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FINDSPINGROUP: Identify Spin Space Group and Related Properties

File NameOriented Spin Space GroupG0 SymbolL0 Symbolitik
\(CoNb3S6.tripleQ\)
\(P ^{3^{1}_{\alpha_{1},\beta_{1},\gamma_{1}}}6_{3} ^{m_{100}}2 ^{m_{\alpha_{2},\beta_{2},\gamma_{2}}}2 \mid (2_{\alpha_{3},\beta_{3},\gamma_{3}},2_{\alpha_{4},\beta_{4},\gamma_{4}},1)\) \((182.4.4.2)\)
\(P6_322\ (182)\)\(P2_1\ (4)\)\(6\)\(4\)
Space Group Magnetic Space Group Spin-space PGConfigurationMagnetic PhaseSOM here means spin-orbit magnet: the core classified the case as AFM, but the SOC-side magnetic-point-group route is FM-like.
\(P6_322\ (182)\)\[\begin{aligned}&P32'1(150.27)\\ &Type\ III\end{aligned}\]\[\begin{aligned}-43m\\ Td\end{aligned}\]\(Noncoplanar\)\(AFM (SOM)\)
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
$$^{1}1^{222}1$$$$1$$\(No\)\(Yes\)
\(\mathrm{forbidden}\)
None
\(\mathrm{s-wave}\)
\(C_{1}\left(\sigma_{z}\right)\)
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
$$62'2'$$ without SOC with SOC
xyz
x$$0$$$$-\sigma_{yx}$$$$0$$
y$$\sigma_{yx}$$$$0$$$$0$$
z$$0$$$$0$$$$0$$
xyz
x$$0$$$$-\sigma_{yx}$$$$0$$
y$$\sigma_{yx}$$$$0$$$$0$$
z$$0$$$$0$$$$0$$
Effective MPG Constraint on rank-3 Tensor
without SOC with SOC
T-odd Quantum metric dipole $$\begin{aligned}Q_{xxz} = Q_{xzx} = Q_{yyz} = Q_{yzy}\\ Q_{zxx} = Q_{zyy}\\ Q_{zzz}\end{aligned}$$ $$\begin{aligned}Q_{xxy} = Q_{xyx} = Q_{yxx} = -Q_{yyy}\\ Q_{xxz} = Q_{xzx} = Q_{yyz} = Q_{yzy}\\ Q_{zxx} = Q_{zyy}\\ Q_{zzz}\end{aligned}$$
T-odd Inversed mass dipole $$\begin{aligned}I_{xxz} = I_{xzx} = I_{yyz} = I_{yzy} = I_{zxx} = I_{zyy}\\ I_{zzz}\end{aligned}$$ $$\begin{aligned}I_{xxy} = I_{xyx} = I_{yxx} = -I_{yyy}\\ I_{xxz} = I_{xzx} = I_{yyz} = I_{yzy} = I_{zxx} = I_{zyy}\\ I_{zzz}\end{aligned}$$
T-even Berry curvature dipole $$\begin{aligned}D_{xyz} = D_{xzy} = -D_{yxz} = -D_{yzx}\end{aligned}$$ $$\begin{aligned}D_{xyz} = D_{xzy} = -D_{yxz} = -D_{yzx}\end{aligned}$$

G0-std Unit Magnetic Cell:

a b c alpha beta gamma
11.498 11.498 11.886 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

Co

[0.1667, 0.3333, 0.25]

[0.0, 0.0, 0.1683]
2

Co

[0.8333, 0.6667, 0.75]

[0.0, 0.0, 0.1683]
3

Co

[0.3333, 0.1667, 0.75]

[-0.0947, 0.0947, -0.0561]
4

Co

[0.6667, 0.8333, 0.25]

[-0.0947, 0.0947, -0.0561]
5

Co

[0.3333, 0.6667, 0.75]

[-0.0947, -0.1894, -0.0561]
6

Co

[0.6667, 0.3333, 0.25]

[-0.0947, -0.1894, -0.0561]
7

Co

[0.1667, 0.8333, 0.25]

[0.1894, 0.0947, -0.0561]
8

Co

[0.8333, 0.1667, 0.75]

[0.1894, 0.0947, -0.0561]
9

Nb

[0.1667, 0.3333, 0.5056]

[0.0, 0.0, 0.0]
10

Nb

[0.8333, 0.6667, 0.4944]

[0.0, 0.0, 0.0]
11

Nb

[0.1667, 0.3333, 0.9944]

[0.0, 0.0, 0.0]
12

Nb

[0.8333, 0.6667, 0.0056]

[0.0, 0.0, 0.0]
13

Nb

[0.1667, 0.8333, 0.5056]

[0.0, 0.0, 0.0]
14

Nb

[0.3333, 0.1667, 0.4944]

[0.0, 0.0, 0.0]
15

Nb

[0.3333, 0.6667, 0.4944]

[0.0, 0.0, 0.0]
16

Nb

[0.6667, 0.3333, 0.5056]

[0.0, 0.0, 0.0]
17

Nb

[0.6667, 0.8333, 0.5056]

[0.0, 0.0, 0.0]
18

Nb

[0.8333, 0.1667, 0.4944]

[0.0, 0.0, 0.0]
19

Nb

[0.1667, 0.8333, 0.9944]

[0.0, 0.0, 0.0]
20

Nb

[0.3333, 0.1667, 0.0056]

[0.0, 0.0, 0.0]
21

Nb

[0.3333, 0.6667, 0.0056]

[0.0, 0.0, 0.0]
22

Nb

[0.6667, 0.3333, 0.9944]

[0.0, 0.0, 0.0]
23

Nb

[0.6667, 0.8333, 0.9944]

[0.0, 0.0, 0.0]
24

Nb

[0.8333, 0.1667, 0.0056]

[0.0, 0.0, 0.0]
25

Nb

[0.5, 0.0, 0.0]

[0.0, 0.0, 0.0]
26

Nb

[0.5, 0.0, 0.5]

[0.0, 0.0, 0.0]
27

Nb

[0.0, 0.0, 0.0]

[0.0, 0.0, 0.0]
28

Nb

[0.0, 0.5, 0.0]

[0.0, 0.0, 0.0]
29

Nb

[0.5, 0.5, 0.0]

[0.0, 0.0, 0.0]
30

Nb

[0.0, 0.0, 0.5]

[0.0, 0.0, 0.0]
31

Nb

[0.0, 0.5, 0.5]

[0.0, 0.0, 0.0]
32

Nb

[0.5, 0.5, 0.5]

[0.0, 0.0, 0.0]
33

S

[0.0004, 0.1661, 0.6306]

[0.0, 0.0, 0.0]
34

S

[0.1656, 0.4995, 0.6306]

[0.0, 0.0, 0.0]
35

S

[0.3339, 0.3343, 0.6306]

[0.0, 0.0, 0.0]
36

S

[0.6661, 0.5004, 0.3694]

[0.0, 0.0, 0.0]
37

S

[0.8344, 0.8339, 0.3694]

[0.0, 0.0, 0.0]
38

S

[0.9996, 0.6656, 0.3694]

[0.0, 0.0, 0.0]
39

S

[0.0004, 0.3343, 0.8694]

[0.0, 0.0, 0.0]
40

S

[0.1656, 0.1661, 0.8694]

[0.0, 0.0, 0.0]
41

S

[0.3339, 0.4995, 0.8694]

[0.0, 0.0, 0.0]
42

S

[0.6661, 0.6656, 0.1306]

[0.0, 0.0, 0.0]
43

S

[0.8344, 0.5004, 0.1306]

[0.0, 0.0, 0.0]
44

S

[0.9996, 0.8339, 0.1306]

[0.0, 0.0, 0.0]
45

S

[0.0004, 0.6661, 0.6306]

[0.0, 0.0, 0.0]
46

S

[0.1661, 0.5004, 0.3694]

[0.0, 0.0, 0.0]
47

S

[0.3343, 0.3339, 0.3694]

[0.0, 0.0, 0.0]
48

S

[0.6656, 0.4995, 0.6306]

[0.0, 0.0, 0.0]
49

S

[0.8339, 0.8344, 0.6306]

[0.0, 0.0, 0.0]
50

S

[0.9996, 0.1656, 0.3694]

[0.0, 0.0, 0.0]
51

S

[0.0004, 0.8344, 0.8694]

[0.0, 0.0, 0.0]
52

S

[0.1661, 0.1656, 0.1306]

[0.0, 0.0, 0.0]
53

S

[0.3343, 0.5004, 0.1306]

[0.0, 0.0, 0.0]
54

S

[0.6656, 0.6661, 0.8694]

[0.0, 0.0, 0.0]
55

S

[0.8339, 0.4995, 0.8694]

[0.0, 0.0, 0.0]
56

S

[0.9996, 0.3339, 0.1306]

[0.0, 0.0, 0.0]
57

S

[0.1656, 0.6661, 0.8694]

[0.0, 0.0, 0.0]
58

S

[0.1661, 0.6656, 0.1306]

[0.0, 0.0, 0.0]
59

S

[0.4995, 0.3339, 0.1306]

[0.0, 0.0, 0.0]
60

S

[0.5004, 0.3343, 0.8694]

[0.0, 0.0, 0.0]
61

S

[0.8339, 0.9996, 0.8694]

[0.0, 0.0, 0.0]
62

S

[0.8344, 0.0004, 0.1306]

[0.0, 0.0, 0.0]
63

S

[0.1656, 0.9996, 0.6306]

[0.0, 0.0, 0.0]
64

S

[0.1661, 0.0004, 0.3694]

[0.0, 0.0, 0.0]
65

S

[0.4995, 0.6656, 0.3694]

[0.0, 0.0, 0.0]
66

S

[0.5004, 0.6661, 0.6306]

[0.0, 0.0, 0.0]
67

S

[0.8339, 0.3343, 0.6306]

[0.0, 0.0, 0.0]
68

S

[0.8344, 0.3339, 0.3694]

[0.0, 0.0, 0.0]
69

S

[0.3339, 0.8344, 0.6306]

[0.0, 0.0, 0.0]
70

S

[0.3343, 0.8339, 0.3694]

[0.0, 0.0, 0.0]
71

S

[0.4995, 0.1656, 0.3694]

[0.0, 0.0, 0.0]
72

S

[0.5004, 0.1661, 0.6306]

[0.0, 0.0, 0.0]
73

S

[0.6656, 0.9996, 0.6306]

[0.0, 0.0, 0.0]
74

S

[0.6661, 0.0004, 0.3694]

[0.0, 0.0, 0.0]
75

S

[0.3339, 0.9996, 0.8694]

[0.0, 0.0, 0.0]
76

S

[0.3343, 0.0004, 0.1306]

[0.0, 0.0, 0.0]
77

S

[0.4995, 0.8339, 0.1306]

[0.0, 0.0, 0.0]
78

S

[0.5004, 0.8344, 0.8694]

[0.0, 0.0, 0.0]
79

S

[0.6656, 0.1661, 0.8694]

[0.0, 0.0, 0.0]
80

S

[0.6661, 0.1656, 0.1306]

[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.
Co8d8d(1)2d(1)
6g(3)
Nb16f16f2d,2d
6g,6g
8a8a1a
1b
3e
3f
S48i48i6g,6g,6g,6g,6g,6g,6g,6g

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\[\begin{bmatrix} 0.333 & 0.0 & 1.126 \\ 0.667 & -1.0 & 0.563 \\ 0.79 & 0.0 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 2_{\alpha,\beta,\gamma} \,\middle\|\, 1 \,\middle|\, 0,\frac{1}{2},0 \right\}\)
3\[\begin{bmatrix} -1.0 & 0.667 & -0.563 \\ 0.0 & 0.333 & -1.125 \\ 0.0 & -0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 2_{\alpha,\beta,\gamma} \,\middle\|\, 1 \,\middle|\, \frac{1}{2},0,0 \right\}\)
4\[\begin{bmatrix} -0.333 & -0.667 & -0.563 \\ -0.667 & -0.333 & 0.563 \\ -0.79 & 0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 2_{\alpha,\beta,\gamma} \,\middle\|\, 1 \,\middle|\, \frac{1}{2},\frac{1}{2},0 \right\}\)
5\[\begin{bmatrix} -0.667 & 1.0 & -0.563 \\ -0.333 & 1.0 & 0.563 \\ 0.79 & 0.0 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 6^{5}_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\)
6\[\begin{bmatrix} 0.667 & 0.333 & -0.563 \\ 0.333 & -0.333 & -1.125 \\ -0.79 & 0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 6^{5}_{001} \,\middle|\, 0,\frac{1}{2},\frac{1}{2} \right\}\)
7\[\begin{bmatrix} 0.0 & -0.333 & 1.126 \\ -1.0 & 0.333 & 0.563 \\ 0.0 & -0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 6^{5}_{001} \,\middle|\, \frac{1}{2},0,\frac{1}{2} \right\}\)
8\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{1}_{001} \,\middle\|\, 6^{5}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
9\[\begin{bmatrix} -0.333 & 0.333 & 1.126 \\ 0.333 & 0.667 & 0.563 \\ -0.79 & 0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 6^{1}_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\)
10\[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{2}_{001} \,\middle\|\, 6^{1}_{001} \,\middle|\, 0,\frac{1}{2},\frac{1}{2} \right\}\)
11\[\begin{bmatrix} 1.0 & -0.333 & -0.563 \\ 1.0 & -0.667 & 0.563 \\ 0.0 & -0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 6^{1}_{001} \,\middle|\, \frac{1}{2},0,\frac{1}{2} \right\}\)
12\[\begin{bmatrix} 0.333 & -1.0 & -0.563 \\ -0.333 & 0.0 & -1.125 \\ 0.79 & 0.0 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 6^{1}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
13\[\begin{bmatrix} -0.333 & 0.333 & 1.126 \\ 0.333 & 0.667 & 0.563 \\ -0.79 & 0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 3^{2}_{001} \,\middle|\, 0,0,0 \right\}\)
14\[\begin{bmatrix} -0.667 & 1.0 & -0.563 \\ -0.333 & 1.0 & 0.563 \\ 0.79 & 0.0 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 3^{1}_{001} \,\middle|\, 0,0,0 \right\}\)
15\[\begin{bmatrix} 0.667 & 0.333 & -0.563 \\ 0.333 & -0.333 & -1.125 \\ -0.79 & 0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 3^{1}_{001} \,\middle|\, 0,\frac{1}{2},0 \right\}\)
16\[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{2}_{001} \,\middle\|\, 3^{2}_{001} \,\middle|\, 0,\frac{1}{2},0 \right\}\)
17\[\begin{bmatrix} 0.0 & -0.333 & 1.126 \\ -1.0 & 0.333 & 0.563 \\ 0.0 & -0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 3^{1}_{001} \,\middle|\, \frac{1}{2},0,0 \right\}\)
18\[\begin{bmatrix} 1.0 & -0.333 & -0.563 \\ 1.0 & -0.667 & 0.563 \\ 0.0 & -0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 3^{2}_{001} \,\middle|\, \frac{1}{2},0,0 \right\}\)
19\[\begin{bmatrix} 0.333 & -1.0 & -0.563 \\ -0.333 & 0.0 & -1.125 \\ 0.79 & 0.0 & -0.333\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 3^{2}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},0 \right\}\)
20\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{1}_{001} \,\middle\|\, 3^{1}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},0 \right\}\)
21\[\begin{bmatrix} 0.667 & 0.333 & -0.563 \\ 0.333 & 0.667 & 0.563 \\ -0.79 & 0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ m_{\alpha,\beta,\gamma} \,\middle\|\, 2_{110} \,\middle|\, 0,0,0 \right\}\)
22\[\begin{bmatrix} -0.667 & 1.0 & -0.563 \\ -0.333 & 0.0 & -1.125 \\ 0.79 & 0.0 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ -4^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 2_{110} \,\middle|\, 0,\frac{1}{2},0 \right\}\)
23\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, 2_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\)
24\[\begin{bmatrix} 0.667 & 0.333 & -0.563 \\ 0.333 & 0.667 & 0.563 \\ -0.79 & 0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ m_{\alpha,\beta,\gamma} \,\middle\|\, 2_{1-10} \,\middle|\, 0,0,\frac{1}{2} \right\}\)
25\[\begin{bmatrix} 0.0 & -0.333 & 1.126 \\ 1.0 & -0.667 & 0.563 \\ 0.0 & -0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ -4^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 2_{110} \,\middle|\, \frac{1}{2},0,0 \right\}\)
26\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ m_{110} \,\middle\|\, 2_{110} \,\middle|\, \frac{1}{2},\frac{1}{2},0 \right\}\)
27\[\begin{bmatrix} 0.333 & 0.0 & 1.126 \\ 0.667 & -1.0 & 0.563 \\ 0.79 & 0.0 & -0.333\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ 2_{\alpha,\beta,\gamma} \,\middle\|\, 2_{001} \,\middle|\, 0,\frac{1}{2},\frac{1}{2} \right\}\)
28\[\begin{bmatrix} -0.667 & 1.0 & -0.563 \\ -0.333 & 0.0 & -1.125 \\ 0.79 & 0.0 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ -4^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 2_{1-10} \,\middle|\, 0,\frac{1}{2},\frac{1}{2} \right\}\)
29\[\begin{bmatrix} 0.0 & -0.333 & 1.126 \\ 1.0 & -0.667 & 0.563 \\ 0.0 & -0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ -4^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 2_{1-10} \,\middle|\, \frac{1}{2},0,\frac{1}{2} \right\}\)
30\[\begin{bmatrix} -1.0 & 0.667 & -0.563 \\ 0.0 & 0.333 & -1.125 \\ 0.0 & -0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ 2_{\alpha,\beta,\gamma} \,\middle\|\, 2_{001} \,\middle|\, \frac{1}{2},0,\frac{1}{2} \right\}\)
31\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ m_{110} \,\middle\|\, 2_{1-10} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
32\[\begin{bmatrix} -0.333 & -0.667 & -0.563 \\ -0.667 & -0.333 & 0.563 \\ -0.79 & 0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ 2_{\alpha,\beta,\gamma} \,\middle\|\, 2_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
33\[\begin{bmatrix} 0.333 & 0.0 & 1.126 \\ -0.333 & 1.0 & 0.563 \\ 0.79 & 0.0 & -0.333\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ m_{\alpha,\beta,\gamma} \,\middle\|\, 2_{010} \,\middle|\, 0,0,0 \right\}\)
34\[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ m_{100} \,\middle\|\, 2_{100} \,\middle|\, 0,0,0 \right\}\)
35\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ m_{010} \,\middle\|\, 2_{010} \,\middle|\, 0,\frac{1}{2},0 \right\}\)
36\[\begin{bmatrix} -0.333 & 0.333 & 1.126 \\ -0.667 & -0.333 & 0.563 \\ -0.79 & 0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ -4^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 2_{100} \,\middle|\, 0,\frac{1}{2},0 \right\}\)
37\[\begin{bmatrix} 1.0 & -0.333 & -0.563 \\ 0.0 & 0.333 & -1.125 \\ 0.0 & -0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ m_{\alpha,\beta,\gamma} \,\middle\|\, 2_{100} \,\middle|\, \frac{1}{2},0,0 \right\}\)
38\[\begin{bmatrix} 0.333 & 0.0 & 1.126 \\ -0.333 & 1.0 & 0.563 \\ 0.79 & 0.0 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ m_{\alpha,\beta,\gamma} \,\middle\|\, 2_{210} \,\middle|\, 0,0,\frac{1}{2} \right\}\)
39\[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ m_{100} \,\middle\|\, 2_{120} \,\middle|\, 0,0,\frac{1}{2} \right\}\)
40\[\begin{bmatrix} -1.0 & 0.667 & -0.563 \\ -1.0 & 0.333 & 0.563 \\ 0.0 & -0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ -4^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 2_{010} \,\middle|\, \frac{1}{2},0,0 \right\}\)
41\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ m_{010} \,\middle\|\, 2_{210} \,\middle|\, 0,\frac{1}{2},\frac{1}{2} \right\}\)
42\[\begin{bmatrix} -0.333 & 0.333 & 1.126 \\ -0.667 & -0.333 & 0.563 \\ -0.79 & 0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ -4^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 2_{120} \,\middle|\, 0,\frac{1}{2},\frac{1}{2} \right\}\)
43\[\begin{bmatrix} -0.333 & -0.667 & -0.563 \\ 0.333 & -0.333 & -1.125 \\ -0.79 & 0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ -4^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 2_{010} \,\middle|\, \frac{1}{2},\frac{1}{2},0 \right\}\)
44\[\begin{bmatrix} 0.333 & -1.0 & -0.563 \\ 0.667 & -1.0 & 0.563 \\ 0.79 & 0.0 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ -4^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 2_{100} \,\middle|\, \frac{1}{2},\frac{1}{2},0 \right\}\)
45\[\begin{bmatrix} 1.0 & -0.333 & -0.563 \\ 0.0 & 0.333 & -1.125 \\ 0.0 & -0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ m_{\alpha,\beta,\gamma} \,\middle\|\, 2_{120} \,\middle|\, \frac{1}{2},0,\frac{1}{2} \right\}\)
46\[\begin{bmatrix} -1.0 & 0.667 & -0.563 \\ -1.0 & 0.333 & 0.563 \\ 0.0 & -0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ -4^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 2_{210} \,\middle|\, \frac{1}{2},0,\frac{1}{2} \right\}\)
47\[\begin{bmatrix} -0.333 & -0.667 & -0.563 \\ 0.333 & -0.333 & -1.125 \\ -0.79 & 0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ -4^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 2_{210} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
48\[\begin{bmatrix} 0.333 & -1.0 & -0.563 \\ 0.667 & -1.0 & 0.563 \\ 0.79 & 0.0 & -0.333\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ -4^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 2_{120} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
No.Spin RotationSpace RotationSpace TranslationSeitz Symbol
9\[\begin{bmatrix} -0.333 & 0.333 & 1.126 \\ 0.333 & 0.667 & 0.563 \\ -0.79 & 0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 6^{1}_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\)
34\[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ m_{100} \,\middle\|\, 2_{100} \,\middle|\, 0,0,0 \right\}\)
38\[\begin{bmatrix} 0.333 & 0.0 & 1.126 \\ -0.333 & 1.0 & 0.563 \\ 0.79 & 0.0 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ m_{\alpha,\beta,\gamma} \,\middle\|\, 2_{210} \,\middle|\, 0,0,\frac{1}{2} \right\}\)
2\[\begin{bmatrix} 0.333 & 0.0 & 1.126 \\ 0.667 & -1.0 & 0.563 \\ 0.79 & 0.0 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 2_{\alpha,\beta,\gamma} \,\middle\|\, 1 \,\middle|\, 0,\frac{1}{2},0 \right\}\)
3\[\begin{bmatrix} -1.0 & 0.667 & -0.563 \\ 0.0 & 0.333 & -1.125 \\ 0.0 & -0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 2_{\alpha,\beta,\gamma} \,\middle\|\, 1 \,\middle|\, \frac{1}{2},0,0 \right\}\)
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\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, 1 \,\middle|\, 0,0,0 \right\}\)
2\[\begin{bmatrix} 0.333 & 0.0 & 1.126 \\ 0.667 & -1.0 & 0.563 \\ 0.79 & 0.0 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 2_{\alpha,\beta,\gamma} \,\middle\|\, 1 \,\middle|\, 0,\frac{1}{2},0 \right\}\)
3\[\begin{bmatrix} -1.0 & 0.667 & -0.563 \\ 0.0 & 0.333 & -1.125 \\ 0.0 & -0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 2_{\alpha,\beta,\gamma} \,\middle\|\, 1 \,\middle|\, \frac{1}{2},0,0 \right\}\)
4\[\begin{bmatrix} -0.333 & -0.667 & -0.563 \\ -0.667 & -0.333 & 0.563 \\ -0.79 & 0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 2_{\alpha,\beta,\gamma} \,\middle\|\, 1 \,\middle|\, \frac{1}{2},\frac{1}{2},0 \right\}\)
No.Spin RotationSpace RotationSpace TranslationSeitz Symbol
1\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, 1 \,\middle|\, 0,0,0 \right\}\)
23\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, 2_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\)
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\[\begin{bmatrix} 0.333 & 0.0 & 1.126 \\ 0.667 & -1.0 & 0.563 \\ 0.79 & 0.0 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 2_{\alpha,\beta,\gamma} \,\middle\|\, 1 \,\middle|\, 0,\frac{1}{2},0 \right\}\)
3\[\begin{bmatrix} -1.0 & 0.667 & -0.563 \\ 0.0 & 0.333 & -1.125 \\ 0.0 & -0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 2_{\alpha,\beta,\gamma} \,\middle\|\, 1 \,\middle|\, \frac{1}{2},0,0 \right\}\)
4\[\begin{bmatrix} -0.333 & -0.667 & -0.563 \\ -0.667 & -0.333 & 0.563 \\ -0.79 & 0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 2_{\alpha,\beta,\gamma} \,\middle\|\, 1 \,\middle|\, \frac{1}{2},\frac{1}{2},0 \right\}\)
5\[\begin{bmatrix} -0.667 & 1.0 & -0.563 \\ -0.333 & 1.0 & 0.563 \\ 0.79 & 0.0 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 6^{5}_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\)
6\[\begin{bmatrix} 0.667 & 0.333 & -0.563 \\ 0.333 & -0.333 & -1.125 \\ -0.79 & 0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 6^{5}_{001} \,\middle|\, 0,\frac{1}{2},\frac{1}{2} \right\}\)
7\[\begin{bmatrix} 0.0 & -0.333 & 1.126 \\ -1.0 & 0.333 & 0.563 \\ 0.0 & -0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 6^{5}_{001} \,\middle|\, \frac{1}{2},0,\frac{1}{2} \right\}\)
8\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{1}_{001} \,\middle\|\, 6^{5}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
9\[\begin{bmatrix} -0.333 & 0.333 & 1.126 \\ 0.333 & 0.667 & 0.563 \\ -0.79 & 0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 6^{1}_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\)
10\[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{2}_{001} \,\middle\|\, 6^{1}_{001} \,\middle|\, 0,\frac{1}{2},\frac{1}{2} \right\}\)
11\[\begin{bmatrix} 1.0 & -0.333 & -0.563 \\ 1.0 & -0.667 & 0.563 \\ 0.0 & -0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 6^{1}_{001} \,\middle|\, \frac{1}{2},0,\frac{1}{2} \right\}\)
12\[\begin{bmatrix} 0.333 & -1.0 & -0.563 \\ -0.333 & 0.0 & -1.125 \\ 0.79 & 0.0 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ 3^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 6^{1}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
13\[\begin{bmatrix} -0.333 & 0.333 & 1.126 \\ 0.333 & 0.667 & 0.563 \\ -0.79 & 0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 3^{2}_{001} \,\middle|\, 0,0,0 \right\}\)
14\[\begin{bmatrix} -0.667 & 1.0 & -0.563 \\ -0.333 & 1.0 & 0.563 \\ 0.79 & 0.0 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 3^{1}_{001} \,\middle|\, 0,0,0 \right\}\)
15\[\begin{bmatrix} 0.667 & 0.333 & -0.563 \\ 0.333 & -0.333 & -1.125 \\ -0.79 & 0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 3^{1}_{001} \,\middle|\, 0,\frac{1}{2},0 \right\}\)
16\[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{2}_{001} \,\middle\|\, 3^{2}_{001} \,\middle|\, 0,\frac{1}{2},0 \right\}\)
17\[\begin{bmatrix} 0.0 & -0.333 & 1.126 \\ -1.0 & 0.333 & 0.563 \\ 0.0 & -0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 3^{1}_{001} \,\middle|\, \frac{1}{2},0,0 \right\}\)
18\[\begin{bmatrix} 1.0 & -0.333 & -0.563 \\ 1.0 & -0.667 & 0.563 \\ 0.0 & -0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 3^{2}_{001} \,\middle|\, \frac{1}{2},0,0 \right\}\)
19\[\begin{bmatrix} 0.333 & -1.0 & -0.563 \\ -0.333 & 0.0 & -1.125 \\ 0.79 & 0.0 & -0.333\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 3^{2}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},0 \right\}\)
20\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{1}_{001} \,\middle\|\, 3^{1}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},0 \right\}\)
21\[\begin{bmatrix} 0.667 & 0.333 & -0.563 \\ 0.333 & 0.667 & 0.563 \\ -0.79 & 0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ m_{\alpha,\beta,\gamma} \,\middle\|\, 2_{110} \,\middle|\, 0,0,0 \right\}\)
22\[\begin{bmatrix} -0.667 & 1.0 & -0.563 \\ -0.333 & 0.0 & -1.125 \\ 0.79 & 0.0 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ -4^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 2_{110} \,\middle|\, 0,\frac{1}{2},0 \right\}\)
23\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, 2_{001} \,\middle|\, 0,0,\frac{1}{2} \right\}\)
24\[\begin{bmatrix} 0.667 & 0.333 & -0.563 \\ 0.333 & 0.667 & 0.563 \\ -0.79 & 0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ m_{\alpha,\beta,\gamma} \,\middle\|\, 2_{1-10} \,\middle|\, 0,0,\frac{1}{2} \right\}\)
25\[\begin{bmatrix} 0.0 & -0.333 & 1.126 \\ 1.0 & -0.667 & 0.563 \\ 0.0 & -0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ -4^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 2_{110} \,\middle|\, \frac{1}{2},0,0 \right\}\)
26\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ m_{110} \,\middle\|\, 2_{110} \,\middle|\, \frac{1}{2},\frac{1}{2},0 \right\}\)
27\[\begin{bmatrix} 0.333 & 0.0 & 1.126 \\ 0.667 & -1.0 & 0.563 \\ 0.79 & 0.0 & -0.333\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ 2_{\alpha,\beta,\gamma} \,\middle\|\, 2_{001} \,\middle|\, 0,\frac{1}{2},\frac{1}{2} \right\}\)
28\[\begin{bmatrix} -0.667 & 1.0 & -0.563 \\ -0.333 & 0.0 & -1.125 \\ 0.79 & 0.0 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ -4^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 2_{1-10} \,\middle|\, 0,\frac{1}{2},\frac{1}{2} \right\}\)
29\[\begin{bmatrix} 0.0 & -0.333 & 1.126 \\ 1.0 & -0.667 & 0.563 \\ 0.0 & -0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ -4^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 2_{1-10} \,\middle|\, \frac{1}{2},0,\frac{1}{2} \right\}\)
30\[\begin{bmatrix} -1.0 & 0.667 & -0.563 \\ 0.0 & 0.333 & -1.125 \\ 0.0 & -0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ 2_{\alpha,\beta,\gamma} \,\middle\|\, 2_{001} \,\middle|\, \frac{1}{2},0,\frac{1}{2} \right\}\)
31\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ m_{110} \,\middle\|\, 2_{1-10} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
32\[\begin{bmatrix} -0.333 & -0.667 & -0.563 \\ -0.667 & -0.333 & 0.563 \\ -0.79 & 0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ 2_{\alpha,\beta,\gamma} \,\middle\|\, 2_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
33\[\begin{bmatrix} 0.333 & 0.0 & 1.126 \\ -0.333 & 1.0 & 0.563 \\ 0.79 & 0.0 & -0.333\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ m_{\alpha,\beta,\gamma} \,\middle\|\, 2_{010} \,\middle|\, 0,0,0 \right\}\)
34\[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ m_{100} \,\middle\|\, 2_{100} \,\middle|\, 0,0,0 \right\}\)
35\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ m_{010} \,\middle\|\, 2_{010} \,\middle|\, 0,\frac{1}{2},0 \right\}\)
36\[\begin{bmatrix} -0.333 & 0.333 & 1.126 \\ -0.667 & -0.333 & 0.563 \\ -0.79 & 0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ -4^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 2_{100} \,\middle|\, 0,\frac{1}{2},0 \right\}\)
37\[\begin{bmatrix} 1.0 & -0.333 & -0.563 \\ 0.0 & 0.333 & -1.125 \\ 0.0 & -0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ m_{\alpha,\beta,\gamma} \,\middle\|\, 2_{100} \,\middle|\, \frac{1}{2},0,0 \right\}\)
38\[\begin{bmatrix} 0.333 & 0.0 & 1.126 \\ -0.333 & 1.0 & 0.563 \\ 0.79 & 0.0 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ m_{\alpha,\beta,\gamma} \,\middle\|\, 2_{210} \,\middle|\, 0,0,\frac{1}{2} \right\}\)
39\[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ m_{100} \,\middle\|\, 2_{120} \,\middle|\, 0,0,\frac{1}{2} \right\}\)
40\[\begin{bmatrix} -1.0 & 0.667 & -0.563 \\ -1.0 & 0.333 & 0.563 \\ 0.0 & -0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ -4^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 2_{010} \,\middle|\, \frac{1}{2},0,0 \right\}\)
41\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ m_{010} \,\middle\|\, 2_{210} \,\middle|\, 0,\frac{1}{2},\frac{1}{2} \right\}\)
42\[\begin{bmatrix} -0.333 & 0.333 & 1.126 \\ -0.667 & -0.333 & 0.563 \\ -0.79 & 0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ -4^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 2_{120} \,\middle|\, 0,\frac{1}{2},\frac{1}{2} \right\}\)
43\[\begin{bmatrix} -0.333 & -0.667 & -0.563 \\ 0.333 & -0.333 & -1.125 \\ -0.79 & 0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ -4^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 2_{010} \,\middle|\, \frac{1}{2},\frac{1}{2},0 \right\}\)
44\[\begin{bmatrix} 0.333 & -1.0 & -0.563 \\ 0.667 & -1.0 & 0.563 \\ 0.79 & 0.0 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ -4^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 2_{100} \,\middle|\, \frac{1}{2},\frac{1}{2},0 \right\}\)
45\[\begin{bmatrix} 1.0 & -0.333 & -0.563 \\ 0.0 & 0.333 & -1.125 \\ 0.0 & -0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ m_{\alpha,\beta,\gamma} \,\middle\|\, 2_{120} \,\middle|\, \frac{1}{2},0,\frac{1}{2} \right\}\)
46\[\begin{bmatrix} -1.0 & 0.667 & -0.563 \\ -1.0 & 0.333 & 0.563 \\ 0.0 & -0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.0 \\ 0.5 \end{bmatrix}\]\(\left\{ -4^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 2_{210} \,\middle|\, \frac{1}{2},0,\frac{1}{2} \right\}\)
47\[\begin{bmatrix} -0.333 & -0.667 & -0.563 \\ 0.333 & -0.333 & -1.125 \\ -0.79 & 0.79 & -0.333\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ -4^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 2_{210} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
48\[\begin{bmatrix} 0.333 & -1.0 & -0.563 \\ 0.667 & -1.0 & 0.563 \\ 0.79 & 0.0 & -0.333\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ -4^{1}_{\alpha,\beta,\gamma} \,\middle\|\, 2_{120} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
No.Spin RotationSpace RotationSpace TranslationSeitz Symbol
1\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, 1 \,\middle|\, 0,0,0 \right\}\)
16\[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{2}_{001} \,\middle\|\, 3^{2}_{001} \,\middle|\, 0,\frac{1}{2},0 \right\}\)
20\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ 3^{1}_{001} \,\middle\|\, 3^{1}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},0 \right\}\)
26\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ m_{110} \,\middle\|\, 2_{110} \,\middle|\, \frac{1}{2},\frac{1}{2},0 \right\}\)
34\[\begin{bmatrix} -1.0 & 1.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & -1.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ m_{100} \,\middle\|\, 2_{100} \,\middle|\, 0,0,0 \right\}\)
35\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 1.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ -1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.5 \\ 0.0 \end{bmatrix}\]\(\left\{ m_{010} \,\middle\|\, 2_{010} \,\middle|\, 0,\frac{1}{2},0 \right\}\)