| File Name | Oriented Spin Space Group | G0 Symbol | L0 Symbol | it | ik |
|---|---|---|---|---|---|
| \(CeAuAl3\) | \(P ^{1}4 ^{1}m ^{1}m \mid ^{4^{1}_{001}}(\frac{1}{2},\frac{1}{2},\frac{1}{4}) ^{m_{001}}1\) \((107.99.4.1.P)\) | \(I4mm\ (107)\) | \(P4mm\ (99)\) | \(1\) | \(4\) |
| Space Group | Magnetic Space Group | Spin-space PG | Configuration | Magnetic Phase |
|---|---|---|---|---|
| \(I4mm\ (107)\) | \[\begin{aligned}&P_c4_1(76.10)\\ &Type\ IV\end{aligned}\] | \[\begin{aligned}4/m\\ C4h\end{aligned}\] | \(Coplanar\) | \(AFM\) |
| Little Cogroup at General Position | Spin Splitting | Spin Texture | |||
|---|---|---|---|---|---|
| without SOC | with SOC | without SOC | with SOC | without SOC | with SOC |
| $$^{1}1^{4}1$$ | $$1$$ | \(k-dependent\) | \(Yes\) | \(\mathrm{p-wave}\) \(C_{1}\left(k_{z}\,\sigma_{z}\right)\) | \(\mathrm{p-wave}\) \(C_{1}\left(k_{y}\,\sigma_{x} - k_{x}\,\sigma_{y}\right) + C_{2}\left(k_{x}\,\sigma_{x} + k_{y}\,\sigma_{y}\right) + C_{3}\left(k_{z}\,\sigma_{z}\right)\) |
| Effective Magnetic Point Group | Anomalous Hall Effect | ||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $$4mm1'$$ | without SOC | with SOC | |||||||||||||||||||||||||||||||
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| Effective MPG Constraint on rank-3 Tensor | |||
|---|---|---|---|
| without SOC | with SOC | ||
| T-odd | Quantum metric dipole | $$0$$ | $$0$$ |
| T-odd | Inversed mass dipole | $$0$$ | $$0$$ |
| T-even | Berry curvature dipole | $$\begin{aligned}D_{xxz} = D_{xzx} = D_{yyz} = D_{yzy} = -1/2D_{zxx} = -1/2D_{zyy}\end{aligned}$$ | $$\begin{aligned}D_{xxz} = D_{xzx} = D_{yyz} = D_{yzy} = -1/2D_{zxx} = -1/2D_{zyy}\\ D_{xyz} = D_{xzy} = -D_{yxz} = -D_{yzx}\end{aligned}$$ |
L0-std Unit Magnetic Cell:
| a | b | c | alpha | beta | gamma |
|---|---|---|---|---|---|
| 4.3172 | 4.3172 | 21.618 | 90.0 | 90.0 | 90.0 |
| Transformation from input to L0-standard setting | Transformation from L0-standard to magnetic primitive setting | ||
|---|---|---|---|
| Transformation Matrix (T) | Origin Shift (τ) | Transformation Matrix (Tp) | Origin Shift (τp) |
| \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] |
| T * r_input + τ = r_L0std | Tp * r_L0std + τp = r_prim | ||
Fractional Coordinates & Moments (in L0-std Unit Magnetic Cell)
| No. | Atom | Site | Moment (Oriented) |
|---|---|---|---|
| 1 | Ce |
[0.0, 0.0, 0.0] |
[0.4633, 0.0, 0.0] |
| 2 | Ce |
[0.0, 0.0, 0.5] |
[-0.4633, 0.0, 0.0] |
| 3 | Ce |
[0.5, 0.5, 0.25] |
[0.0, 0.4633, 0.0] |
| 4 | Ce |
[0.5, 0.5, 0.75] |
[0.0, -0.4633, 0.0] |
| 5 | Al |
[0.0, 0.0, 0.2035] |
[0.0, 0.0, 0.0] |
| 6 | Al |
[0.0, 0.0, 0.7035] |
[0.0, 0.0, 0.0] |
| 7 | Al |
[0.0, 0.5, 0.129] |
[0.0, 0.0, 0.0] |
| 8 | Al |
[0.0, 0.5, 0.379] |
[0.0, 0.0, 0.0] |
| 9 | Al |
[0.0, 0.5, 0.629] |
[0.0, 0.0, 0.0] |
| 10 | Al |
[0.0, 0.5, 0.879] |
[0.0, 0.0, 0.0] |
| 11 | Al |
[0.5, 0.0, 0.129] |
[0.0, 0.0, 0.0] |
| 12 | Al |
[0.5, 0.0, 0.379] |
[0.0, 0.0, 0.0] |
| 13 | Al |
[0.5, 0.0, 0.629] |
[0.0, 0.0, 0.0] |
| 14 | Al |
[0.5, 0.0, 0.879] |
[0.0, 0.0, 0.0] |
| 15 | Al |
[0.5, 0.5, 0.4535] |
[0.0, 0.0, 0.0] |
| 16 | Al |
[0.5, 0.5, 0.9535] |
[0.0, 0.0, 0.0] |
| 17 | Au |
[0.0, 0.0, 0.3155] |
[0.0, 0.0, 0.0] |
| 18 | Au |
[0.0, 0.0, 0.8155] |
[0.0, 0.0, 0.0] |
| 19 | Au |
[0.5, 0.5, 0.0655] |
[0.0, 0.0, 0.0] |
| 20 | Au |
[0.5, 0.5, 0.5655] |
[0.0, 0.0, 0.0] |
| Element | SG Wyckoff | SSG Wyckoff | MSG Wyckoff |
|---|---|---|---|
| Al | 4a | 4a | 4c |
| 8b | 8b | 4a | |
| 4b | |||
| Au | 4a | 4a | 4c |
| Ce | 4a | 4a(2) | 4c(2) |
Elements of the Spin Space Group (L0-std Unit Magnetic Cell):
Real Space Elements are under the basis of the lattice given above.
Spin components are expressed in the lattice basis (a, b, c).
| No. | Spin Rotation | Space Rotation | Space Translation | Seitz Symbol |
|---|---|---|---|---|
| 1 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, 1 \,\middle|\, 0,0,0 \right\}\) |
| 2 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ m_{001} \,\middle\|\, 1 \,\middle|\, 0,0,0 \right\}\) |
| 3 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 2_{001} \,\middle\|\, 1 \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 4 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ -1 \,\middle\|\, 1 \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 5 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.25 \end{bmatrix}\] | \(\left\{ 4^{1}_{001} \,\middle\|\, 1 \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{4} \right\}\) |
| 6 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.25 \end{bmatrix}\] | \(\left\{ -4^{3}_{001} \,\middle\|\, 1 \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{4} \right\}\) |
| 7 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.75 \end{bmatrix}\] | \(\left\{ 4^{3}_{001} \,\middle\|\, 1 \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{3}{4} \right\}\) |
| 8 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.75 \end{bmatrix}\] | \(\left\{ -4^{1}_{001} \,\middle\|\, 1 \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{3}{4} \right\}\) |
| 9 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, 4^{1}_{001} \,\middle|\, 0,0,0 \right\}\) |
| 10 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, 4^{3}_{001} \,\middle|\, 0,0,0 \right\}\) |
| 11 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, m_{100} \,\middle|\, 0,0,0 \right\}\) |
| 12 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, m_{1-10} \,\middle|\, 0,0,0 \right\}\) |
| 13 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, m_{010} \,\middle|\, 0,0,0 \right\}\) |
| 14 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, m_{110} \,\middle|\, 0,0,0 \right\}\) |
| 15 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ m_{001} \,\middle\|\, 4^{1}_{001} \,\middle|\, 0,0,0 \right\}\) |
| 16 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ m_{001} \,\middle\|\, 4^{3}_{001} \,\middle|\, 0,0,0 \right\}\) |
| 17 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ m_{001} \,\middle\|\, m_{100} \,\middle|\, 0,0,0 \right\}\) |
| 18 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ m_{001} \,\middle\|\, m_{1-10} \,\middle|\, 0,0,0 \right\}\) |
| 19 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ m_{001} \,\middle\|\, m_{010} \,\middle|\, 0,0,0 \right\}\) |
| 20 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ m_{001} \,\middle\|\, m_{110} \,\middle|\, 0,0,0 \right\}\) |
| 21 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 2_{001} \,\middle\|\, 4^{1}_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 22 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 2_{001} \,\middle\|\, 4^{3}_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 23 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 2_{001} \,\middle\|\, m_{100} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 24 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 2_{001} \,\middle\|\, m_{1-10} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 25 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 2_{001} \,\middle\|\, m_{010} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 26 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 2_{001} \,\middle\|\, m_{110} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 27 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ -1 \,\middle\|\, 4^{1}_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 28 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ -1 \,\middle\|\, 4^{3}_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 29 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ -1 \,\middle\|\, m_{100} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 30 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ -1 \,\middle\|\, m_{1-10} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 31 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ -1 \,\middle\|\, m_{010} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 32 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ -1 \,\middle\|\, m_{110} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 33 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.25 \end{bmatrix}\] | \(\left\{ 4^{1}_{001} \,\middle\|\, 4^{1}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{4} \right\}\) |
| 34 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.25 \end{bmatrix}\] | \(\left\{ 4^{1}_{001} \,\middle\|\, 4^{3}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{4} \right\}\) |
| 35 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.25 \end{bmatrix}\] | \(\left\{ 4^{1}_{001} \,\middle\|\, m_{100} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{4} \right\}\) |
| 36 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.25 \end{bmatrix}\] | \(\left\{ 4^{1}_{001} \,\middle\|\, m_{1-10} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{4} \right\}\) |
| 37 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.25 \end{bmatrix}\] | \(\left\{ 4^{1}_{001} \,\middle\|\, m_{010} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{4} \right\}\) |
| 38 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.25 \end{bmatrix}\] | \(\left\{ 4^{1}_{001} \,\middle\|\, m_{110} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{4} \right\}\) |
| 39 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.25 \end{bmatrix}\] | \(\left\{ -4^{3}_{001} \,\middle\|\, 4^{1}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{4} \right\}\) |
| 40 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.25 \end{bmatrix}\] | \(\left\{ -4^{3}_{001} \,\middle\|\, 4^{3}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{4} \right\}\) |
| 41 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.25 \end{bmatrix}\] | \(\left\{ -4^{3}_{001} \,\middle\|\, m_{100} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{4} \right\}\) |
| 42 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.25 \end{bmatrix}\] | \(\left\{ -4^{3}_{001} \,\middle\|\, m_{1-10} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{4} \right\}\) |
| 43 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.25 \end{bmatrix}\] | \(\left\{ -4^{3}_{001} \,\middle\|\, m_{010} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{4} \right\}\) |
| 44 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.25 \end{bmatrix}\] | \(\left\{ -4^{3}_{001} \,\middle\|\, m_{110} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{4} \right\}\) |
| 45 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.75 \end{bmatrix}\] | \(\left\{ 4^{3}_{001} \,\middle\|\, 4^{1}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{3}{4} \right\}\) |
| 46 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.75 \end{bmatrix}\] | \(\left\{ 4^{3}_{001} \,\middle\|\, 4^{3}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{3}{4} \right\}\) |
| 47 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.75 \end{bmatrix}\] | \(\left\{ 4^{3}_{001} \,\middle\|\, m_{100} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{3}{4} \right\}\) |
| 48 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.75 \end{bmatrix}\] | \(\left\{ 4^{3}_{001} \,\middle\|\, m_{1-10} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{3}{4} \right\}\) |
| 49 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.75 \end{bmatrix}\] | \(\left\{ 4^{3}_{001} \,\middle\|\, m_{010} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{3}{4} \right\}\) |
| 50 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.75 \end{bmatrix}\] | \(\left\{ 4^{3}_{001} \,\middle\|\, m_{110} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{3}{4} \right\}\) |
| 51 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.75 \end{bmatrix}\] | \(\left\{ -4^{1}_{001} \,\middle\|\, 4^{1}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{3}{4} \right\}\) |
| 52 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.75 \end{bmatrix}\] | \(\left\{ -4^{1}_{001} \,\middle\|\, 4^{3}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{3}{4} \right\}\) |
| 53 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.75 \end{bmatrix}\] | \(\left\{ -4^{1}_{001} \,\middle\|\, m_{100} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{3}{4} \right\}\) |
| 54 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.75 \end{bmatrix}\] | \(\left\{ -4^{1}_{001} \,\middle\|\, m_{1-10} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{3}{4} \right\}\) |
| 55 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.75 \end{bmatrix}\] | \(\left\{ -4^{1}_{001} \,\middle\|\, m_{010} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{3}{4} \right\}\) |
| 56 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.75 \end{bmatrix}\] | \(\left\{ -4^{1}_{001} \,\middle\|\, m_{110} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{3}{4} \right\}\) |
| 57 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, 2_{001} \,\middle|\, 0,0,0 \right\}\) |
| 58 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ m_{001} \,\middle\|\, 2_{001} \,\middle|\, 0,0,0 \right\}\) |
| 59 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 2_{001} \,\middle\|\, 2_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 60 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ -1 \,\middle\|\, 2_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 61 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.25 \end{bmatrix}\] | \(\left\{ 4^{1}_{001} \,\middle\|\, 2_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{4} \right\}\) |
| 62 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.25 \end{bmatrix}\] | \(\left\{ -4^{3}_{001} \,\middle\|\, 2_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{4} \right\}\) |
| 63 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.75 \end{bmatrix}\] | \(\left\{ 4^{3}_{001} \,\middle\|\, 2_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{3}{4} \right\}\) |
| 64 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.75 \end{bmatrix}\] | \(\left\{ -4^{1}_{001} \,\middle\|\, 2_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{3}{4} \right\}\) |
| No. | Spin Rotation | Space Rotation | Space Translation | Seitz Symbol |
|---|---|---|---|---|
| 9 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, 4^{1}_{001} \,\middle|\, 0,0,0 \right\}\) |
| 11 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, m_{100} \,\middle|\, 0,0,0 \right\}\) |
| 14 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, m_{110} \,\middle|\, 0,0,0 \right\}\) |
| 5 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.25 \end{bmatrix}\] | \(\left\{ 4^{1}_{001} \,\middle\|\, 1 \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{4} \right\}\) |
| 2 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ m_{001} \,\middle\|\, 1 \,\middle|\, 0,0,0 \right\}\) |
| 3 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 2_{001} \,\middle\|\, 1 \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 6 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.25 \end{bmatrix}\] | \(\left\{ -4^{3}_{001} \,\middle\|\, 1 \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{4} \right\}\) |
| No. | Spin Rotation | Space Rotation | Space Translation | Seitz Symbol |
|---|---|---|---|---|
| 1 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, 1 \,\middle|\, 0,0,0 \right\}\) |
| No. | Spin Rotation | Space Rotation | Space Translation | Seitz Symbol |
|---|---|---|---|---|
| 1 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, 1 \,\middle|\, 0,0,0 \right\}\) |
| 2 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ m_{001} \,\middle\|\, 1 \,\middle|\, 0,0,0 \right\}\) |
| 3 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 2_{001} \,\middle\|\, 1 \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 4 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ -1 \,\middle\|\, 1 \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 5 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.25 \end{bmatrix}\] | \(\left\{ 4^{1}_{001} \,\middle\|\, 1 \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{4} \right\}\) |
| 6 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.25 \end{bmatrix}\] | \(\left\{ -4^{3}_{001} \,\middle\|\, 1 \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{4} \right\}\) |
| 7 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.75 \end{bmatrix}\] | \(\left\{ 4^{3}_{001} \,\middle\|\, 1 \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{3}{4} \right\}\) |
| 8 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.75 \end{bmatrix}\] | \(\left\{ -4^{1}_{001} \,\middle\|\, 1 \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{3}{4} \right\}\) |
| No. | Spin Rotation | Space Rotation | Space Translation | Seitz Symbol |
|---|---|---|---|---|
| 1 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, 1 \,\middle|\, 0,0,0 \right\}\) |
| 9 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, 4^{1}_{001} \,\middle|\, 0,0,0 \right\}\) |
| 10 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, 4^{3}_{001} \,\middle|\, 0,0,0 \right\}\) |
| 11 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, m_{100} \,\middle|\, 0,0,0 \right\}\) |
| 12 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, m_{1-10} \,\middle|\, 0,0,0 \right\}\) |
| 13 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, m_{010} \,\middle|\, 0,0,0 \right\}\) |
| 14 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, m_{110} \,\middle|\, 0,0,0 \right\}\) |
| 57 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, 2_{001} \,\middle|\, 0,0,0 \right\}\) |
| No. | Spin Rotation | Space Rotation | Space Translation | Seitz Symbol |
|---|---|---|---|---|
| 1 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, 1 \,\middle|\, 0,0,0 \right\}\) |
| 3 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 2_{001} \,\middle\|\, 1 \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 5 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.25 \end{bmatrix}\] | \(\left\{ 4^{1}_{001} \,\middle\|\, 1 \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{4} \right\}\) |
| 7 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.75 \end{bmatrix}\] | \(\left\{ 4^{3}_{001} \,\middle\|\, 1 \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{3}{4} \right\}\) |
| 9 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, 4^{1}_{001} \,\middle|\, 0,0,0 \right\}\) |
| 10 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, 4^{3}_{001} \,\middle|\, 0,0,0 \right\}\) |
| 11 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, m_{100} \,\middle|\, 0,0,0 \right\}\) |
| 12 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, m_{1-10} \,\middle|\, 0,0,0 \right\}\) |
| 13 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, m_{010} \,\middle|\, 0,0,0 \right\}\) |
| 14 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, m_{110} \,\middle|\, 0,0,0 \right\}\) |
| 21 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 2_{001} \,\middle\|\, 4^{1}_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 22 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 2_{001} \,\middle\|\, 4^{3}_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 23 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 2_{001} \,\middle\|\, m_{100} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 24 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 2_{001} \,\middle\|\, m_{1-10} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 25 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 2_{001} \,\middle\|\, m_{010} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 26 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 2_{001} \,\middle\|\, m_{110} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 33 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.25 \end{bmatrix}\] | \(\left\{ 4^{1}_{001} \,\middle\|\, 4^{1}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{4} \right\}\) |
| 34 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.25 \end{bmatrix}\] | \(\left\{ 4^{1}_{001} \,\middle\|\, 4^{3}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{4} \right\}\) |
| 35 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.25 \end{bmatrix}\] | \(\left\{ 4^{1}_{001} \,\middle\|\, m_{100} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{4} \right\}\) |
| 36 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.25 \end{bmatrix}\] | \(\left\{ 4^{1}_{001} \,\middle\|\, m_{1-10} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{4} \right\}\) |
| 37 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.25 \end{bmatrix}\] | \(\left\{ 4^{1}_{001} \,\middle\|\, m_{010} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{4} \right\}\) |
| 38 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.25 \end{bmatrix}\] | \(\left\{ 4^{1}_{001} \,\middle\|\, m_{110} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{4} \right\}\) |
| 45 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.75 \end{bmatrix}\] | \(\left\{ 4^{3}_{001} \,\middle\|\, 4^{1}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{3}{4} \right\}\) |
| 46 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.75 \end{bmatrix}\] | \(\left\{ 4^{3}_{001} \,\middle\|\, 4^{3}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{3}{4} \right\}\) |
| 47 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.75 \end{bmatrix}\] | \(\left\{ 4^{3}_{001} \,\middle\|\, m_{100} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{3}{4} \right\}\) |
| 48 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.75 \end{bmatrix}\] | \(\left\{ 4^{3}_{001} \,\middle\|\, m_{1-10} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{3}{4} \right\}\) |
| 49 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.75 \end{bmatrix}\] | \(\left\{ 4^{3}_{001} \,\middle\|\, m_{010} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{3}{4} \right\}\) |
| 50 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.75 \end{bmatrix}\] | \(\left\{ 4^{3}_{001} \,\middle\|\, m_{110} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{3}{4} \right\}\) |
| 57 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, 2_{001} \,\middle|\, 0,0,0 \right\}\) |
| 59 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 2_{001} \,\middle\|\, 2_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 61 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.25 \end{bmatrix}\] | \(\left\{ 4^{1}_{001} \,\middle\|\, 2_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{4} \right\}\) |
| 63 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.75 \end{bmatrix}\] | \(\left\{ 4^{3}_{001} \,\middle\|\, 2_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{3}{4} \right\}\) |
| No. | Spin Rotation | Space Rotation | Space Translation | Seitz Symbol |
|---|---|---|---|---|
| 1 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, 1 \,\middle|\, 0,0,0 \right\}\) |
| 4 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ -1 \,\middle\|\, 1 \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 33 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.25 \end{bmatrix}\] | \(\left\{ 4^{1}_{001} \,\middle\|\, 4^{1}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{4} \right\}\) |
| 40 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.25 \end{bmatrix}\] | \(\left\{ -4^{3}_{001} \,\middle\|\, 4^{3}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{4} \right\}\) |
| 46 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.75 \end{bmatrix}\] | \(\left\{ 4^{3}_{001} \,\middle\|\, 4^{3}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{3}{4} \right\}\) |
| 51 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.75 \end{bmatrix}\] | \(\left\{ -4^{1}_{001} \,\middle\|\, 4^{1}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{3}{4} \right\}\) |
| 58 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ m_{001} \,\middle\|\, 2_{001} \,\middle|\, 0,0,0 \right\}\) |
| 59 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 2_{001} \,\middle\|\, 2_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |