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FINDSPINGROUP: Identify Spin Space Group and Related Properties

File NameOriented Spin Space GroupG0 SymbolL0 Symbolitik
\(0.20.MnTe2\)
\(P ^{2_{100}}a ^{3^{1}_{111}}\bar{3}\) \((205.2.1.1)\)
\(Pa-3\ (205)\)\(P-1\ (2)\)\(12\)\(1\)
Space Group Magnetic Space Group Spin-space PGConfigurationMagnetic Phase
\(Pa-3\ (205)\)\[\begin{aligned}&Pa-3(205.33)\\ &Type\ I\end{aligned}\]\[\begin{aligned}23\\ T\end{aligned}\]\(Noncoplanar\)\(AFM\)
Little Cogroup at General PositionSpin SplittingSpin TextureExpressions are written in the special Cartesian frame used by the SSG convention: x is parallel to a, y lies in the ab plane perpendicular to x and chosen toward b, and z = x × y. The listed spin textures are symmetry-allowed leading terms. They describe the lowest-order terms not forbidden by symmetry; the actual low-energy or Fermi-surface spin texture may start at a higher order. For example, a symmetry-allowed p-wave term may vanish for microscopic reasons, and the leading observable texture near the Fermi surface may instead be f-wave or higher-order.
without SOCwith SOCwithout SOCwith SOCwithout SOCwith SOC
$$^{1}1$$$$1$$\(k-dependent\)\(Yes\)
\(\mathrm{d-wave}\)
\(C_{1}\left(k_{y}k_{z}\,\sigma_{x} + k_{x}k_{z}\,\sigma_{y} + k_{x}k_{y}\,\sigma_{z}\right)\)
\(\mathrm{d-wave}\)
\(C_{1}\left(k_{y}k_{z}\,\sigma_{x} + k_{x}k_{z}\,\sigma_{y} + k_{x}k_{y}\,\sigma_{z}\right)\)
Effective Magnetic Point GroupEffective Magnetic Point Group is obtained from the spin space group by mapping spin rotations with det = -1 to time reversal and then removing the translation part. Anomalous Hall Effect
$$m-3$$ without SOC with SOC
xyz
x$$0$$$$0$$$$0$$
y$$0$$$$0$$$$0$$
z$$0$$$$0$$$$0$$
xyz
x$$0$$$$0$$$$0$$
y$$0$$$$0$$$$0$$
z$$0$$$$0$$$$0$$
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
6.9 6.9 6.9 90.0 90.0 90.0
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

[0.0, 0.0, 0.0]

[0.358, 0.358, 0.358]
2

Mn

[0.0, 0.5, 0.5]

[-0.358, 0.358, -0.358]
3

Mn

[0.5, 0.0, 0.5]

[-0.358, -0.358, 0.358]
4

Mn

[0.5, 0.5, 0.0]

[0.358, -0.358, -0.358]
5

Te

[0.1153, 0.6153, 0.8847]

[0.0, 0.0, 0.0]
6

Te

[0.1153, 0.8847, 0.3847]

[0.0, 0.0, 0.0]
7

Te

[0.3847, 0.1153, 0.8847]

[0.0, 0.0, 0.0]
8

Te

[0.3847, 0.3847, 0.3847]

[0.0, 0.0, 0.0]
9

Te

[0.6153, 0.6153, 0.6153]

[0.0, 0.0, 0.0]
10

Te

[0.6153, 0.8847, 0.1153]

[0.0, 0.0, 0.0]
11

Te

[0.8847, 0.1153, 0.6153]

[0.0, 0.0, 0.0]
12

Te

[0.8847, 0.3847, 0.1153]

[0.0, 0.0, 0.0]
ElementSG WyckoffSSG WyckoffFor magnetic atoms, the number in parentheses after an SSG or MSG Wyckoff symbol is the constrained magnetic freedom degree of that magnetic site.MSG WyckoffFor magnetic atoms, the number in parentheses after an SSG or MSG Wyckoff symbol is the constrained magnetic freedom degree of that magnetic site.
Mn4a4a(1)4a(1)
Te8c8c8c

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 RotationSpace RotationSpace TranslationSeitz 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\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 2_{100} \,\middle\|\, m_{100} \,\middle|\, \frac{1}{2},\frac{1}{2},0 \right\}\)
3\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ 2_{001} \,\middle\|\, m_{001} \,\middle|\, \frac{1}{2},0,\frac{1}{2} \right\}\)
4\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ 2_{010} \,\middle\|\, m_{010} \,\middle|\, 0,\frac{1}{2},\frac{1}{2} \right\}\)
5\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \\ 1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \\ 1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{2}_{111} \,\middle\|\, 3^{2}_{111} \,\middle|\, 0,0,0 \right\}\)
6\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \\ 1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0 \\ -1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{2}_{111} \,\middle\|\, -3^{2}_{111} \,\middle|\, 0,0,0 \right\}\)
7\[\begin{bmatrix} 0.0 & 0.0 & 1.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 0.0 & 1.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{1}_{111} \,\middle\|\, 3^{1}_{111} \,\middle|\, 0,0,0 \right\}\)
8\[\begin{bmatrix} 0.0 & 0.0 & 1.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 0.0 & -1.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{1}_{111} \,\middle\|\, -3^{1}_{111} \,\middle|\, 0,0,0 \right\}\)
9\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \\ -1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \\ -1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{2}_{1-1-1} \,\middle\|\, 3^{2}_{1-1-1} \,\middle|\, 0,\frac{1}{2},\frac{1}{2} \right\}\)
10\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \\ -1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0 \\ 1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{2}_{1-1-1} \,\middle\|\, -3^{2}_{1-1-1} \,\middle|\, 0,\frac{1}{2},\frac{1}{2} \right\}\)
11\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0 \\ 1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0 \\ 1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{1}_{1-11} \,\middle\|\, 3^{1}_{1-11} \,\middle|\, \frac{1}{2},0,\frac{1}{2} \right\}\)
12\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0 \\ 1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \\ -1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{1}_{1-11} \,\middle\|\, 3^{1}_{1-11} \,\middle|\, \frac{1}{2},0,\frac{1}{2} \right\}\)
13\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0 \\ -1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0 \\ -1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{1}_{11-1} \,\middle\|\, 3^{1}_{11-1} \,\middle|\, \frac{1}{2},\frac{1}{2},0 \right\}\)
14\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0 \\ -1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \\ 1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{1}_{11-1} \,\middle\|\, -3^{1}_{11-1} \,\middle|\, \frac{1}{2},\frac{1}{2},0 \right\}\)
15\[\begin{bmatrix} 0.0 & 0.0 & 1.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 0.0 & 1.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{2}_{1-11} \,\middle\|\, 3^{2}_{1-11} \,\middle|\, \frac{1}{2},\frac{1}{2},0 \right\}\)
16\[\begin{bmatrix} 0.0 & 0.0 & 1.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 0.0 & -1.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{2}_{1-11} \,\middle\|\, 3^{2}_{1-11} \,\middle|\, \frac{1}{2},\frac{1}{2},0 \right\}\)
17\[\begin{bmatrix} 0.0 & 0.0 & -1.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 0.0 & -1.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{2}_{11-1} \,\middle\|\, 3^{2}_{11-1} \,\middle|\, 0,\frac{1}{2},\frac{1}{2} \right\}\)
18\[\begin{bmatrix} 0.0 & 0.0 & -1.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 0.0 & 1.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{2}_{11-1} \,\middle\|\, -3^{2}_{11-1} \,\middle|\, 0,\frac{1}{2},\frac{1}{2} \right\}\)
19\[\begin{bmatrix} 0.0 & 0.0 & -1.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 0.0 & -1.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{1}_{1-1-1} \,\middle\|\, 3^{1}_{1-1-1} \,\middle|\, \frac{1}{2},0,\frac{1}{2} \right\}\)
20\[\begin{bmatrix} 0.0 & 0.0 & -1.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 0.0 & 1.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{1}_{1-1-1} \,\middle\|\, -3^{1}_{1-1-1} \,\middle|\, \frac{1}{2},0,\frac{1}{2} \right\}\)
21\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 2_{100} \,\middle\|\, 2_{100} \,\middle|\, \frac{1}{2},\frac{1}{2},0 \right\}\)
22\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ 2_{001} \,\middle\|\, 2_{001} \,\middle|\, \frac{1}{2},0,\frac{1}{2} \right\}\)
23\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ 2_{010} \,\middle\|\, 2_{010} \,\middle|\, 0,\frac{1}{2},\frac{1}{2} \right\}\)
24\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, -1 \,\middle|\, 0,0,0 \right\}\)
No.Spin RotationSpace RotationSpace TranslationSeitz Symbol
2\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 2_{100} \,\middle\|\, m_{100} \,\middle|\, \frac{1}{2},\frac{1}{2},0 \right\}\)
8\[\begin{bmatrix} 0.0 & 0.0 & 1.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 0.0 & -1.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{1}_{111} \,\middle\|\, -3^{1}_{111} \,\middle|\, 0,0,0 \right\}\)
No.Spin RotationSpace RotationSpace TranslationSeitz 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 RotationSpace RotationSpace TranslationSeitz 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 RotationSpace RotationSpace TranslationSeitz Symbol
1\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, 1 \,\middle|\, 0,0,0 \right\}\)
24\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, -1 \,\middle|\, 0,0,0 \right\}\)
No.Spin RotationSpace RotationSpace TranslationSeitz 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\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 2_{100} \,\middle\|\, m_{100} \,\middle|\, \frac{1}{2},\frac{1}{2},0 \right\}\)
3\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ 2_{001} \,\middle\|\, m_{001} \,\middle|\, \frac{1}{2},0,\frac{1}{2} \right\}\)
4\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ 2_{010} \,\middle\|\, m_{010} \,\middle|\, 0,\frac{1}{2},\frac{1}{2} \right\}\)
5\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \\ 1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \\ 1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{2}_{111} \,\middle\|\, 3^{2}_{111} \,\middle|\, 0,0,0 \right\}\)
6\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \\ 1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0 \\ -1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{2}_{111} \,\middle\|\, -3^{2}_{111} \,\middle|\, 0,0,0 \right\}\)
7\[\begin{bmatrix} 0.0 & 0.0 & 1.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 0.0 & 1.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{1}_{111} \,\middle\|\, 3^{1}_{111} \,\middle|\, 0,0,0 \right\}\)
8\[\begin{bmatrix} 0.0 & 0.0 & 1.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 0.0 & -1.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{1}_{111} \,\middle\|\, -3^{1}_{111} \,\middle|\, 0,0,0 \right\}\)
9\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \\ -1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \\ -1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{2}_{1-1-1} \,\middle\|\, 3^{2}_{1-1-1} \,\middle|\, 0,\frac{1}{2},\frac{1}{2} \right\}\)
10\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \\ -1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0 \\ 1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{2}_{1-1-1} \,\middle\|\, -3^{2}_{1-1-1} \,\middle|\, 0,\frac{1}{2},\frac{1}{2} \right\}\)
11\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0 \\ 1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0 \\ 1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{1}_{1-11} \,\middle\|\, 3^{1}_{1-11} \,\middle|\, \frac{1}{2},0,\frac{1}{2} \right\}\)
12\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0 \\ 1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \\ -1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{1}_{1-11} \,\middle\|\, 3^{1}_{1-11} \,\middle|\, \frac{1}{2},0,\frac{1}{2} \right\}\)
13\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0 \\ -1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0 \\ -1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{1}_{11-1} \,\middle\|\, 3^{1}_{11-1} \,\middle|\, \frac{1}{2},\frac{1}{2},0 \right\}\)
14\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0 \\ -1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \\ 1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{1}_{11-1} \,\middle\|\, -3^{1}_{11-1} \,\middle|\, \frac{1}{2},\frac{1}{2},0 \right\}\)
15\[\begin{bmatrix} 0.0 & 0.0 & 1.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 0.0 & 1.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{2}_{1-11} \,\middle\|\, 3^{2}_{1-11} \,\middle|\, \frac{1}{2},\frac{1}{2},0 \right\}\)
16\[\begin{bmatrix} 0.0 & 0.0 & 1.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 0.0 & -1.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{2}_{1-11} \,\middle\|\, 3^{2}_{1-11} \,\middle|\, \frac{1}{2},\frac{1}{2},0 \right\}\)
17\[\begin{bmatrix} 0.0 & 0.0 & -1.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 0.0 & -1.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{2}_{11-1} \,\middle\|\, 3^{2}_{11-1} \,\middle|\, 0,\frac{1}{2},\frac{1}{2} \right\}\)
18\[\begin{bmatrix} 0.0 & 0.0 & -1.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 0.0 & 1.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{2}_{11-1} \,\middle\|\, -3^{2}_{11-1} \,\middle|\, 0,\frac{1}{2},\frac{1}{2} \right\}\)
19\[\begin{bmatrix} 0.0 & 0.0 & -1.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 0.0 & -1.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{1}_{1-1-1} \,\middle\|\, 3^{1}_{1-1-1} \,\middle|\, \frac{1}{2},0,\frac{1}{2} \right\}\)
20\[\begin{bmatrix} 0.0 & 0.0 & -1.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 0.0 & 1.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{1}_{1-1-1} \,\middle\|\, -3^{1}_{1-1-1} \,\middle|\, \frac{1}{2},0,\frac{1}{2} \right\}\)
21\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 2_{100} \,\middle\|\, 2_{100} \,\middle|\, \frac{1}{2},\frac{1}{2},0 \right\}\)
22\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ 2_{001} \,\middle\|\, 2_{001} \,\middle|\, \frac{1}{2},0,\frac{1}{2} \right\}\)
23\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ 2_{010} \,\middle\|\, 2_{010} \,\middle|\, 0,\frac{1}{2},\frac{1}{2} \right\}\)
24\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, -1 \,\middle|\, 0,0,0 \right\}\)
No.Spin RotationSpace RotationSpace TranslationSeitz 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\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 2_{100} \,\middle\|\, m_{100} \,\middle|\, \frac{1}{2},\frac{1}{2},0 \right\}\)
3\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ 2_{001} \,\middle\|\, m_{001} \,\middle|\, \frac{1}{2},0,\frac{1}{2} \right\}\)
4\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ 2_{010} \,\middle\|\, m_{010} \,\middle|\, 0,\frac{1}{2},\frac{1}{2} \right\}\)
5\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \\ 1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \\ 1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{2}_{111} \,\middle\|\, 3^{2}_{111} \,\middle|\, 0,0,0 \right\}\)
6\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \\ 1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0 \\ -1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{2}_{111} \,\middle\|\, -3^{2}_{111} \,\middle|\, 0,0,0 \right\}\)
7\[\begin{bmatrix} 0.0 & 0.0 & 1.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 0.0 & 1.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{1}_{111} \,\middle\|\, 3^{1}_{111} \,\middle|\, 0,0,0 \right\}\)
8\[\begin{bmatrix} 0.0 & 0.0 & 1.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 0.0 & -1.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{1}_{111} \,\middle\|\, -3^{1}_{111} \,\middle|\, 0,0,0 \right\}\)
9\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \\ -1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \\ -1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{2}_{1-1-1} \,\middle\|\, 3^{2}_{1-1-1} \,\middle|\, 0,\frac{1}{2},\frac{1}{2} \right\}\)
10\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \\ -1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0 \\ 1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{2}_{1-1-1} \,\middle\|\, -3^{2}_{1-1-1} \,\middle|\, 0,\frac{1}{2},\frac{1}{2} \right\}\)
11\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0 \\ 1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0 \\ 1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{1}_{1-11} \,\middle\|\, 3^{1}_{1-11} \,\middle|\, \frac{1}{2},0,\frac{1}{2} \right\}\)
12\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0 \\ 1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \\ -1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{1}_{1-11} \,\middle\|\, 3^{1}_{1-11} \,\middle|\, \frac{1}{2},0,\frac{1}{2} \right\}\)
13\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0 \\ -1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0 \\ -1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{1}_{11-1} \,\middle\|\, 3^{1}_{11-1} \,\middle|\, \frac{1}{2},\frac{1}{2},0 \right\}\)
14\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0 \\ -1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \\ 1.0 & 0.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{1}_{11-1} \,\middle\|\, -3^{1}_{11-1} \,\middle|\, \frac{1}{2},\frac{1}{2},0 \right\}\)
15\[\begin{bmatrix} 0.0 & 0.0 & 1.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 0.0 & 1.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{2}_{1-11} \,\middle\|\, 3^{2}_{1-11} \,\middle|\, \frac{1}{2},\frac{1}{2},0 \right\}\)
16\[\begin{bmatrix} 0.0 & 0.0 & 1.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 0.0 & -1.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{2}_{1-11} \,\middle\|\, 3^{2}_{1-11} \,\middle|\, \frac{1}{2},\frac{1}{2},0 \right\}\)
17\[\begin{bmatrix} 0.0 & 0.0 & -1.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 0.0 & -1.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{2}_{11-1} \,\middle\|\, 3^{2}_{11-1} \,\middle|\, 0,\frac{1}{2},\frac{1}{2} \right\}\)
18\[\begin{bmatrix} 0.0 & 0.0 & -1.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 0.0 & 1.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{2}_{11-1} \,\middle\|\, -3^{2}_{11-1} \,\middle|\, 0,\frac{1}{2},\frac{1}{2} \right\}\)
19\[\begin{bmatrix} 0.0 & 0.0 & -1.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 0.0 & -1.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{1}_{1-1-1} \,\middle\|\, 3^{1}_{1-1-1} \,\middle|\, \frac{1}{2},0,\frac{1}{2} \right\}\)
20\[\begin{bmatrix} 0.0 & 0.0 & -1.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 0.0 & 1.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{1}_{1-1-1} \,\middle\|\, -3^{1}_{1-1-1} \,\middle|\, \frac{1}{2},0,\frac{1}{2} \right\}\)
21\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 2_{100} \,\middle\|\, 2_{100} \,\middle|\, \frac{1}{2},\frac{1}{2},0 \right\}\)
22\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ 2_{001} \,\middle\|\, 2_{001} \,\middle|\, \frac{1}{2},0,\frac{1}{2} \right\}\)
23\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ 2_{010} \,\middle\|\, 2_{010} \,\middle|\, 0,\frac{1}{2},\frac{1}{2} \right\}\)
24\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, -1 \,\middle|\, 0,0,0 \right\}\)