Information of spin space group $I\ce{^{1}{4_{1}/}}\ce{^{1}{a}}\ce{^{\text{-}1}{c}}\ce{^{\text{-}1}{d}}\ce{^{\infty m}{1}}$ ( nSSG:88.142.1.1 )

General Positions

The continuously infinite group ^∞1 is omitted.

Coordinates	Seitz symbol
a, b, c | x, y, z	{ 1 ‖ 1 | 0 }
-b+1/4, a+3/4, c+1/4 | x, y, z	{ 1 ‖ 4+001 | 1/4 3/4 1/4 }
b+3/4, -a+3/4, c+1/4 | x, y, z	{ 1 ‖ 4-001 | 3/4 3/4 1/4 }
-a, -b+1/2, c | x, y, z	{ 1 ‖ 2001 | 0 1/2 0 }
-a, -b, -c | x, y, z	{ 1 ‖ -1 | 0 }
b+3/4, -a+1/4, -c+3/4 | x, y, z	{ 1 ‖ -4+001 | 3/4 1/4 3/4 }
-b+1/4, a+1/4, -c+3/4 | x, y, z	{ 1 ‖ -4-001 | 1/4 1/4 3/4 }
a, b+1/2, -c | x, y, z	{ 1 ‖ m001 | 0 1/2 0 }
a+1/2, -b+1/2, -c | -x, -y, -z	{ -1 ‖ 2100 | 1/2 1/2 0 }
-a+1/2, b, -c | -x, -y, -z	{ -1 ‖ 2010 | 1/2 0 0 }
b+3/4, a+1/4, -c+1/4 | -x, -y, -z	{ -1 ‖ 2110 | 3/4 1/4 1/4 }
-b+1/4, -a+1/4, -c+1/4 | -x, -y, -z	{ -1 ‖ 21-10 | 1/4 1/4 1/4 }
-a, b, c+1/2 | -x, -y, -z	{ -1 ‖ m100 | 0 0 1/2 }
a, -b+1/2, c+1/2 | -x, -y, -z	{ -1 ‖ m010 | 0 1/2 1/2 }
-b+3/4, -a+1/4, c+1/4 | -x, -y, -z	{ -1 ‖ m110 | 3/4 1/4 1/4 }
b+1/4, a+1/4, c+1/4 | -x, -y, -z	{ -1 ‖ m1-10 | 1/4 1/4 1/4 }
a+1/2, b+1/2, c+1/2 | x, y, z	{ 1 ‖ 1 | 1/2 1/2 1/2 }
-b+3/4, a+1/4, c+3/4 | x, y, z	{ 1 ‖ 4+001 | 3/4 1/4 3/4 }
b+1/4, -a+1/4, c+3/4 | x, y, z	{ 1 ‖ 4-001 | 1/4 1/4 3/4 }
-a+1/2, -b, c+1/2 | x, y, z	{ 1 ‖ 2001 | 1/2 0 1/2 }
-a+1/2, -b+1/2, -c+1/2 | x, y, z	{ 1 ‖ -1 | 1/2 1/2 1/2 }
b+1/4, -a+3/4, -c+1/4 | x, y, z	{ 1 ‖ -4+001 | 1/4 3/4 1/4 }
-b+3/4, a+3/4, -c+1/4 | x, y, z	{ 1 ‖ -4-001 | 3/4 3/4 1/4 }
a+1/2, b, -c+1/2 | x, y, z	{ 1 ‖ m001 | 1/2 0 1/2 }
a, -b, -c+1/2 | -x, -y, -z	{ -1 ‖ 2100 | 0 0 1/2 }
-a, b+1/2, -c+1/2 | -x, -y, -z	{ -1 ‖ 2010 | 0 1/2 1/2 }
b+1/4, a+3/4, -c+3/4 | -x, -y, -z	{ -1 ‖ 2110 | 1/4 3/4 3/4 }
-b+3/4, -a+3/4, -c+3/4 | -x, -y, -z	{ -1 ‖ 21-10 | 3/4 3/4 3/4 }
-a+1/2, b+1/2, c | -x, -y, -z	{ -1 ‖ m100 | 1/2 1/2 0 }
a+1/2, -b, c | -x, -y, -z	{ -1 ‖ m010 | 1/2 0 0 }
-b+1/4, -a+3/4, c+3/4 | -x, -y, -z	{ -1 ‖ m110 | 1/4 3/4 3/4 }
b+3/4, a+3/4, c+3/4 | -x, -y, -z	{ -1 ‖ m1-10 | 3/4 3/4 3/4 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
-b+1/4, a+3/4, c+1/4 | -x, y, z	{ m ‖ 4+001 | 1/4 3/4 1/4 }
b+3/4, -a+3/4, c+1/4 | -x, y, z	{ m ‖ 4-001 | 3/4 3/4 1/4 }
-a, -b+1/2, c | -x, y, z	{ m ‖ 2001 | 0 1/2 0 }
-a, -b, -c | -x, y, z	{ m ‖ -1 | 0 }
b+3/4, -a+1/4, -c+3/4 | -x, y, z	{ m ‖ -4+001 | 3/4 1/4 3/4 }
-b+1/4, a+1/4, -c+3/4 | -x, y, z	{ m ‖ -4-001 | 1/4 1/4 3/4 }
a, b+1/2, -c | -x, y, z	{ m ‖ m001 | 0 1/2 0 }
a+1/2, -b+1/2, -c | x, -y, -z	{ 2 ‖ 2100 | 1/2 1/2 0 }
-a+1/2, b, -c | x, -y, -z	{ 2 ‖ 2010 | 1/2 0 0 }
b+3/4, a+1/4, -c+1/4 | x, -y, -z	{ 2 ‖ 2110 | 3/4 1/4 1/4 }
-b+1/4, -a+1/4, -c+1/4 | x, -y, -z	{ 2 ‖ 21-10 | 1/4 1/4 1/4 }
-a, b, c+1/2 | x, -y, -z	{ 2 ‖ m100 | 0 0 1/2 }
a, -b+1/2, c+1/2 | x, -y, -z	{ 2 ‖ m010 | 0 1/2 1/2 }
-b+3/4, -a+1/4, c+1/4 | x, -y, -z	{ 2 ‖ m110 | 3/4 1/4 1/4 }
b+1/4, a+1/4, c+1/4 | x, -y, -z	{ 2 ‖ m1-10 | 1/4 1/4 1/4 }
a+1/2, b+1/2, c+1/2 | -x, y, z	{ m ‖ 1 | 1/2 1/2 1/2 }
-b+3/4, a+1/4, c+3/4 | -x, y, z	{ m ‖ 4+001 | 3/4 1/4 3/4 }
b+1/4, -a+1/4, c+3/4 | -x, y, z	{ m ‖ 4-001 | 1/4 1/4 3/4 }
-a+1/2, -b, c+1/2 | -x, y, z	{ m ‖ 2001 | 1/2 0 1/2 }
-a+1/2, -b+1/2, -c+1/2 | -x, y, z	{ m ‖ -1 | 1/2 1/2 1/2 }
b+1/4, -a+3/4, -c+1/4 | -x, y, z	{ m ‖ -4+001 | 1/4 3/4 1/4 }
-b+3/4, a+3/4, -c+1/4 | -x, y, z	{ m ‖ -4-001 | 3/4 3/4 1/4 }
a+1/2, b, -c+1/2 | -x, y, z	{ m ‖ m001 | 1/2 0 1/2 }
a, -b, -c+1/2 | x, -y, -z	{ 2 ‖ 2100 | 0 0 1/2 }
-a, b+1/2, -c+1/2 | x, -y, -z	{ 2 ‖ 2010 | 0 1/2 1/2 }
b+1/4, a+3/4, -c+3/4 | x, -y, -z	{ 2 ‖ 2110 | 1/4 3/4 3/4 }
-b+3/4, -a+3/4, -c+3/4 | x, -y, -z	{ 2 ‖ 21-10 | 3/4 3/4 3/4 }
-a+1/2, b+1/2, c | x, -y, -z	{ 2 ‖ m100 | 1/2 1/2 0 }
a+1/2, -b, c | x, -y, -z	{ 2 ‖ m010 | 1/2 0 0 }
-b+1/4, -a+3/4, c+3/4 | x, -y, -z	{ 2 ‖ m110 | 1/4 3/4 3/4 }
b+3/4, a+3/4, c+3/4 | x, -y, -z	{ 2 ‖ m1-10 | 3/4 3/4 3/4 }

Wyckoff Positions

WP	Site symmetry	Representative
8a	$\ce{^{1}{-4}}..\ce{^{\infty m}{1}} $	(0,1/4,3/8 | 0,0,z)
8b	$\ce{^{1}{2}}.\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(0,1/4,1/8 | 0,0,0)
16c	$\ce{^{1}{-1}}\ce{^{\infty m}{1}} $	(0,0,0 | 0,0,z)
16d	$\ce{^{1}{2}}..\ce{^{\infty m}{1}} $	(0,1/4,c | 0,0,z)
16e	$.\ce{^{-1}{2}}.\ce{^{\infty m}{1}} $	(a,0,1/4 | 0,0,0)
16f	$..\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(a,a+1/4,1/8 | 0,0,0)
32g	$\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}{4/}}\ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
M:(1,1,1)	$\ce{^{1}{4/}}\ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
P:(1/2,1/2,1/2)	$\ce{^{m}{4/}}\ce{^{m}{m}}\ce{^{-1}{m}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
X:(1/2,1/2,0)	$\ce{^{2}{m}}\ce{^{2}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
N:(1/2,0,1/2)	$\ce{^{2}{2/}}\ce{^{2}{m}}\ce{^{\infty m}{1}} $
Λ:(0,0,w)	$\ce{^{1}{4/}}\ce{^{m}{m}}\ce{^{2}{m}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
Δ:(u,u,0)	$\ce{^{2}{m}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
Q:(1/2,v,1/2)	$\ce{^{2}{2/}}\ce{^{-1}{m}}\ce{^{\infty}{1}} $
Σ:(u,0,0)	$\ce{^{-1}{m}}\ce{^{2}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
W:(1/2,1/2,w)	$\ce{^{2}{m}}\ce{^{2}{m}}\ce{^{m}{m}}\ce{^{\infty}{1}} $
Y:(u,1-u,0)	$\ce{^{-1}{m}}\ce{^{2}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
A:(u,u,w)	$\ce{^{-1}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
B:(u,0,w)	$\ce{^{-1}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
C:(u,v,0)	$\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)

32g

k-vector	k-vector-G↑	A↑G(16)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S,+}(2)⊕Γ_{1}^{S,-}(2)⊕Γ_{2}^{S,+}(2)⊕Γ_{2}^{S,-}(2)⊕Γ_{3}^{S,+}Γ_{4}^{S,+}(4)⊕Γ_{3}^{S,-}Γ_{4}^{S,-}(4) $
M:(1,1,1)	M:(1,1,1)	$2M_{1}^{S}(4)⊕2M_{2}^{S}(4) $
N:(1/2,0,1/2)	N:(1/2,0,1/2)	$4N_{1}^{S,+}(2)⊕4N_{1}^{S,-}(2) $
P:(1/2,1/2,1/2)	P:(1/2,1/2,1/2)	$2P_{1}^{S}P_{4}^{S}(4)⊕2P_{2}^{S}P_{3}^{S}(4) $
X:(1/2,1/2,0)	X:(1/2,1/2,0)	$4X_{1}^{S}(4) $
Δ:(u,u,0)	Δ:(u,u,0)	$4Δ_{1}^{S}(2)⊕4Δ_{2}^{S}(2) $
Λ:(0,0,w)	Λ:(0,0,w)	$2Λ_{1}^{S}(2)⊕2Λ_{2}^{S}(2)⊕2Λ_{3}^{S}Λ_{4}^{S}(4) $
Q:(1/2,v,1/2)	Q:(1/2,v,1/2)	$8Q_{1}^{S}(2) $
Σ:(u,0,0)	Σ:(u,0,0)	$4Σ_{1}^{S}(2)⊕4Σ_{2}^{S}(2) $
W:(1/2,1/2,w)	W:(1/2,1/2,w)	$4W_{1}^{S}W_{2}^{S}(4) $
Y:(u,1-u,0)	Y:(u,1-u,0)	$4Y_{1}^{S}(2)⊕4Y_{2}^{S}(2) $
A:(u,u,w)	A:(u,u,w)	$8A_{1}^{S}(2) $
B:(u,0,w)	B:(u,0,w)	$8B_{1}^{S}(2) $
C:(u,v,0)	C:(u,v,0)	$4C_{1}^{S}(2)⊕4C_{2}^{S}(2) $
GP:(u,v,w)	GP:(u,v,w)	$8GP_{1}^{S}(2) $

8a

k-vector	k-vector-G↑	A↑G(4)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S,+}(2)⊕Γ_{2}^{S,-}(2) $
M:(1,1,1)	M:(1,1,1)	$M_{1}^{S}(4) $
N:(1/2,0,1/2)	N:(1/2,0,1/2)	$N_{1}^{S,+}(2)⊕N_{1}^{S,-}(2) $
P:(1/2,1/2,1/2)	P:(1/2,1/2,1/2)	$P_{1}^{S}P_{4}^{S}(4) $
X:(1/2,1/2,0)	X:(1/2,1/2,0)	$X_{1}^{S}(4) $
Δ:(u,u,0)	Δ:(u,u,0)	$Δ_{1}^{S}(2)⊕Δ_{2}^{S}(2) $
Λ:(0,0,w)	Λ:(0,0,w)	$Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2) $
Q:(1/2,v,1/2)	Q:(1/2,v,1/2)	$2Q_{1}^{S}(2) $
Σ:(u,0,0)	Σ:(u,0,0)	$Σ_{1}^{S}(2)⊕Σ_{2}^{S}(2) $
W:(1/2,1/2,w)	W:(1/2,1/2,w)	$W_{1}^{S}W_{2}^{S}(4) $
Y:(u,1-u,0)	Y:(u,1-u,0)	$Y_{1}^{S}(2)⊕Y_{2}^{S}(2) $
A:(u,u,w)	A:(u,u,w)	$2A_{1}^{S}(2) $
B:(u,0,w)	B:(u,0,w)	$2B_{1}^{S}(2) $
C:(u,v,0)	C:(u,v,0)	$C_{1}^{S}(2)⊕C_{2}^{S}(2) $
GP:(u,v,w)	GP:(u,v,w)	$2GP_{1}^{S}(2) $

16c

k-vector	k-vector-G↑	Ag↑G(8)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S,+}(2)⊕Γ_{2}^{S,+}(2)⊕Γ_{3}^{S,+}Γ_{4}^{S,+}(4) $
M:(1,1,1)	M:(1,1,1)	$M_{1}^{S}(4)⊕M_{2}^{S}(4) $
N:(1/2,0,1/2)	N:(1/2,0,1/2)	$3N_{1}^{S,+}(2)⊕N_{1}^{S,-}(2) $
P:(1/2,1/2,1/2)	P:(1/2,1/2,1/2)	$P_{1}^{S}P_{4}^{S}(4)⊕P_{2}^{S}P_{3}^{S}(4) $
X:(1/2,1/2,0)	X:(1/2,1/2,0)	$2X_{1}^{S}(4) $
Δ:(u,u,0)	Δ:(u,u,0)	$2Δ_{1}^{S}(2)⊕2Δ_{2}^{S}(2) $
Λ:(0,0,w)	Λ:(0,0,w)	$Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2)⊕Λ_{3}^{S}Λ_{4}^{S}(4) $
Q:(1/2,v,1/2)	Q:(1/2,v,1/2)	$4Q_{1}^{S}(2) $
Σ:(u,0,0)	Σ:(u,0,0)	$2Σ_{1}^{S}(2)⊕2Σ_{2}^{S}(2) $
W:(1/2,1/2,w)	W:(1/2,1/2,w)	$2W_{1}^{S}W_{2}^{S}(4) $
Y:(u,1-u,0)	Y:(u,1-u,0)	$2Y_{1}^{S}(2)⊕2Y_{2}^{S}(2) $
A:(u,u,w)	A:(u,u,w)	$4A_{1}^{S}(2) $
B:(u,0,w)	B:(u,0,w)	$4B_{1}^{S}(2) $
C:(u,v,0)	C:(u,v,0)	$2C_{1}^{S}(2)⊕2C_{2}^{S}(2) $
GP:(u,v,w)	GP:(u,v,w)	$4GP_{1}^{S}(2) $

16d

k-vector	k-vector-G↑	A↑G(8)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S,+}(2)⊕Γ_{1}^{S,-}(2)⊕Γ_{2}^{S,+}(2)⊕Γ_{2}^{S,-}(2) $
M:(1,1,1)	M:(1,1,1)	$2M_{1}^{S}(4) $
N:(1/2,0,1/2)	N:(1/2,0,1/2)	$2N_{1}^{S,+}(2)⊕2N_{1}^{S,-}(2) $
P:(1/2,1/2,1/2)	P:(1/2,1/2,1/2)	$P_{1}^{S}P_{4}^{S}(4)⊕P_{2}^{S}P_{3}^{S}(4) $
X:(1/2,1/2,0)	X:(1/2,1/2,0)	$2X_{1}^{S}(4) $
Δ:(u,u,0)	Δ:(u,u,0)	$2Δ_{1}^{S}(2)⊕2Δ_{2}^{S}(2) $
Λ:(0,0,w)	Λ:(0,0,w)	$2Λ_{1}^{S}(2)⊕2Λ_{2}^{S}(2) $
Q:(1/2,v,1/2)	Q:(1/2,v,1/2)	$4Q_{1}^{S}(2) $
Σ:(u,0,0)	Σ:(u,0,0)	$2Σ_{1}^{S}(2)⊕2Σ_{2}^{S}(2) $
W:(1/2,1/2,w)	W:(1/2,1/2,w)	$2W_{1}^{S}W_{2}^{S}(4) $
Y:(u,1-u,0)	Y:(u,1-u,0)	$2Y_{1}^{S}(2)⊕2Y_{2}^{S}(2) $
A:(u,u,w)	A:(u,u,w)	$4A_{1}^{S}(2) $
B:(u,0,w)	B:(u,0,w)	$4B_{1}^{S}(2) $
C:(u,v,0)	C:(u,v,0)	$2C_{1}^{S}(2)⊕2C_{2}^{S}(2) $
GP:(u,v,w)	GP:(u,v,w)	$4GP_{1}^{S}(2) $