Find Spin Group - Program for Identifying Spin Space Group

File Name	Spin Space Group	G0 Symbol	L0 Symbol	it	ik
$C e A u A l 3$	$P^{1} 4^{1} m^{1} m^{4_{001}^{1}} (1 / 2 1 / 2 1 / 4)^{m} 1$	$I 4 m m (107)$	$P 4 m m (99)$	$1$	$4$

Space Group	Magnetic Space Group	Spin Part PG	Configuration	Magnetic Phase
$I 4 m m (107)$	$\begin{aligned} P_{c} 4_{1} (76.10) \\ T y p e I V \end{aligned}$	$\begin{aligned} C 4 \times C s & = C 4 h \\ 4 \times m & = 4 / m \end{aligned}$	$C o p l a n a r$	$A F M$

Little Cogroup at General Position		Spin Splitting
without SOC	with SOC	without SOC	with SOC
$^{4} 1$	$1$	$k - d e p e n d e n t$	$Y e s$

Effective Magnetic Point Group	Anomalous Hall Effect
$4 m m 1^{'}$	without SOC	with SOC
$4 m m 1^{'}$	$N o$	$N o$

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

L0-std Unit Magnetic Cell:

a	b	c	alpha	beta	gamma
4.3172	4.3172	21.618	90.0	90.0	90.0

Fractional Coordinates & Moments (in L0-std Unit Magnetic Cell)

No.	Atom	Site	Moment (in lattice)
1	Al	[0.0, 0.0, 0.2035]	[0.0, 0.0, 0.0]
2	Al	[0.5, 0.5, 0.4535]	[0.0, 0.0, 0.0]
3	Al	[0.0, 0.0, 0.7035]	[0.0, 0.0, 0.0]
4	Al	[0.5, 0.5, 0.9535]	[0.0, 0.0, 0.0]
5	Al	[0.0, 0.5, 0.129]	[0.0, 0.0, 0.0]
6	Al	[0.5, 0.0, 0.129]	[0.0, 0.0, 0.0]
7	Al	[0.5, 0.0, 0.379]	[0.0, 0.0, 0.0]
8	Al	[0.0, 0.5, 0.379]	[0.0, 0.0, 0.0]
9	Al	[0.0, 0.5, 0.629]	[0.0, 0.0, 0.0]
10	Al	[0.5, 0.0, 0.629]	[0.0, 0.0, 0.0]
11	Al	[0.5, 0.0, 0.879]	[0.0, 0.0, 0.0]
12	Al	[0.0, 0.5, 0.879]	[0.0, 0.0, 0.0]
13	Au	[0.0, 0.0, 0.3155]	[0.0, 0.0, 0.0]
14	Au	[0.5, 0.5, 0.5655]	[0.0, 0.0, 0.0]
15	Au	[0.0, 0.0, 0.8155]	[0.0, 0.0, 0.0]
16	Au	[0.5, 0.5, 0.0655]	[0.0, 0.0, 0.0]
17	Ce	[0.0, 0.0, 0.0]	[2.0, 0.0, 0.0]
18	Ce	[0.0, 0.0, 0.5]	[-2.0, 0.0, 0.0]
19	Ce	[0.5, 0.5, 0.25]	[0.0, 2.0, 0.0]
20	Ce	[0.5, 0.5, 0.75]	[-0.0, -2.0, 0.0]

Elements of the Spin Space Group (L0-std Unit Magnetic Cell):

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
1	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${1 \| \| 1 \| τ_{(0.0, 0.0, 0.0)}}$
2	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.25 \end{matrix}]$	${4_{001}^{1} \| \| 1 \| τ_{(0.5, 0.5, 0.25)}}$
3	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.5 \end{matrix}]$	${2_{001} \| \| 1 \| τ_{(0.0, 0.0, 0.5)}}$
4	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.75 \end{matrix}]$	${4_{001}^{3} \| \| 1 \| τ_{(0.5, 0.5, 0.75)}}$
5	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${m_{001} \| \| 1 \| τ_{(0.0, 0.0, 0.0)}}$
6	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.25 \end{matrix}]$	${- 4_{001}^{3} \| \| 1 \| τ_{(0.5, 0.5, 0.25)}}$
7	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.5 \end{matrix}]$	${- 1 \| \| 1 \| τ_{(0.0, 0.0, 0.5)}}$
8	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.75 \end{matrix}]$	${- 4_{001}^{1} \| \| 1 \| τ_{(0.5, 0.5, 0.75)}}$
9	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${1 \| \| m_{010} \| τ_{(0.0, 0.0, 0.0)}}$
10	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.25 \end{matrix}]$	${4_{001}^{1} \| \| m_{010} \| τ_{(0.5, 0.5, 0.25)}}$
11	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.5 \end{matrix}]$	${2_{001} \| \| m_{010} \| τ_{(0.0, 0.0, 0.5)}}$
12	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.75 \end{matrix}]$	${4_{001}^{3} \| \| m_{010} \| τ_{(0.5, 0.5, 0.75)}}$
13	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${m_{001} \| \| m_{010} \| τ_{(0.0, 0.0, 0.0)}}$
14	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.25 \end{matrix}]$	${- 4_{001}^{3} \| \| m_{010} \| τ_{(0.5, 0.5, 0.25)}}$
15	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.5 \end{matrix}]$	${- 1 \| \| m_{010} \| τ_{(0.0, 0.0, 0.5)}}$
16	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.75 \end{matrix}]$	${- 4_{001}^{1} \| \| m_{010} \| τ_{(0.5, 0.5, 0.75)}}$
17	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${1 \| \| m_{1 - 10} \| τ_{(0.0, 0.0, 0.0)}}$
18	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.25 \end{matrix}]$	${4_{001}^{1} \| \| m_{1 - 10} \| τ_{(0.5, 0.5, 0.25)}}$
19	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.5 \end{matrix}]$	${2_{001} \| \| m_{1 - 10} \| τ_{(0.0, 0.0, 0.5)}}$
20	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.75 \end{matrix}]$	${4_{001}^{3} \| \| m_{1 - 10} \| τ_{(0.5, 0.5, 0.75)}}$
21	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${m_{001} \| \| m_{1 - 10} \| τ_{(0.0, 0.0, 0.0)}}$
22	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.25 \end{matrix}]$	${- 4_{001}^{3} \| \| m_{1 - 10} \| τ_{(0.5, 0.5, 0.25)}}$
23	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.5 \end{matrix}]$	${- 1 \| \| m_{1 - 10} \| τ_{(0.0, 0.0, 0.5)}}$
24	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.75 \end{matrix}]$	${- 4_{001}^{1} \| \| m_{1 - 10} \| τ_{(0.5, 0.5, 0.75)}}$
25	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${1 \| \| 4_{001}^{3} \| τ_{(0.0, 0.0, 0.0)}}$
26	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.25 \end{matrix}]$	${4_{001}^{1} \| \| 4_{001}^{3} \| τ_{(0.5, 0.5, 0.25)}}$
27	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.5 \end{matrix}]$	${2_{001} \| \| 4_{001}^{3} \| τ_{(0.0, 0.0, 0.5)}}$
28	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.75 \end{matrix}]$	${4_{001}^{3} \| \| 4_{001}^{3} \| τ_{(0.5, 0.5, 0.75)}}$
29	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${m_{001} \| \| 4_{001}^{3} \| τ_{(0.0, 0.0, 0.0)}}$
30	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.25 \end{matrix}]$	${- 4_{001}^{3} \| \| 4_{001}^{3} \| τ_{(0.5, 0.5, 0.25)}}$
31	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.5 \end{matrix}]$	${- 1 \| \| 4_{001}^{3} \| τ_{(0.0, 0.0, 0.5)}}$
32	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.75 \end{matrix}]$	${- 4_{001}^{1} \| \| 4_{001}^{3} \| τ_{(0.5, 0.5, 0.75)}}$
33	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${1 \| \| 4_{001}^{1} \| τ_{(0.0, 0.0, 0.0)}}$
34	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.25 \end{matrix}]$	${4_{001}^{1} \| \| 4_{001}^{1} \| τ_{(0.5, 0.5, 0.25)}}$
35	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.5 \end{matrix}]$	${2_{001} \| \| 4_{001}^{1} \| τ_{(0.0, 0.0, 0.5)}}$
36	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.75 \end{matrix}]$	${4_{001}^{3} \| \| 4_{001}^{1} \| τ_{(0.5, 0.5, 0.75)}}$
37	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${m_{001} \| \| 4_{001}^{1} \| τ_{(0.0, 0.0, 0.0)}}$
38	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.25 \end{matrix}]$	${- 4_{001}^{3} \| \| 4_{001}^{1} \| τ_{(0.5, 0.5, 0.25)}}$
39	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.5 \end{matrix}]$	${- 1 \| \| 4_{001}^{1} \| τ_{(0.0, 0.0, 0.5)}}$
40	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.75 \end{matrix}]$	${- 4_{001}^{1} \| \| 4_{001}^{1} \| τ_{(0.5, 0.5, 0.75)}}$
41	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${1 \| \| m_{110} \| τ_{(0.0, 0.0, 0.0)}}$
42	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.25 \end{matrix}]$	${4_{001}^{1} \| \| m_{110} \| τ_{(0.5, 0.5, 0.25)}}$
43	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.5 \end{matrix}]$	${2_{001} \| \| m_{110} \| τ_{(0.0, 0.0, 0.5)}}$
44	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.75 \end{matrix}]$	${4_{001}^{3} \| \| m_{110} \| τ_{(0.5, 0.5, 0.75)}}$
45	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${m_{001} \| \| m_{110} \| τ_{(0.0, 0.0, 0.0)}}$
46	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.25 \end{matrix}]$	${- 4_{001}^{3} \| \| m_{110} \| τ_{(0.5, 0.5, 0.25)}}$
47	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.5 \end{matrix}]$	${- 1 \| \| m_{110} \| τ_{(0.0, 0.0, 0.5)}}$
48	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.75 \end{matrix}]$	${- 4_{001}^{1} \| \| m_{110} \| τ_{(0.5, 0.5, 0.75)}}$
49	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${1 \| \| m_{100} \| τ_{(0.0, 0.0, 0.0)}}$
50	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.25 \end{matrix}]$	${4_{001}^{1} \| \| m_{100} \| τ_{(0.5, 0.5, 0.25)}}$
51	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.5 \end{matrix}]$	${2_{001} \| \| m_{100} \| τ_{(0.0, 0.0, 0.5)}}$
52	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.75 \end{matrix}]$	${4_{001}^{3} \| \| m_{100} \| τ_{(0.5, 0.5, 0.75)}}$
53	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${m_{001} \| \| m_{100} \| τ_{(0.0, 0.0, 0.0)}}$
54	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.25 \end{matrix}]$	${- 4_{001}^{3} \| \| m_{100} \| τ_{(0.5, 0.5, 0.25)}}$
55	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.5 \end{matrix}]$	${- 1 \| \| m_{100} \| τ_{(0.0, 0.0, 0.5)}}$
56	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.75 \end{matrix}]$	${- 4_{001}^{1} \| \| m_{100} \| τ_{(0.5, 0.5, 0.75)}}$
57	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${1 \| \| 2_{001} \| τ_{(0.0, 0.0, 0.0)}}$
58	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.25 \end{matrix}]$	${4_{001}^{1} \| \| 2_{001} \| τ_{(0.5, 0.5, 0.25)}}$
59	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.5 \end{matrix}]$	${2_{001} \| \| 2_{001} \| τ_{(0.0, 0.0, 0.5)}}$
60	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.75 \end{matrix}]$	${4_{001}^{3} \| \| 2_{001} \| τ_{(0.5, 0.5, 0.75)}}$
61	$[\begin{matrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.0 \end{matrix}]$	${m_{001} \| \| 2_{001} \| τ_{(0.0, 0.0, 0.0)}}$
62	$[\begin{matrix} 0.0 & - 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.25 \end{matrix}]$	${- 4_{001}^{3} \| \| 2_{001} \| τ_{(0.5, 0.5, 0.25)}}$
63	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.0 \\ 0.0 \\ 0.5 \end{matrix}]$	${- 1 \| \| 2_{001} \| τ_{(0.0, 0.0, 0.5)}}$
64	$[\begin{matrix} 0.0 & 1.0 & 0.0 \\ - 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & - 1.0 \end{matrix}]$	$[\begin{matrix} - 1.0 & 0.0 & 0.0 \\ 0.0 & - 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \\ 0.75 \end{matrix}]$	${- 4_{001}^{1} \| \| 2_{001} \| τ_{(0.5, 0.5, 0.75)}}$

FINDSPINGROUP: Identify Spin Group and Related Properties