1-11}||3^{1}_{-1-11}|0 \frac{1}{2} \frac{1}{2}\right\}$ $(-z,x + \frac{1}{2},\frac{1}{2} - y,+1,-w,u,-v)$ $$\begin{bmatrix}0 & 0 & -1 \\ 1 & 0 & 0 \\ 0 & -1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 0 & -1 & 0 \\ 1 & 0 & 0 & \frac{1}{2} \\ 0 & -1 & 0 & \frac{1}{2}\end{array}\right]$$
6 $\left\{3^{2}_{111}||3^{2}_{111}|0 0 0\right\}$ $(y,z,x,+1,v,w,u)$ $$\begin{bmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0\end{array}\right]$$
7 $\left\{3^{2}_{1-1-1}||-3^{2}_{1-1-1}|0 \frac{1}{2} \frac{1}{2}\right\}$ $(y,\frac{1}{2} - z,x + \frac{1}{2},+1,-v,w,-u)$ $$\begin{bmatrix}0 & -1 & 0 \\ 0 & 0 & 1 \\ -1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2} \\ 1 & 0 & 0 & \frac{1}{2}\end{array}\right]$$
8 $\left\{3^{2}_{-11-1}||-3^{2}_{-11-1}|\frac{1}{2} 0 \frac{1}{2}\right\}$ $(y + \frac{1}{2},z,\frac{1}{2} - x,+1,-v,-w,u)$ $$\begin{bmatrix}0 & -1 & 0 \\ 0 & 0 & -1 \\ 1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & \frac{1}{2} \\ 0 & 0 & 1 & 0 \\ -1 & 0 & 0 & \frac{1}{2}\end{array}\right]$$
9 $\left\{1||-1|0 0 0\right\}$ $(-x,-y,-z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
10 $\left\{2_{001}||2_{001}|\frac{1}{2} 0 \frac{1}{2}\right\}$ $(\frac{1}{2} - x,-y,z + \frac{1}{2},+1,-u,-v,w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & \frac{1}{2} \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
11 $\left\{3^{1}_{111}||3^{1}_{111}|0 0 0\right\}$ $(z,x,y,+1,w,u,v)$ $$\begin{bmatrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\end{array}\right]$$
12 $\left\{3^{1}_{-11-1}||-3^{1}_{-11-1}|\frac{1}{2} \frac{1}{2} 0\right\}$ $(\frac{1}{2} - z,x + \frac{1}{2},y,+1,w,-u,-v)$ $$\begin{bmatrix}0 & 0 & 1 \\ -1 & 0 & 0 \\ 0 & -1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 0 & -1 & \frac{1}{2} \\ 1 & 0 & 0 & \frac{1}{2} \\ 0 & 1 & 0 & 0\end{array}\right]$$
13 $\left\{2_{100}||2_{100}|\frac{1}{2} \frac{1}{2} 0\right\}$ $(x + \frac{1}{2},\frac{1}{2} - y,-z,+1,u,-v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & \frac{1}{2} \\ 0 & -1 & 0 & \frac{1}{2} \\ 0 & 0 & -1 & 0\end{array}\right]$$
14 $\left\{3^{2}_{111}||-3^{2}_{111}|0 0 0\right\}$ $(-y,-z,-x,+1,v,w,u)$ $$\begin{bmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ -1 & 0 & 0 & 0\end{array}\right]$$
15 $\left\{3^{2}_{1-1-1}||3^{2}_{1-1-1}|0 \frac{1}{2} \frac{1}{2}\right\}$ $(-y,z + \frac{1}{2},\frac{1}{2} - x,+1,-v,w,-u)$ $$\begin{bmatrix}0 & -1 & 0 \\ 0 & 0 & 1 \\ -1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2} \\ -1 & 0 & 0 & \frac{1}{2}\end{array}\right]$$
16 $\left\{3^{1}_{-1-11}||-3^{1}_{-1-11}|0 \frac{1}{2} \frac{1}{2}\right\}$ $(z,\frac{1}{2} - x,y + \frac{1}{2},+1,-w,u,-v)$ $$\begin{bmatrix}0 & 0 & -1 \\ 1 & 0 & 0 \\ 0 & -1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 0 & 1 & 0 \\ -1 & 0 & 0 & \frac{1}{2} \\ 0 & 1 & 0 & \frac{1}{2}\end{array}\right]$$
17 $\left\{2_{001}||m_{001}|\frac{1}{2} 0 \frac{1}{2}\right\}$ $(x + \frac{1}{2},y,\frac{1}{2} - z,+1,-u,-v,w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & \frac{1}{2} \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
18 $\left\{3^{2}_{-11-1}||3^{2}_{-11-1}|\frac{1}{2} 0 \frac{1}{2}\right\}$ $(\frac{1}{2} - y,-z,x + \frac{1}{2},+1,-v,-w,u)$ $$\begin{bmatrix}0 & -1 & 0 \\ 0 & 0 & -1 \\ 1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & \frac{1}{2} \\ 0 & 0 & -1 & 0 \\ 1 & 0 & 0 & \frac{1}{2}\end{array}\right]$$
19 $\left\{3^{1}_{1-1-1}||-3^{1}_{1-1-1}|\frac{1}{2} 0 \frac{1}{2}\right\}$ $(z + \frac{1}{2},x,\frac{1}{2} - y,+1,-w,-u,v)$ $$\begin{bmatrix}0 & 0 & -1 \\ -1 & 0 & 0 \\ 0 & 1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 0 & 1 & \frac{1}{2} \\ 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & \frac{1}{2}\end{array}\right]$$
20 $\left\{3^{2}_{-1-11}||-3^{2}_{-1-11}|\frac{1}{2} \frac{1}{2} 0\right\}$ $(\frac{1}{2} - y,z + \frac{1}{2},x,+1,v,-w,-u)$ $$\begin{bmatrix}0 & 1 & 0 \\ 0 & 0 & -1 \\ -1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & \frac{1}{2} \\ 0 & 0 & 1 & \frac{1}{2} \\ 1 & 0 & 0 & 0\end{array}\right]$$
21 $\left\{2_{010}||m_{010}|0 \frac{1}{2} \frac{1}{2}\right\}$ $(x,\frac{1}{2} - y,z + \frac{1}{2},+1,-u,v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & -1 & 0 & \frac{1}{2} \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
22 $\left\{3^{2}_{-1-11}||3^{2}_{-1-11}|\frac{1}{2} \frac{1}{2} 0\right\}$ $(y + \frac{1}{2},\frac{1}{2} - z,-x,+1,v,-w,-u)$ $$\begin{bmatrix}0 & 1 & 0 \\ 0 & 0 & -1 \\ -1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & \frac{1}{2} \\ 0 & 0 & -1 & \frac{1}{2} \\ -1 & 0 & 0 & 0\end{array}\right]$$
23 $\left\{3^{1}_{1-1-1}||3^{1}_{1-1-1}|\frac{1}{2} 0 \frac{1}{2}\right\}$ $(\frac{1}{2} - z,-x,y + \frac{1}{2},+1,-w,-u,v)$ $$\begin{bmatrix}0 & 0 & -1 \\ -1 & 0 & 0 \\ 0 & 1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 0 & -1 & \frac{1}{2} \\ -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & \frac{1}{2}\end{array}\right]$$
24 $\left\{2_{010}||2_{010}|0 \frac{1}{2} \frac{1}{2}\right\}$ $(-x,y + \frac{1}{2},\frac{1}{2} - z,+1,-u,v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ 0 & 1 & 0 & \frac{1}{2} \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
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 $\left\{2_{100}||m_{100}|\frac{1}{2} \frac{1}{2} 0\right\}$ $(\frac{1}{2} - x,y + \frac{1}{2},z,+1,u,-v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & \frac{1}{2} \\ 0 & 1 & 0 & \frac{1}{2} \\ 0 & 0 & 1 & 0\end{array}\right]$$
3 $\left\{3^{1}_{111}||-3^{1}_{111}|0 0 0\right\}$ $(-z,-x,-y,+1,w,u,v)$ $$\begin{bmatrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 0 & -1 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0\end{array}\right]$$
4 $\left\{3^{1}_{-11-1}||3^{1}_{-11-1}|\frac{1}{2} \frac{1}{2} 0\right\}$ $(z + \frac{1}{2},\frac{1}{2} - x,-y,+1,w,-u,-v)$ $$\begin{bmatrix}0 & 0 & 1 \\ -1 & 0 & 0 \\ 0 & -1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 0 & 1 & \frac{1}{2} \\ -1 & 0 & 0 & \frac{1}{2} \\ 0 & -1 & 0 & 0\end{array}\right]$$
5 $\left\{3^{1}_{-1-11}||3^{1}_{-1-11}|0 \frac{1}{2} \frac{1}{2}\right\}$ $(-z,x + \frac{1}{2},\frac{1}{2} - y,+1,-w,u,-v)$ $$\begin{bmatrix}0 & 0 & -1 \\ 1 & 0 & 0 \\ 0 & -1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 0 & -1 & 0 \\ 1 & 0 & 0 & \frac{1}{2} \\ 0 & -1 & 0 & \frac{1}{2}\end{array}\right]$$
6 $\left\{3^{2}_{111}||3^{2}_{111}|0 0 0\right\}$ $(y,z,x,+1,v,w,u)$ $$\begin{bmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0\end{array}\right]$$
7 $\left\{3^{2}_{1-1-1}||-3^{2}_{1-1-1}|0 \frac{1}{2} \frac{1}{2}\right\}$ $(y,\frac{1}{2} - z,x + \frac{1}{2},+1,-v,w,-u)$ $$\begin{bmatrix}0 & -1 & 0 \\ 0 & 0 & 1 \\ -1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2} \\ 1 & 0 & 0 & \frac{1}{2}\end{array}\right]$$
8 $\left\{3^{2}_{-11-1}||-3^{2}_{-11-1}|\frac{1}{2} 0 \frac{1}{2}\right\}$ $(y + \frac{1}{2},z,\frac{1}{2} - x,+1,-v,-w,u)$ $$\begin{bmatrix}0 & -1 & 0 \\ 0 & 0 & -1 \\ 1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & \frac{1}{2} \\ 0 & 0 & 1 & 0 \\ -1 & 0 & 0 & \frac{1}{2}\end{array}\right]$$
9 $\left\{1||-1|0 0 0\right\}$ $(-x,-y,-z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
10 $\left\{2_{001}||2_{001}|\frac{1}{2} 0 \frac{1}{2}\right\}$ $(\frac{1}{2} - x,-y,z + \frac{1}{2},+1,-u,-v,w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & \frac{1}{2} \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
11 $\left\{3^{1}_{111}||3^{1}_{111}|0 0 0\right\}$ $(z,x,y,+1,w,u,v)$ $$\begin{bmatrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\end{array}\right]$$
12 $\left\{3^{1}_{-11-1}||-3^{1}_{-11-1}|\frac{1}{2} \frac{1}{2} 0\right\}$ $(\frac{1}{2} - z,x + \frac{1}{2},y,+1,w,-u,-v)$ $$\begin{bmatrix}0 & 0 & 1 \\ -1 & 0 & 0 \\ 0 & -1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 0 & -1 & \frac{1}{2} \\ 1 & 0 & 0 & \frac{1}{2} \\ 0 & 1 & 0 & 0\end{array}\right]$$
13 $\left\{2_{100}||2_{100}|\frac{1}{2} \frac{1}{2} 0\right\}$ $(x + \frac{1}{2},\frac{1}{2} - y,-z,+1,u,-v,-w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & \frac{1}{2} \\ 0 & -1 & 0 & \frac{1}{2} \\ 0 & 0 & -1 & 0\end{array}\right]$$
14 $\left\{3^{2}_{111}||-3^{2}_{111}|0 0 0\right\}$ $(-y,-z,-x,+1,v,w,u)$ $$\begin{bmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ -1 & 0 & 0 & 0\end{array}\right]$$
15 $\left\{3^{2}_{1-1-1}||3^{2}_{1-1-1}|0 \frac{1}{2} \frac{1}{2}\right\}$ $(-y,z + \frac{1}{2},\frac{1}{2} - x,+1,-v,w,-u)$ $$\begin{bmatrix}0 & -1 & 0 \\ 0 & 0 & 1 \\ -1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2} \\ -1 & 0 & 0 & \frac{1}{2}\end{array}\right]$$
16 $\left\{3^{1}_{-1-11}||-3^{1}_{-1-11}|0 \frac{1}{2} \frac{1}{2}\right\}$ $(z,\frac{1}{2} - x,y + \frac{1}{2},+1,-w,u,-v)$ $$\begin{bmatrix}0 & 0 & -1 \\ 1 & 0 & 0 \\ 0 & -1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 0 & 1 & 0 \\ -1 & 0 & 0 & \frac{1}{2} \\ 0 & 1 & 0 & \frac{1}{2}\end{array}\right]$$
17 $\left\{2_{001}||m_{001}|\frac{1}{2} 0 \frac{1}{2}\right\}$ $(x + \frac{1}{2},y,\frac{1}{2} - z,+1,-u,-v,w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & \frac{1}{2} \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
18 $\left\{3^{2}_{-11-1}||3^{2}_{-11-1}|\frac{1}{2} 0 \frac{1}{2}\right\}$ $(\frac{1}{2} - y,-z,x + \frac{1}{2},+1,-v,-w,u)$ $$\begin{bmatrix}0 & -1 & 0 \\ 0 & 0 & -1 \\ 1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & \frac{1}{2} \\ 0 & 0 & -1 & 0 \\ 1 & 0 & 0 & \frac{1}{2}\end{array}\right]$$
19 $\left\{3^{1}_{1-1-1}||-3^{1}_{1-1-1}|\frac{1}{2} 0 \frac{1}{2}\right\}$ $(z + \frac{1}{2},x,\frac{1}{2} - y,+1,-w,-u,v)$ $$\begin{bmatrix}0 & 0 & -1 \\ -1 & 0 & 0 \\ 0 & 1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 0 & 1 & \frac{1}{2} \\ 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & \frac{1}{2}\end{array}\right]$$
20 $\left\{3^{2}_{-1-11}||-3^{2}_{-1-11}|\frac{1}{2} \frac{1}{2} 0\right\}$ $(\frac{1}{2} - y,z + \frac{1}{2},x,+1,v,-w,-u)$ $$\begin{bmatrix}0 & 1 & 0 \\ 0 & 0 & -1 \\ -1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & \frac{1}{2} \\ 0 & 0 & 1 & \frac{1}{2} \\ 1 & 0 & 0 & 0\end{array}\right]$$
21 $\left\{2_{010}||m_{010}|0 \frac{1}{2} \frac{1}{2}\right\}$ $(x,\frac{1}{2} - y,z + \frac{1}{2},+1,-u,v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & -1 & 0 & \frac{1}{2} \\ 0 & 0 & 1 & \frac{1}{2}\end{array}\right]$$
22 $\left\{3^{2}_{-1-11}||3^{2}_{-1-11}|\frac{1}{2} \frac{1}{2} 0\right\}$ $(y + \frac{1}{2},\frac{1}{2} - z,-x,+1,v,-w,-u)$ $$\begin{bmatrix}0 & 1 & 0 \\ 0 & 0 & -1 \\ -1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & \frac{1}{2} \\ 0 & 0 & -1 & \frac{1}{2} \\ -1 & 0 & 0 & 0\end{array}\right]$$
23 $\left\{3^{1}_{1-1-1}||3^{1}_{1-1-1}|\frac{1}{2} 0 \frac{1}{2}\right\}$ $(\frac{1}{2} - z,-x,y + \frac{1}{2},+1,-w,-u,v)$ $$\begin{bmatrix}0 & 0 & -1 \\ -1 & 0 & 0 \\ 0 & 1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 0 & -1 & \frac{1}{2} \\ -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & \frac{1}{2}\end{array}\right]$$
24 $\left\{2_{010}||2_{010}|0 \frac{1}{2} \frac{1}{2}\right\}$ $(-x,y + \frac{1}{2},\frac{1}{2} - z,+1,-u,v,-w)$ $$\begin{bmatrix}-1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ 0 & 1 & 0 & \frac{1}{2} \\ 0 & 0 & -1 & \frac{1}{2}\end{array}\right]$$
wyckoff position site symmetry Coordinates
$(0,0,0\mid mx,my,mz)$
24d $(x,y,z\mid mx,my,mz),(\frac{1}{2} - x,-y,z + \frac{1}{2}\mid -mx,-my,mz),(-x,y + \frac{1}{2},\frac{1}{2} - z\mid -mx,my,-mz),(x + \frac{1}{2},\frac{1}{2} - y,-z\mid mx,-my,-mz),(z,x,y\mid mz,mx,my),(z + \frac{1}{2},\frac{1}{2} - x,-y\mid mz,-mx,-my),(\frac{1}{2} - z,-x,y + \frac{1}{2}\mid -mz,-mx,my),(-z,x + \frac{1}{2},\frac{1}{2} - y\mid -mz,mx,-my),(y,z,x\mid my,mz,mx),(-y,z + \frac{1}{2},\frac{1}{2} - x\mid -my,mz,-mx),(y + \frac{1}{2},\frac{1}{2} - z,-x\mid my,-mz,-mx),(\frac{1}{2} - y,-z,x + \frac{1}{2}\mid -my,-mz,mx),(-x,-y,-z\mid mx,my,mz),(x + \frac{1}{2},y,\frac{1}{2} - z\mid -mx,-my,mz),(x,\frac{1}{2} - y,z + \frac{1}{2}\mid -mx,my,-mz),(\frac{1}{2} - x,y + \frac{1}{2},z\mid mx,-my,-mz),(-z,-x,-y\mid mz,mx,my),(\frac{1}{2} - z,x + \frac{1}{2},y\mid mz,-mx,-my),(z + \frac{1}{2},x,\frac{1}{2} - y\mid -mz,-mx,my),(z,\frac{1}{2} - x,y + \frac{1}{2}\mid -mz,mx,-my),(-y,-z,-x\mid my,mz,mx),(y,\frac{1}{2} - z,x + \frac{1}{2}\mid -my,mz,-mx),(\frac{1}{2} - y,z + \frac{1}{2},x\mid my,-mz,-mx),(y + \frac{1}{2},z,\frac{1}{2} - x\mid -my,-mz,mx)$
8c $(x,x,x\mid \sqrt{3} mx/3,\sqrt{3} mx/3,\sqrt{3} mx/3),(\frac{1}{2} - x,-x,x + \frac{1}{2}\mid -\sqrt{3} mx/3,-\sqrt{3} mx/3,\sqrt{3} mx/3),(-x,x + \frac{1}{2},\frac{1}{2} - x\mid -\sqrt{3} mx/3,\sqrt{3} mx/3,-\sqrt{3} mx/3),(x + \frac{1}{2},\frac{1}{2} - x,-x\mid \sqrt{3} mx/3,-\sqrt{3} mx/3,-\sqrt{3} mx/3),(-x,-x,-x\mid \sqrt{3} mx/3,\sqrt{3} mx/3,\sqrt{3} mx/3),(x + \frac{1}{2},x,\frac{1}{2} - x\mid -\sqrt{3} mx/3,-\sqrt{3} mx/3,\sqrt{3} mx/3),(x,\frac{1}{2} - x,x + \frac{1}{2}\mid -\sqrt{3} mx/3,\sqrt{3} mx/3,-\sqrt{3} mx/3),(\frac{1}{2} - x,x + \frac{1}{2},x\mid \sqrt{3} mx/3,-\sqrt{3} mx/3,-\sqrt{3} mx/3)$
4b $(\frac{1}{2},\frac{1}{2},\frac{1}{2}\mid \sqrt{3} mx/3,\sqrt{3} mx/3,\sqrt{3} mx/3),(0,\frac{1}{2},0\mid -\sqrt{3} mx/3,-\sqrt{3} mx/3,\sqrt{3} mx/3),(\frac{1}{2},0,0\mid -\sqrt{3} mx/3,\sqrt{3} mx/3,-\sqrt{3} mx/3),(0,0,\frac{1}{2}\mid \sqrt{3} mx/3,-\sqrt{3} mx/3,-\sqrt{3} mx/3)$
4a $(0,0,0\mid \sqrt{3} mx/3,\sqrt{3} mx/3,\sqrt{3} mx/3),(\frac{1}{2},0,\frac{1}{2}\mid -\sqrt{3} mx/3,-\sqrt{3} mx/3,\sqrt{3} mx/3),(0,\frac{1}{2},\frac{1}{2}\mid -\sqrt{3} mx/3,\sqrt{3} mx/3,-\sqrt{3} mx/3),(\frac{1}{2},\frac{1}{2},0\mid \sqrt{3} mx/3,-\sqrt{3} mx/3,-\sqrt{3} mx/3)$
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: $.^{3^{1}_{111}}3.$
op_index Seitz coordinate spin_expr space_expr
1 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
6 $\left\{3^{2}_{111}||3^{2}_{111}|0 0 0\right\}$ $(y,z,x,+1,v,w,u)$ $$\begin{bmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0\end{array}\right]$$
11 $\left\{3^{1}_{111}||3^{1}_{111}|0 0 0\right\}$ $(z,x,y,+1,w,u,v)$ $$\begin{bmatrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\end{array}\right]$$
Site symmetry: $.^{3^{1}_{111}}\!-\!3.$
op_index Seitz coordinate spin_expr space_expr
1 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
3 $\left\{3^{1}_{111}||-3^{1}_{111}|0 0 0\right\}$ $(-z,-x,-y,+1,w,u,v)$ $$\begin{bmatrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 0 & -1 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0\end{array}\right]$$
6 $\left\{3^{2}_{111}||3^{2}_{111}|0 0 0\right\}$ $(y,z,x,+1,v,w,u)$ $$\begin{bmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0\end{array}\right]$$
9 $\left\{1||-1|0 0 0\right\}$ $(-x,-y,-z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
11 $\left\{3^{1}_{111}||3^{1}_{111}|0 0 0\right\}$ $(z,x,y,+1,w,u,v)$ $$\begin{bmatrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\end{array}\right]$$
14 $\left\{3^{2}_{111}||-3^{2}_{111}|0 0 0\right\}$ $(-y,-z,-x,+1,v,w,u)$ $$\begin{bmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ -1 & 0 & 0 & 0\end{array}\right]$$
Site symmetry: $.^{3^{1}_{111}}\!-\!3.$
op_index Seitz coordinate spin_expr space_expr
1 $\left\{1||1|0 0 0\right\}$ $(x,y,z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right]$$
3 $\left\{3^{1}_{111}||-3^{1}_{111}|0 0 0\right\}$ $(-z,-x,-y,+1,w,u,v)$ $$\begin{bmatrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 0 & -1 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0\end{array}\right]$$
6 $\left\{3^{2}_{111}||3^{2}_{111}|0 0 0\right\}$ $(y,z,x,+1,v,w,u)$ $$\begin{bmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0\end{array}\right]$$
9 $\left\{1||-1|0 0 0\right\}$ $(-x,-y,-z,+1,u,v,w)$ $$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}-1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0\end{array}\right]$$
11 $\left\{3^{1}_{111}||3^{1}_{111}|0 0 0\right\}$ $(z,x,y,+1,w,u,v)$ $$\begin{bmatrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\end{array}\right]$$
14 $\left\{3^{2}_{111}||-3^{2}_{111}|0 0 0\right\}$ $(-y,-z,-x,+1,v,w,u)$ $$\begin{bmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{bmatrix}$$ $$\left[\begin{array}{ccc|c}0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ -1 & 0 & 0 & 0\end{array}\right]$$