Information of spin space group $P\ce{^{1}{b}}\ce{^{1}{c}}\ce{^{1}{a}}\ce{^{\infty m}{1}}$ ( nSSG:61.61.1.1 )

General Positions

The continuously infinite group ^∞1 is omitted.

Coordinates	Seitz symbol
a, b, c | x, y, z	{ 1 ‖ 1 | 0 }
a+1/2, -b+1/2, -c | x, y, z	{ 1 ‖ 2100 | 1/2 1/2 0 }
-a, b+1/2, -c+1/2 | x, y, z	{ 1 ‖ 2010 | 0 1/2 1/2 }
-a+1/2, -b, c+1/2 | x, y, z	{ 1 ‖ 2001 | 1/2 0 1/2 }
-a, -b, -c | x, y, z	{ 1 ‖ -1 | 0 }
-a+1/2, b+1/2, c | x, y, z	{ 1 ‖ m100 | 1/2 1/2 0 }
a, -b+1/2, c+1/2 | x, y, z	{ 1 ‖ m010 | 0 1/2 1/2 }
a+1/2, b, -c+1/2 | x, y, z	{ 1 ‖ m001 | 1/2 0 1/2 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
a+1/2, -b+1/2, -c | -x, y, z	{ m ‖ 2100 | 1/2 1/2 0 }
-a, b+1/2, -c+1/2 | -x, y, z	{ m ‖ 2010 | 0 1/2 1/2 }
-a+1/2, -b, c+1/2 | -x, y, z	{ m ‖ 2001 | 1/2 0 1/2 }
-a, -b, -c | -x, y, z	{ m ‖ -1 | 0 }
-a+1/2, b+1/2, c | -x, y, z	{ m ‖ m100 | 1/2 1/2 0 }
a, -b+1/2, c+1/2 | -x, y, z	{ m ‖ m010 | 0 1/2 1/2 }
a+1/2, b, -c+1/2 | -x, y, z	{ m ‖ m001 | 1/2 0 1/2 }

Wyckoff Positions

WP	Site symmetry	Representative
4a	$\ce{^{1}{-1}}\ce{^{\infty m}{1}} $	(0,0,0 |0,0,z)
4b	$\ce{^{1}{-1}}\ce{^{\infty m}{1}} $	(0,0,1/2 |0,0,z)
8c	$\ce{^{1}{1}}\ce{^{\infty m}{1}} $	(a,b,c |0,0,z)

Spin Brillouin Zone & k-little co-groups

Wavevector-k	Little co-group
Γ:(0,0,0)	$ \ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
R:(1/2,1/2,1/2)	$ \ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
S:(1/2,1/2,0)	$ \ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
T:(0,1/2,1/2)	$ \ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
U:(1/2,0,1/2)	$ \ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
X:(1/2,0,0)	$ \ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
Y:(0,1/2,0)	$ \ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
Z:(0,0,1/2)	$ \ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
A:(u,0,1/2)	$ \ce{^{m}{m}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
B:(0,v,1/2)	$ \ce{^{1}{m}}\ce{^{m}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
C:(u,1/2,0)	$ \ce{^{m}{m}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
D:(1/2,v,0)	$ \ce{^{1}{m}}\ce{^{m}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
Δ:(0,v,0)	$ \ce{^{1}{m}}\ce{^{m}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
E:(u,1/2,1/2)	$ \ce{^{m}{m}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
G:(1/2,0,w)	$ \ce{^{1}{m}}\ce{^{1}{m}}\ce{^{m}{m}}\ce{^{\infty}{1}} $
H:(0,1/2,w)	$ \ce{^{1}{m}}\ce{^{1}{m}}\ce{^{m}{m}}\ce{^{\infty}{1}} $
Λ:(0,0,w)	$ \ce{^{1}{m}}\ce{^{1}{m}}\ce{^{m}{m}}\ce{^{\infty}{1}} $
P:(1/2,v,1/2)	$ \ce{^{1}{m}}\ce{^{m}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
Q:(1/2,1/2,w)	$ \ce{^{1}{m}}\ce{^{1}{m}}\ce{^{m}{m}}\ce{^{\infty}{1}} $
Σ:(u,0,0)	$ \ce{^{m}{m}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
K:(0,v,w)	$ \ce{^{m}{2/}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
L:(1/2,v,w)	$ \ce{^{m}{2/}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
M:(u,0,w)	$ \ce{^{m}{2/}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
N:(u,1/2,w)	$ \ce{^{m}{2/}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
V:(u,v,0)	$ \ce{^{m}{2/}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
W:(u,v,1/2)	$ \ce{^{m}{2/}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
GP:(u,v,w)	$ \ce{^{m}{-1}}\ce{^{\infty}{1}} $

Spin Brillouin Zone

Band Representations (Magnon)

4a

k-vector	Ag↑G(4)
Γ:(0,0,0)	$Γ_{1}^{S,+}(1)⊕Γ_{2}^{S,+}(1)⊕Γ_{3}^{S,+}(1)⊕Γ_{4}^{S,+}(1) $
R:(1/2,1/2,1/2)	$R_{1}^{S,+}R_{1}^{S,+}(4) $
S:(1/2,1/2,0)	$S_{1}^{S}S_{2}^{S}(4) $
T:(0,1/2,1/2)	$T_{1}^{S}T_{2}^{S}(4) $
U:(1/2,0,1/2)	$U_{1}^{S}U_{2}^{S}(4) $
X:(1/2,0,0)	$X_{1}^{S}(2)⊕X_{2}^{S}(2) $
Y:(0,1/2,0)	$Y_{1}^{S}(2)⊕Y_{2}^{S}(2) $
Z:(0,0,1/2)	$Z_{1}^{S}(2)⊕Z_{2}^{S}(2) $
A:(u,0,1/2)	$2A_{1}^{S}(2) $
B:(0,v,1/2)	$B_{1}^{S}B_{3}^{S}(2)⊕B_{2}^{S}B_{4}^{S}(2) $
C:(u,1/2,0)	$C_{1}^{S}C_{4}^{S}(2)⊕C_{2}^{S}C_{3}^{S}(2) $
D:(1/2,v,0)	$2D_{1}^{S}(2) $
Δ:(0,v,0)	$Δ_{1}^{S}(1)⊕Δ_{2}^{S}(1)⊕Δ_{3}^{S}(1)⊕Δ_{4}^{S}(1) $
E:(u,1/2,1/2)	$E_{1}^{S}E_{1}^{S}(4) $
G:(1/2,0,w)	$G_{1}^{S}G_{4}^{S}(2)⊕G_{2}^{S}G_{3}^{S}(2) $
H:(0,1/2,w)	$2H_{1}^{S}(2) $
Λ:(0,0,w)	$Λ_{1}^{S}(1)⊕Λ_{2}^{S}(1)⊕Λ_{3}^{S}(1)⊕Λ_{4}^{S}(1) $
P:(1/2,v,1/2)	$P_{1}^{S}P_{1}^{S}(4) $
Q:(1/2,1/2,w)	$Q_{1}^{S}Q_{1}^{S}(4) $
Σ:(u,0,0)	$Σ_{1}^{S}(1)⊕Σ_{2}^{S}(1)⊕Σ_{3}^{S}(1)⊕Σ_{4}^{S}(1) $
K:(0,v,w)	$2K_{1}^{S}(1)⊕2K_{2}^{S}(1) $
L:(1/2,v,w)	$2L_{1}^{S}L_{2}^{S}(2) $
M:(u,0,w)	$2M_{1}^{S}(1)⊕2M_{2}^{S}(1) $
N:(u,1/2,w)	$2N_{1}^{S}N_{2}^{S}(2) $
V:(u,v,0)	$2V_{1}^{S}(1)⊕2V_{2}^{S}(1) $
W:(u,v,1/2)	$2W_{1}^{S}W_{2}^{S}(2) $
GP:(u,v,w)	$4GP_{1}^{S}(1) $

4b

k-vector	Ag↑G(4)
Γ:(0,0,0)	$Γ_{1}^{S,+}(1)⊕Γ_{2}^{S,+}(1)⊕Γ_{3}^{S,+}(1)⊕Γ_{4}^{S,+}(1) $
R:(1/2,1/2,1/2)	$R_{1}^{S,-}R_{1}^{S,-}(4) $
S:(1/2,1/2,0)	$S_{1}^{S}S_{2}^{S}(4) $
T:(0,1/2,1/2)	$T_{1}^{S}T_{2}^{S}(4) $
U:(1/2,0,1/2)	$U_{1}^{S}U_{2}^{S}(4) $
X:(1/2,0,0)	$X_{1}^{S}(2)⊕X_{2}^{S}(2) $
Y:(0,1/2,0)	$Y_{1}^{S}(2)⊕Y_{2}^{S}(2) $
Z:(0,0,1/2)	$Z_{1}^{S}(2)⊕Z_{2}^{S}(2) $
A:(u,0,1/2)	$2A_{1}^{S}(2) $
B:(0,v,1/2)	$B_{1}^{S}B_{3}^{S}(2)⊕B_{2}^{S}B_{4}^{S}(2) $
C:(u,1/2,0)	$C_{1}^{S}C_{4}^{S}(2)⊕C_{2}^{S}C_{3}^{S}(2) $
D:(1/2,v,0)	$2D_{1}^{S}(2) $
Δ:(0,v,0)	$Δ_{1}^{S}(1)⊕Δ_{2}^{S}(1)⊕Δ_{3}^{S}(1)⊕Δ_{4}^{S}(1) $
E:(u,1/2,1/2)	$E_{1}^{S}E_{1}^{S}(4) $
G:(1/2,0,w)	$G_{1}^{S}G_{4}^{S}(2)⊕G_{2}^{S}G_{3}^{S}(2) $
H:(0,1/2,w)	$2H_{1}^{S}(2) $
Λ:(0,0,w)	$Λ_{1}^{S}(1)⊕Λ_{2}^{S}(1)⊕Λ_{3}^{S}(1)⊕Λ_{4}^{S}(1) $
P:(1/2,v,1/2)	$P_{1}^{S}P_{1}^{S}(4) $
Q:(1/2,1/2,w)	$Q_{1}^{S}Q_{1}^{S}(4) $
Σ:(u,0,0)	$Σ_{1}^{S}(1)⊕Σ_{2}^{S}(1)⊕Σ_{3}^{S}(1)⊕Σ_{4}^{S}(1) $
K:(0,v,w)	$2K_{1}^{S}(1)⊕2K_{2}^{S}(1) $
L:(1/2,v,w)	$2L_{1}^{S}L_{2}^{S}(2) $
M:(u,0,w)	$2M_{1}^{S}(1)⊕2M_{2}^{S}(1) $
N:(u,1/2,w)	$2N_{1}^{S}N_{2}^{S}(2) $
V:(u,v,0)	$2V_{1}^{S}(1)⊕2V_{2}^{S}(1) $
W:(u,v,1/2)	$2W_{1}^{S}W_{2}^{S}(2) $
GP:(u,v,w)	$4GP_{1}^{S}(1) $

8c

k-vector	A↑G(8)
Γ:(0,0,0)	$Γ_{1}^{S,+}(1)⊕Γ_{1}^{S,-}(1)⊕Γ_{2}^{S,+}(1)⊕Γ_{2}^{S,-}(1)⊕Γ_{3}^{S,+}(1)⊕Γ_{3}^{S,-}(1)⊕Γ_{4}^{S,+}(1)⊕Γ_{4}^{S,-}(1) $
R:(1/2,1/2,1/2)	$R_{1}^{S,+}R_{1}^{S,+}(4)⊕R_{1}^{S,-}R_{1}^{S,-}(4) $
S:(1/2,1/2,0)	$2S_{1}^{S}S_{2}^{S}(4) $
T:(0,1/2,1/2)	$2T_{1}^{S}T_{2}^{S}(4) $
U:(1/2,0,1/2)	$2U_{1}^{S}U_{2}^{S}(4) $
X:(1/2,0,0)	$2X_{1}^{S}(2)⊕2X_{2}^{S}(2) $
Y:(0,1/2,0)	$2Y_{1}^{S}(2)⊕2Y_{2}^{S}(2) $
Z:(0,0,1/2)	$2Z_{1}^{S}(2)⊕2Z_{2}^{S}(2) $
A:(u,0,1/2)	$4A_{1}^{S}(2) $
B:(0,v,1/2)	$2B_{1}^{S}B_{3}^{S}(2)⊕2B_{2}^{S}B_{4}^{S}(2) $
C:(u,1/2,0)	$2C_{1}^{S}C_{4}^{S}(2)⊕2C_{2}^{S}C_{3}^{S}(2) $
D:(1/2,v,0)	$4D_{1}^{S}(2) $
Δ:(0,v,0)	$2Δ_{1}^{S}(1)⊕2Δ_{2}^{S}(1)⊕2Δ_{3}^{S}(1)⊕2Δ_{4}^{S}(1) $
E:(u,1/2,1/2)	$2E_{1}^{S}E_{1}^{S}(4) $
G:(1/2,0,w)	$2G_{1}^{S}G_{4}^{S}(2)⊕2G_{2}^{S}G_{3}^{S}(2) $
H:(0,1/2,w)	$4H_{1}^{S}(2) $
Λ:(0,0,w)	$2Λ_{1}^{S}(1)⊕2Λ_{2}^{S}(1)⊕2Λ_{3}^{S}(1)⊕2Λ_{4}^{S}(1) $
P:(1/2,v,1/2)	$2P_{1}^{S}P_{1}^{S}(4) $
Q:(1/2,1/2,w)	$2Q_{1}^{S}Q_{1}^{S}(4) $
Σ:(u,0,0)	$2Σ_{1}^{S}(1)⊕2Σ_{2}^{S}(1)⊕2Σ_{3}^{S}(1)⊕2Σ_{4}^{S}(1) $
K:(0,v,w)	$4K_{1}^{S}(1)⊕4K_{2}^{S}(1) $
L:(1/2,v,w)	$4L_{1}^{S}L_{2}^{S}(2) $
M:(u,0,w)	$4M_{1}^{S}(1)⊕4M_{2}^{S}(1) $
N:(u,1/2,w)	$4N_{1}^{S}N_{2}^{S}(2) $
V:(u,v,0)	$4V_{1}^{S}(1)⊕4V_{2}^{S}(1) $
W:(u,v,1/2)	$4W_{1}^{S}W_{2}^{S}(2) $
GP:(u,v,w)	$8GP_{1}^{S}(1) $