| File Name | Oriented Spin Space Group | G0 Symbol | L0 Symbol | it | ik |
|---|---|---|---|---|---|
| \(0.199.Mn3Sn\) | \(P ^{3^{1}_{001}}6_{3}/ ^{1}m ^{2_{110}}m ^{2_{010}}c ^{m_{001}}1\) \((194.11.1.1.P)\) | \(P6_3/mmc\ (194)\) | \(P2_1/m\ (11)\) | \(6\) | \(1\) |
| Space Group | Magnetic Space Group | Spin-space PG | Configuration | Magnetic Phase |
|---|---|---|---|---|
| \(P6_3/mmc\ (194)\) | \[\begin{aligned}&Cmc'm'(63.463)\\ &Type\ III\end{aligned}\] | \[\begin{aligned}-62m\\ D3h\end{aligned}\] | \(Coplanar\) | \(AFM (SOM)\) |
| Little Cogroup at General Position | Spin Splitting | Spin Texture | |||
|---|---|---|---|---|---|
| without SOC | with SOC | without SOC | with SOC | without SOC | with SOC |
| $$^{m}-1$$ | $$1$$ | \(k-dependent\) | \(Yes\) | \(\mathrm{d-wave}\) \(C_{1}\left(\frac{\sqrt{3}}{6}k_{y}^{2}\,\sigma_{x} + k_{x}k_{y}\,\sigma_{x} - \frac{\sqrt{3}}{6}k_{x}^{2}\,\sigma_{x} + \frac{1}{2}k_{y}^{2}\,\sigma_{y} - \frac{\sqrt{3}}{3}k_{x}k_{y}\,\sigma_{y} - \frac{1}{2}k_{x}^{2}\,\sigma_{y}\right)\) | \(\mathrm{s-wave}\) \(C_{1}\left(-\frac{\sqrt{3}}{3}\,\sigma_{x} + \sigma_{y}\right)\) |
| Effective Magnetic Point Group | Anomalous Hall Effect | ||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $$6/mmm1'$$ | 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 | $$0$$ | $$0$$ |
G0-std Unit Magnetic Cell:
| a | b | c | alpha | beta | gamma |
|---|---|---|---|---|---|
| 5.665 | 5.665 | 4.531 | 90.0 | 90.0 | 120.0 |
| Transformation from input to G0-standard setting | Transformation from G0-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_G0std | Tp * r_G0std + τp = r_prim | ||
Fractional Coordinates & Moments (in G0-std Unit Magnetic Cell)
| No. | Atom | Site | Moment (Oriented) |
|---|---|---|---|
| 1 | Mn |
[0.3224, 0.1612, 0.25] |
[0.0, -0.5296, 0.0] |
| 2 | Mn |
[0.6776, 0.8388, 0.75] |
[0.0, -0.5296, 0.0] |
| 3 | Mn |
[0.1612, 0.8388, 0.75] |
[-0.5296, 0.0, 0.0] |
| 4 | Mn |
[0.8388, 0.1612, 0.25] |
[-0.5296, 0.0, 0.0] |
| 5 | Mn |
[0.1612, 0.3224, 0.75] |
[0.5296, 0.5296, 0.0] |
| 6 | Mn |
[0.8388, 0.6776, 0.25] |
[0.5296, 0.5296, 0.0] |
| 7 | Sn |
[0.3333, 0.6667, 0.25] |
[0.0, 0.0, 0.0] |
| 8 | Sn |
[0.6667, 0.3333, 0.75] |
[0.0, 0.0, 0.0] |
| Element | SG Wyckoff | SSG Wyckoff | MSG Wyckoff |
|---|---|---|---|
| Mn | 6h | 6h(1) | 2c(1) |
| 4g(2) | |||
| Sn | 2c | 2c | 2c |
Elements of the Spin Space Group (G0-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 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 3^{1}_{001} \,\middle\|\, 6^{1}_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 4 | \[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 3^{2}_{001} \,\middle\|\, 6^{5}_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 5 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ -6^{5}_{001} \,\middle\|\, 6^{1}_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 6 | \[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ -6^{1}_{001} \,\middle\|\, 6^{5}_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 7 | \[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ m_{120} \,\middle\|\, m_{110} \,\middle|\, 0,0,0 \right\}\) |
| 8 | \[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 2_{100} \,\middle\|\, m_{110} \,\middle|\, 0,0,0 \right\}\) |
| 9 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, m_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 10 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ m_{001} \,\middle\|\, m_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 11 | \[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ m_{120} \,\middle\|\, m_{1-10} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 12 | \[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 2_{100} \,\middle\|\, m_{1-10} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 13 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ m_{1-10} \,\middle\|\, m_{100} \,\middle|\, 0,0,0 \right\}\) |
| 14 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ m_{210} \,\middle\|\, m_{010} \,\middle|\, 0,0,0 \right\}\) |
| 15 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 2_{110} \,\middle\|\, m_{100} \,\middle|\, 0,0,0 \right\}\) |
| 16 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 2_{010} \,\middle\|\, m_{010} \,\middle|\, 0,0,0 \right\}\) |
| 17 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ m_{1-10} \,\middle\|\, m_{120} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 18 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ m_{210} \,\middle\|\, m_{210} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 19 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 2_{110} \,\middle\|\, m_{120} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 20 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 2_{010} \,\middle\|\, m_{210} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 21 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 3^{1}_{001} \,\middle\|\, -3^{2}_{001} \,\middle|\, 0,0,0 \right\}\) |
| 22 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 3^{1}_{001} \,\middle\|\, 3^{2}_{001} \,\middle|\, 0,0,0 \right\}\) |
| 23 | \[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 3^{2}_{001} \,\middle\|\, 3^{1}_{001} \,\middle|\, 0,0,0 \right\}\) |
| 24 | \[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 3^{2}_{001} \,\middle\|\, -3^{1}_{001} \,\middle|\, 0,0,0 \right\}\) |
| 25 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ -6^{5}_{001} \,\middle\|\, -3^{2}_{001} \,\middle|\, 0,0,0 \right\}\) |
| 26 | \[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ -6^{1}_{001} \,\middle\|\, 3^{1}_{001} \,\middle|\, 0,0,0 \right\}\) |
| 27 | \[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ -6^{1}_{001} \,\middle\|\, -3^{1}_{001} \,\middle|\, 0,0,0 \right\}\) |
| 28 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ -6^{5}_{001} \,\middle\|\, 3^{2}_{001} \,\middle|\, 0,0,0 \right\}\) |
| 29 | \[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ m_{120} \,\middle\|\, 2_{110} \,\middle|\, 0,0,0 \right\}\) |
| 30 | \[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 2_{100} \,\middle\|\, 2_{110} \,\middle|\, 0,0,0 \right\}\) |
| 31 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, 2_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 32 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ m_{001} \,\middle\|\, 2_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 33 | \[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ m_{120} \,\middle\|\, 2_{1-10} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 34 | \[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 2_{100} \,\middle\|\, 2_{1-10} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 35 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ m_{1-10} \,\middle\|\, 2_{100} \,\middle|\, 0,0,0 \right\}\) |
| 36 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ m_{210} \,\middle\|\, 2_{010} \,\middle|\, 0,0,0 \right\}\) |
| 37 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 2_{110} \,\middle\|\, 2_{100} \,\middle|\, 0,0,0 \right\}\) |
| 38 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 2_{010} \,\middle\|\, 2_{010} \,\middle|\, 0,0,0 \right\}\) |
| 39 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ m_{1-10} \,\middle\|\, 2_{120} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 40 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ m_{210} \,\middle\|\, 2_{210} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 41 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 2_{110} \,\middle\|\, 2_{120} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 42 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 2_{010} \,\middle\|\, 2_{210} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 43 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 3^{1}_{001} \,\middle\|\, -6^{1}_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 44 | \[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 3^{2}_{001} \,\middle\|\, -6^{5}_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 45 | \[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ -6^{1}_{001} \,\middle\|\, -6^{5}_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 46 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ -6^{5}_{001} \,\middle\|\, -6^{1}_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 47 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, -1 \,\middle|\, 0,0,0 \right\}\) |
| 48 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ m_{001} \,\middle\|\, -1 \,\middle|\, 0,0,0 \right\}\) |
| No. | Spin Rotation | Space Rotation | Space Translation | Seitz Symbol |
|---|---|---|---|---|
| 3 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 3^{1}_{001} \,\middle\|\, 6^{1}_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 9 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, m_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 15 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 2_{110} \,\middle\|\, m_{100} \,\middle|\, 0,0,0 \right\}\) |
| 20 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 2_{010} \,\middle\|\, m_{210} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 2 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ m_{001} \,\middle\|\, 1 \,\middle|\, 0,0,0 \right\}\) |
| 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 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ m_{001} \,\middle\|\, 1 \,\middle|\, 0,0,0 \right\}\) |
| 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 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, m_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 31 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, 2_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 47 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, -1 \,\middle|\, 0,0,0 \right\}\) |
| No. | Spin Rotation | Space Rotation | Space Translation | Seitz Symbol |
|---|---|---|---|---|
| 1 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, 1 \,\middle|\, 0,0,0 \right\}\) |
| 3 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 3^{1}_{001} \,\middle\|\, 6^{1}_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 4 | \[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 3^{2}_{001} \,\middle\|\, 6^{5}_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 8 | \[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 2_{100} \,\middle\|\, m_{110} \,\middle|\, 0,0,0 \right\}\) |
| 9 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, m_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 12 | \[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 2_{100} \,\middle\|\, m_{1-10} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 15 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 2_{110} \,\middle\|\, m_{100} \,\middle|\, 0,0,0 \right\}\) |
| 16 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 2_{010} \,\middle\|\, m_{010} \,\middle|\, 0,0,0 \right\}\) |
| 19 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 2_{110} \,\middle\|\, m_{120} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 20 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 2_{010} \,\middle\|\, m_{210} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 21 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 3^{1}_{001} \,\middle\|\, -3^{2}_{001} \,\middle|\, 0,0,0 \right\}\) |
| 22 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 3^{1}_{001} \,\middle\|\, 3^{2}_{001} \,\middle|\, 0,0,0 \right\}\) |
| 23 | \[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 3^{2}_{001} \,\middle\|\, 3^{1}_{001} \,\middle|\, 0,0,0 \right\}\) |
| 24 | \[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 3^{2}_{001} \,\middle\|\, -3^{1}_{001} \,\middle|\, 0,0,0 \right\}\) |
| 30 | \[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 2_{100} \,\middle\|\, 2_{110} \,\middle|\, 0,0,0 \right\}\) |
| 31 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, 2_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 34 | \[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 2_{100} \,\middle\|\, 2_{1-10} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 37 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 2_{110} \,\middle\|\, 2_{100} \,\middle|\, 0,0,0 \right\}\) |
| 38 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 2_{010} \,\middle\|\, 2_{010} \,\middle|\, 0,0,0 \right\}\) |
| 41 | \[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 2_{110} \,\middle\|\, 2_{120} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 42 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 2_{010} \,\middle\|\, 2_{210} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 43 | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 3^{1}_{001} \,\middle\|\, -6^{1}_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 44 | \[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ 3^{2}_{001} \,\middle\|\, -6^{5}_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 47 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, -1 \,\middle|\, 0,0,0 \right\}\) |
| No. | Spin Rotation | Space Rotation | Space Translation | Seitz Symbol |
|---|---|---|---|---|
| 1 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, 1 \,\middle|\, 0,0,0 \right\}\) |
| 10 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ m_{001} \,\middle\|\, m_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 16 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 2_{010} \,\middle\|\, m_{010} \,\middle|\, 0,0,0 \right\}\) |
| 18 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ m_{210} \,\middle\|\, m_{210} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 32 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ m_{001} \,\middle\|\, 2_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 38 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 2_{010} \,\middle\|\, 2_{010} \,\middle|\, 0,0,0 \right\}\) |
| 40 | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\] | \(\left\{ m_{210} \,\middle\|\, 2_{210} \,\middle|\, 0,0,\frac{1}{2} \right\}\) |
| 47 | \[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] | \[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\] | \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] | \(\left\{ 1 \,\middle\|\, -1 \,\middle|\, 0,0,0 \right\}\) |