Find Spin Group - Program for Identifying Spin Space Group

File Name	Spin Space Group	G0 Symbol	L0 Symbol	it	ik
$C o N b 3 S 6 / t r i p l e Q$	$P^{3_{1 - 1 - 1}^{1}} {6_{3}}^{m_{110}} 2^{m_{101}} 2 \| (2_{001}, 2_{010}, 1)$	$P 6_{3} 22 (182)$	$P 2_{1} (4)$	$6$	$4$

Space Group	Magnetic Space Group	Spin Part PG	Configuration	Magnetic Phase
$P 6_{3} 22 (182)$	$\begin{aligned} P 32^{'} 1 (150.27) \\ T y p e I I I \end{aligned}$	$\begin{array}{r} T d \\ - 43 m \end{array}$	$N o n c o p l a n a r$	$C a n t i n g A F M$

Little Cogroup at General Position		Spin Splitting
without SOC	with SOC	without SOC	with SOC
$^{222} 1$	$1$	$N o$	$Y e s$

Effective Magnetic Point Group	Anomalous Hall Effect
$62^{'} 2^{'}$	without SOC	with SOC
$62^{'} 2^{'}$	$Y e s$	$Y e s$

Effective MPG Constraint on rank-3 Tensor
T-odd	Quantum metric dipole	$σ_{x x z} = σ_{y z y} \| σ_{x z x} = σ_{y z y} \| σ_{y y z} = σ_{y z y} \| σ_{z x x} = σ_{z y y}$
T-odd	Inversed mass dipole	$σ_{x x z} = σ_{z y y} \| σ_{x z x} = σ_{z y y} \| σ_{y y z} = σ_{z y y} \| σ_{y z y} = σ_{z y y} \| σ_{z x x} = σ_{z y y}$
T-even	Berry curvature dipole	$σ_{x y z} = - σ_{y z x} \| σ_{x z y} = - σ_{y z x} \| σ_{y x z} = σ_{y z x}$

G0-std Unit Magnetic Cell:

a	b	c	alpha	beta	gamma
11.498	11.498	11.886	90.0	90.0	120.0

Fractional Coordinates & Moments (in G0-std Unit Magnetic Cell)

No.	Atom	Site	Moment (in lattice)
1	Co	[0.3333, 0.1667, 0.75]	[1.6329, 0.9428, -0.6667]
2	Co	[0.6667, 0.8333, 0.25]	[1.6329, 0.9428, -0.6667]
3	Co	[0.8333, 0.1667, 0.75]	[-1.6329, 0.9428, -0.6667]
4	Co	[0.1667, 0.8333, 0.25]	[-1.6329, 0.9428, -0.6667]
5	Co	[0.1667, 0.3333, 0.25]	[0.0, -1.8855, -0.6667]
6	Co	[0.8333, 0.6667, 0.75]	[0.0, -1.8855, -0.6667]
7	Co	[0.6667, 0.3333, 0.25]	[-0.0, -0.0, 2.0]
8	Co	[0.3333, 0.6667, 0.75]	[-0.0, -0.0, 2.0]
9	S	[0.1661, 0.0005, 0.3694]	[0.0, 0.0, 0.0]
10	S	[0.4995, 0.1656, 0.3694]	[0.0, 0.0, 0.0]
11	S	[0.3344, 0.3339, 0.3694]	[0.0, 0.0, 0.0]
12	S	[0.1656, 0.4995, 0.6306]	[0.0, 0.0, 0.0]
13	S	[0.0005, 0.1661, 0.6306]	[0.0, 0.0, 0.0]
14	S	[0.3339, 0.3344, 0.6306]	[0.0, 0.0, 0.0]
15	S	[0.3339, 0.4995, 0.8694]	[0.0, 0.0, 0.0]
16	S	[0.0005, 0.3344, 0.8694]	[0.0, 0.0, 0.0]
17	S	[0.1656, 0.1661, 0.8694]	[0.0, 0.0, 0.0]
18	S	[0.3344, 0.0005, 0.1306]	[0.0, 0.0, 0.0]
19	S	[0.4995, 0.3339, 0.1306]	[0.0, 0.0, 0.0]
20	S	[0.1661, 0.1656, 0.1306]	[0.0, 0.0, 0.0]
21	S	[0.6661, 0.0005, 0.3694]	[0.0, 0.0, 0.0]
22	S	[0.9996, 0.1656, 0.3694]	[0.0, 0.0, 0.0]
23	S	[0.8344, 0.3339, 0.3694]	[0.0, 0.0, 0.0]
24	S	[0.6656, 0.4995, 0.6306]	[0.0, 0.0, 0.0]
25	S	[0.5005, 0.1661, 0.6306]	[0.0, 0.0, 0.0]
26	S	[0.8339, 0.3344, 0.6306]	[0.0, 0.0, 0.0]
27	S	[0.8339, 0.4995, 0.8694]	[0.0, 0.0, 0.0]
28	S	[0.5005, 0.3344, 0.8694]	[0.0, 0.0, 0.0]
29	S	[0.6656, 0.1661, 0.8694]	[0.0, 0.0, 0.0]
30	S	[0.8344, 0.0005, 0.1306]	[0.0, 0.0, 0.0]
31	S	[0.9996, 0.3339, 0.1306]	[0.0, 0.0, 0.0]
32	S	[0.6661, 0.1656, 0.1306]	[0.0, 0.0, 0.0]
33	S	[0.1661, 0.5005, 0.3694]	[0.0, 0.0, 0.0]
34	S	[0.4995, 0.6656, 0.3694]	[0.0, 0.0, 0.0]
35	S	[0.3344, 0.8339, 0.3694]	[0.0, 0.0, 0.0]
36	S	[0.1656, 0.9996, 0.6306]	[0.0, 0.0, 0.0]
37	S	[0.0005, 0.6661, 0.6306]	[0.0, 0.0, 0.0]
38	S	[0.3339, 0.8344, 0.6306]	[0.0, 0.0, 0.0]
39	S	[0.3339, 0.9996, 0.8694]	[0.0, 0.0, 0.0]
40	S	[0.0005, 0.8344, 0.8694]	[0.0, 0.0, 0.0]
41	S	[0.1656, 0.6661, 0.8694]	[0.0, 0.0, 0.0]
42	S	[0.3344, 0.5005, 0.1306]	[0.0, 0.0, 0.0]
43	S	[0.4995, 0.8339, 0.1306]	[0.0, 0.0, 0.0]
44	S	[0.1661, 0.6656, 0.1306]	[0.0, 0.0, 0.0]
45	S	[0.6661, 0.5005, 0.3694]	[0.0, 0.0, 0.0]
46	S	[0.9996, 0.6656, 0.3694]	[0.0, 0.0, 0.0]
47	S	[0.8344, 0.8339, 0.3694]	[0.0, 0.0, 0.0]
48	S	[0.6656, 0.9996, 0.6306]	[0.0, 0.0, 0.0]
49	S	[0.5005, 0.6661, 0.6306]	[0.0, 0.0, 0.0]
50	S	[0.8339, 0.8344, 0.6306]	[0.0, 0.0, 0.0]
51	S	[0.8339, 0.9996, 0.8694]	[0.0, 0.0, 0.0]
52	S	[0.5005, 0.8344, 0.8694]	[0.0, 0.0, 0.0]
53	S	[0.6656, 0.6661, 0.8694]	[0.0, 0.0, 0.0]
54	S	[0.8344, 0.5005, 0.1306]	[0.0, 0.0, 0.0]
55	S	[0.9996, 0.8339, 0.1306]	[0.0, 0.0, 0.0]
56	S	[0.6661, 0.6656, 0.1306]	[0.0, 0.0, 0.0]
57	Nb	[0.0, 0.0, 0.0]	[0.0, 0.0, 0.0]
58	Nb	[0.0, 0.0, 0.5]	[0.0, 0.0, 0.0]
59	Nb	[0.1667, 0.3333, 0.9944]	[0.0, 0.0, 0.0]
60	Nb	[0.3333, 0.1667, 0.0056]	[0.0, 0.0, 0.0]
61	Nb	[0.3333, 0.1667, 0.4944]	[0.0, 0.0, 0.0]
62	Nb	[0.1667, 0.3333, 0.5056]	[0.0, 0.0, 0.0]
63	Nb	[0.5, 0.0, 0.0]	[0.0, 0.0, 0.0]
64	Nb	[0.5, 0.0, 0.5]	[0.0, 0.0, 0.0]
65	Nb	[0.6667, 0.3333, 0.9944]	[0.0, 0.0, 0.0]
66	Nb	[0.8333, 0.1667, 0.0056]	[0.0, 0.0, 0.0]
67	Nb	[0.8333, 0.1667, 0.4944]	[0.0, 0.0, 0.0]
68	Nb	[0.6667, 0.3333, 0.5056]	[0.0, 0.0, 0.0]
69	Nb	[0.0, 0.5, 0.0]	[0.0, 0.0, 0.0]
70	Nb	[0.0, 0.5, 0.5]	[0.0, 0.0, 0.0]
71	Nb	[0.1667, 0.8333, 0.9944]	[0.0, 0.0, 0.0]
72	Nb	[0.3333, 0.6667, 0.0056]	[0.0, 0.0, 0.0]
73	Nb	[0.3333, 0.6667, 0.4944]	[0.0, 0.0, 0.0]
74	Nb	[0.1667, 0.8333, 0.5056]	[0.0, 0.0, 0.0]
75	Nb	[0.5, 0.5, 0.0]	[0.0, 0.0, 0.0]
76	Nb	[0.5, 0.5, 0.5]	[0.0, 0.0, 0.0]
77	Nb	[0.6667, 0.8333, 0.9944]	[0.0, 0.0, 0.0]
78	Nb	[0.8333, 0.6667, 0.0056]	[0.0, 0.0, 0.0]
79	Nb	[0.8333, 0.6667, 0.4944]	[0.0, 0.0, 0.0]
80	Nb	[0.6667, 0.8333, 0.5056]	[0.0, 0.0, 0.0]

Elements of the Spin Space Group (G0-std-setting):

Real Space Elements are under the basis of the lattice given above.

Spin Part Elements are represented corresponding to the SpinPartPG.

No.	Spin Rotation	Space Rotation	Space Translation	Seitz Symbol
1	$[\begin{matrix} 0.5 & 0.866 & 0.0 \\ 0.866 & - 0.5 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 1.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.5 \end{matrix}]$	${m_{101} \| \| 2_{210} \| τ_{(0.0, 0.0, 0.5)}}$
2	$[\begin{matrix} 0.5 & - 0.288 & 0.816 \\ - 0.288 & 0.834 & 0.471 \\ 0.816 & 0.471 & - 0.334 \end{matrix}]$	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 1.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.5 \\ 0.5 \end{matrix}]$	${m_{- 101} \| \| 2_{210} \| τ_{(0.0, 0.5, 0.5)}}$
3	$[\begin{matrix} - 0.5 & - 0.866 & 0.0 \\ 0.288 & - 0.167 & - 0.943 \\ - 0.816 & 0.472 & - 0.333 \end{matrix}]$	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 1.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.0 \\ 0.5 \end{matrix}]$	${- 4_{010}^{3} \| \| 2_{210} \| τ_{(0.5, 0.0, 0.5)}}$
4	$[\begin{matrix} - 0.5 & 0.288 & - 0.816 \\ - 0.866 & - 0.167 & 0.472 \\ 0.0 & - 0.943 & - 0.333 \end{matrix}]$	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 1.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.5 \end{matrix}]$	${- 4_{010}^{1} \| \| 2_{210} \| τ_{(0.5, 0.5, 0.5)}}$
5	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${1 \| \| 1 \| τ_{(0.0, 0.0, 0.0)}}$
6	$[\begin{matrix} 0.0 & 0.577 & 0.816 \\ 0.577 & - 0.666 & 0.472 \\ 0.816 & 0.472 & - 0.333 \end{matrix}]$	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.5 \\ 0.0 \end{matrix}]$	${2_{010} \| \| 1 \| τ_{(0.0, 0.5, 0.0)}}$
7	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.333 & - 0.943 \\ 0.0 & - 0.943 & - 0.333 \end{matrix}]$	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.0 \\ 0.0 \end{matrix}]$	${2_{001} \| \| 1 \| τ_{(0.5, 0.0, 0.0)}}$
8	$[\begin{matrix} 0.0 & - 0.577 & - 0.816 \\ - 0.577 & - 0.666 & 0.472 \\ - 0.816 & 0.472 & - 0.333 \end{matrix}]$	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.0 \end{matrix}]$	${2_{100} \| \| 1 \| τ_{(0.5, 0.5, 0.0)}}$
9	$[\begin{matrix} - 0.5 & 0.866 & 0.0 \\ - 0.866 & - 0.5 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 1.0 & - 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.5 \end{matrix}]$	${3_{1 - 1 - 1}^{1} \| \| 6_{001}^{1} \| τ_{(0.0, 0.0, 0.5)}}$
10	$[\begin{matrix} - 0.5 & - 0.289 & 0.816 \\ 0.289 & 0.834 & 0.472 \\ - 0.816 & 0.472 & - 0.333 \end{matrix}]$	$[\begin{matrix} 1.0 & - 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.5 \\ 0.5 \end{matrix}]$	${3_{- 11 - 1}^{1} \| \| 6_{001}^{1} \| τ_{(0.0, 0.5, 0.5)}}$
11	$[\begin{matrix} 0.5 & - 0.866 & 0.0 \\ - 0.288 & - 0.167 & - 0.943 \\ 0.816 & 0.472 & - 0.333 \end{matrix}]$	$[\begin{matrix} 1.0 & - 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.0 \\ 0.5 \end{matrix}]$	${3_{111}^{1} \| \| 6_{001}^{1} \| τ_{(0.5, 0.0, 0.5)}}$
12	$[\begin{matrix} 0.5 & 0.288 & - 0.816 \\ 0.866 & - 0.167 & 0.472 \\ 0.0 & - 0.943 & - 0.333 \end{matrix}]$	$[\begin{matrix} 1.0 & - 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.5 \end{matrix}]$	${3_{- 1 - 11}^{1} \| \| 6_{001}^{1} \| τ_{(0.5, 0.5, 0.5)}}$
13	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 1.0 & - 1.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${m_{110} \| \| 2_{100} \| τ_{(0.0, 0.0, 0.0)}}$
14	$[\begin{matrix} 0.0 & 0.577 & 0.816 \\ - 0.577 & - 0.666 & 0.472 \\ - 0.816 & 0.472 & - 0.333 \end{matrix}]$	$[\begin{matrix} 1.0 & - 1.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.5 \\ 0.0 \end{matrix}]$	${- 4_{001}^{1} \| \| 2_{100} \| τ_{(0.0, 0.5, 0.0)}}$
15	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 0.333 & - 0.943 \\ 0.0 & - 0.943 & - 0.333 \end{matrix}]$	$[\begin{matrix} 1.0 & - 1.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.0 \\ 0.0 \end{matrix}]$	${m_{1 - 10} \| \| 2_{100} \| τ_{(0.5, 0.0, 0.0)}}$
16	$[\begin{matrix} 0.0 & - 0.577 & - 0.816 \\ 0.578 & - 0.666 & 0.471 \\ 0.816 & 0.472 & - 0.334 \end{matrix}]$	$[\begin{matrix} 1.0 & - 1.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.0 \end{matrix}]$	${- 4_{001}^{3} \| \| 2_{100} \| τ_{(0.5, 0.5, 0.0)}}$
17	$[\begin{matrix} 0.5 & - 0.866 & 0.0 \\ - 0.866 & - 0.5 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${m_{01 - 1} \| \| 2_{110} \| τ_{(0.0, 0.0, 0.0)}}$
18	$[\begin{matrix} - 0.5 & - 0.288 & 0.816 \\ 0.866 & - 0.167 & 0.472 \\ 0.0 & - 0.943 & - 0.333 \end{matrix}]$	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.5 \\ 0.0 \end{matrix}]$	${- 4_{100}^{1} \| \| 2_{110} \| τ_{(0.0, 0.5, 0.0)}}$
19	$[\begin{matrix} - 0.5 & 0.866 & 0.0 \\ - 0.288 & - 0.167 & - 0.943 \\ 0.816 & 0.472 & - 0.333 \end{matrix}]$	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.0 \\ 0.0 \end{matrix}]$	${- 4_{100}^{3} \| \| 2_{110} \| τ_{(0.5, 0.0, 0.0)}}$
20	$[\begin{matrix} 0.5 & 0.288 & - 0.816 \\ 0.288 & 0.834 & 0.471 \\ - 0.816 & 0.471 & - 0.334 \end{matrix}]$	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.0 \end{matrix}]$	${m_{011} \| \| 2_{110} \| τ_{(0.5, 0.5, 0.0)}}$
21	$[\begin{matrix} - 0.5 & - 0.866 & 0.0 \\ 0.866 & - 0.5 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ - 1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.5 \end{matrix}]$	${3_{1 - 1 - 1}^{2} \| \| 6_{001}^{5} \| τ_{(0.0, 0.0, 0.5)}}$
22	$[\begin{matrix} 0.5 & - 0.288 & 0.816 \\ - 0.866 & - 0.167 & 0.472 \\ 0.0 & - 0.943 & - 0.333 \end{matrix}]$	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ - 1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.5 \\ 0.5 \end{matrix}]$	${3_{111}^{2} \| \| 6_{001}^{5} \| τ_{(0.0, 0.5, 0.5)}}$
23	$[\begin{matrix} 0.5 & 0.866 & 0.0 \\ 0.288 & - 0.167 & - 0.943 \\ - 0.816 & 0.472 & - 0.333 \end{matrix}]$	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ - 1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.0 \\ 0.5 \end{matrix}]$	${3_{- 1 - 11}^{2} \| \| 6_{001}^{5} \| τ_{(0.5, 0.0, 0.5)}}$
24	$[\begin{matrix} - 0.5 & 0.288 & - 0.816 \\ - 0.288 & 0.834 & 0.471 \\ 0.816 & 0.471 & - 0.334 \end{matrix}]$	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ - 1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.5 \end{matrix}]$	${3_{- 11 - 1}^{2} \| \| 6_{001}^{5} \| τ_{(0.5, 0.5, 0.5)}}$
25	$[\begin{matrix} - 0.5 & - 0.866 & 0.0 \\ 0.866 & - 0.5 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ 1.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${3_{1 - 1 - 1}^{2} \| \| 3_{001}^{1} \| τ_{(0.0, 0.0, 0.0)}}$
26	$[\begin{matrix} 0.5 & - 0.288 & 0.816 \\ - 0.866 & - 0.167 & 0.472 \\ 0.0 & - 0.943 & - 0.333 \end{matrix}]$	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ 1.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.5 \\ 0.0 \end{matrix}]$	${3_{111}^{2} \| \| 3_{001}^{1} \| τ_{(0.0, 0.5, 0.0)}}$
27	$[\begin{matrix} 0.5 & 0.866 & 0.0 \\ 0.288 & - 0.167 & - 0.943 \\ - 0.816 & 0.472 & - 0.333 \end{matrix}]$	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ 1.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.0 \\ 0.0 \end{matrix}]$	${3_{- 1 - 11}^{2} \| \| 3_{001}^{1} \| τ_{(0.5, 0.0, 0.0)}}$
28	$[\begin{matrix} - 0.5 & 0.288 & - 0.816 \\ - 0.288 & 0.834 & 0.471 \\ 0.816 & 0.471 & - 0.334 \end{matrix}]$	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ 1.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.0 \end{matrix}]$	${3_{- 11 - 1}^{2} \| \| 3_{001}^{1} \| τ_{(0.5, 0.5, 0.0)}}$
29	$[\begin{matrix} 0.5 & - 0.866 & 0.0 \\ - 0.866 & - 0.5 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.5 \end{matrix}]$	${m_{01 - 1} \| \| 2_{1 - 10} \| τ_{(0.0, 0.0, 0.5)}}$
30	$[\begin{matrix} - 0.5 & - 0.288 & 0.816 \\ 0.866 & - 0.167 & 0.472 \\ 0.0 & - 0.943 & - 0.333 \end{matrix}]$	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.5 \\ 0.5 \end{matrix}]$	${- 4_{100}^{1} \| \| 2_{1 - 10} \| τ_{(0.0, 0.5, 0.5)}}$
31	$[\begin{matrix} - 0.5 & 0.866 & 0.0 \\ - 0.288 & - 0.167 & - 0.943 \\ 0.816 & 0.472 & - 0.333 \end{matrix}]$	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.0 \\ 0.5 \end{matrix}]$	${- 4_{100}^{3} \| \| 2_{1 - 10} \| τ_{(0.5, 0.0, 0.5)}}$
32	$[\begin{matrix} 0.5 & 0.288 & - 0.816 \\ 0.288 & 0.834 & 0.471 \\ - 0.816 & 0.471 & - 0.334 \end{matrix}]$	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.5 \end{matrix}]$	${m_{011} \| \| 2_{1 - 10} \| τ_{(0.5, 0.5, 0.5)}}$
33	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} - 1.0 & 1.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.5 \end{matrix}]$	${m_{110} \| \| 2_{120} \| τ_{(0.0, 0.0, 0.5)}}$
34	$[\begin{matrix} 0.0 & 0.577 & 0.816 \\ - 0.577 & - 0.666 & 0.472 \\ - 0.816 & 0.472 & - 0.333 \end{matrix}]$	$[\begin{matrix} - 1.0 & 1.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.5 \\ 0.5 \end{matrix}]$	${- 4_{001}^{1} \| \| 2_{120} \| τ_{(0.0, 0.5, 0.5)}}$
35	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 0.333 & - 0.943 \\ 0.0 & - 0.943 & - 0.333 \end{matrix}]$	$[\begin{matrix} - 1.0 & 1.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.0 \\ 0.5 \end{matrix}]$	${m_{1 - 10} \| \| 2_{120} \| τ_{(0.5, 0.0, 0.5)}}$
36	$[\begin{matrix} 0.0 & - 0.577 & - 0.816 \\ 0.578 & - 0.666 & 0.471 \\ 0.816 & 0.472 & - 0.334 \end{matrix}]$	$[\begin{matrix} - 1.0 & 1.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.5 \end{matrix}]$	${- 4_{001}^{3} \| \| 2_{120} \| τ_{(0.5, 0.5, 0.5)}}$
37	$[\begin{matrix} - 0.5 & 0.866 & 0.0 \\ - 0.866 & - 0.5 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} - 1.0 & 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${3_{1 - 1 - 1}^{1} \| \| 3_{001}^{2} \| τ_{(0.0, 0.0, 0.0)}}$
38	$[\begin{matrix} - 0.5 & - 0.289 & 0.816 \\ 0.289 & 0.834 & 0.472 \\ - 0.816 & 0.472 & - 0.333 \end{matrix}]$	$[\begin{matrix} - 1.0 & 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.5 \\ 0.0 \end{matrix}]$	${3_{- 11 - 1}^{1} \| \| 3_{001}^{2} \| τ_{(0.0, 0.5, 0.0)}}$
39	$[\begin{matrix} 0.5 & - 0.866 & 0.0 \\ - 0.288 & - 0.167 & - 0.943 \\ 0.816 & 0.472 & - 0.333 \end{matrix}]$	$[\begin{matrix} - 1.0 & 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.0 \\ 0.0 \end{matrix}]$	${3_{111}^{1} \| \| 3_{001}^{2} \| τ_{(0.5, 0.0, 0.0)}}$
40	$[\begin{matrix} 0.5 & 0.288 & - 0.816 \\ 0.866 & - 0.167 & 0.472 \\ 0.0 & - 0.943 & - 0.333 \end{matrix}]$	$[\begin{matrix} - 1.0 & 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.0 \end{matrix}]$	${3_{- 1 - 11}^{1} \| \| 3_{001}^{2} \| τ_{(0.5, 0.5, 0.0)}}$
41	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.5 \end{matrix}]$	${1 \| \| 2_{001} \| τ_{(0.0, 0.0, 0.5)}}$
42	$[\begin{matrix} 0.0 & 0.577 & 0.816 \\ 0.577 & - 0.666 & 0.472 \\ 0.816 & 0.472 & - 0.333 \end{matrix}]$	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.5 \\ 0.5 \end{matrix}]$	${2_{010} \| \| 2_{001} \| τ_{(0.0, 0.5, 0.5)}}$
43	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.333 & - 0.943 \\ 0.0 & - 0.943 & - 0.333 \end{matrix}]$	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.0 \\ 0.5 \end{matrix}]$	${2_{001} \| \| 2_{001} \| τ_{(0.5, 0.0, 0.5)}}$
44	$[\begin{matrix} 0.0 & - 0.577 & - 0.816 \\ - 0.577 & - 0.666 & 0.472 \\ - 0.816 & 0.472 & - 0.333 \end{matrix}]$	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.5 \end{matrix}]$	${2_{100} \| \| 2_{001} \| τ_{(0.5, 0.5, 0.5)}}$
45	$[\begin{matrix} 0.5 & 0.866 & 0.0 \\ 0.866 & - 0.5 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ - 1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${m_{101} \| \| 2_{010} \| τ_{(0.0, 0.0, 0.0)}}$
46	$[\begin{matrix} 0.5 & - 0.288 & 0.816 \\ - 0.288 & 0.834 & 0.471 \\ 0.816 & 0.471 & - 0.334 \end{matrix}]$	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ - 1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.5 \\ 0.0 \end{matrix}]$	${m_{- 101} \| \| 2_{010} \| τ_{(0.0, 0.5, 0.0)}}$
47	$[\begin{matrix} - 0.5 & - 0.866 & 0.0 \\ 0.288 & - 0.167 & - 0.943 \\ - 0.816 & 0.472 & - 0.333 \end{matrix}]$	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ - 1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.0 \\ 0.0 \end{matrix}]$	${- 4_{010}^{3} \| \| 2_{010} \| τ_{(0.5, 0.0, 0.0)}}$
48	$[\begin{matrix} - 0.5 & 0.288 & - 0.816 \\ - 0.866 & - 0.167 & 0.472 \\ 0.0 & - 0.943 & - 0.333 \end{matrix}]$	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ - 1.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.0 \end{matrix}]$	${- 4_{010}^{1} \| \| 2_{010} \| τ_{(0.5, 0.5, 0.0)}}$

FINDSPINGROUP: Identify Spin Group and Related Properties