Information of spin space group $I\ce{^{\text{-}1}{a}}\ce{^{\text{-}1}{\text{-}3}}\ce{^{\infty m}{1}}$ ( nSSG:199.206.1.1 )

General Positions

The continuously infinite group ^∞1 is omitted.

Coordinates	Seitz symbol
a, b, c | x, y, z	{ 1 ‖ 1 | 0 }
a, -b, -c+1/2 | x, y, z	{ 1 ‖ 2100 | 0 0 1/2 }
-a+1/2, b, -c | x, y, z	{ 1 ‖ 2010 | 1/2 0 0 }
-a, -b+1/2, c | x, y, z	{ 1 ‖ 2001 | 0 1/2 0 }
c, a, b | x, y, z	{ 1 ‖ 3+111 | 0 }
b, c, a | x, y, z	{ 1 ‖ 3-111 | 0 }
-b+1/2, c, -a | x, y, z	{ 1 ‖ 3-1-1-1 | 1/2 0 0 }
-c, -a+1/2, b | x, y, z	{ 1 ‖ 3+1-1-1 | 0 1/2 0 }
-b, -c+1/2, a | x, y, z	{ 1 ‖ 3--11-1 | 0 1/2 0 }
c, -a, -b+1/2 | x, y, z	{ 1 ‖ 3+-11-1 | 0 0 1/2 }
b, -c, -a+1/2 | x, y, z	{ 1 ‖ 3--1-11 | 0 0 1/2 }
-c+1/2, a, -b | x, y, z	{ 1 ‖ 3+-1-11 | 1/2 0 0 }
-a, -b, -c | -x, -y, -z	{ -1 ‖ -1 | 0 }
-a, b, c+1/2 | -x, -y, -z	{ -1 ‖ m100 | 0 0 1/2 }
a+1/2, -b, c | -x, -y, -z	{ -1 ‖ m010 | 1/2 0 0 }
a, b+1/2, -c | -x, -y, -z	{ -1 ‖ m001 | 0 1/2 0 }
-c, -a, -b | -x, -y, -z	{ -1 ‖ -3+111 | 0 }
-b, -c, -a | -x, -y, -z	{ -1 ‖ -3-111 | 0 }
b+1/2, -c, a | -x, -y, -z	{ -1 ‖ -3-1-1-1 | 1/2 0 0 }
c, a+1/2, -b | -x, -y, -z	{ -1 ‖ -3+1-1-1 | 0 1/2 0 }
b, c+1/2, -a | -x, -y, -z	{ -1 ‖ -3--11-1 | 0 1/2 0 }
-c, a, b+1/2 | -x, -y, -z	{ -1 ‖ -3+-11-1 | 0 0 1/2 }
-b, c, a+1/2 | -x, -y, -z	{ -1 ‖ -3--1-11 | 0 0 1/2 }
c+1/2, -a, b | -x, -y, -z	{ -1 ‖ -3+-1-11 | 1/2 0 0 }
a+1/2, b+1/2, c+1/2 | x, y, z	{ 1 ‖ 1 | 1/2 1/2 1/2 }
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 }
c+1/2, a+1/2, b+1/2 | x, y, z	{ 1 ‖ 3+111 | 1/2 1/2 1/2 }
b+1/2, c+1/2, a+1/2 | x, y, z	{ 1 ‖ 3-111 | 1/2 1/2 1/2 }
-b, c+1/2, -a+1/2 | x, y, z	{ 1 ‖ 3-1-1-1 | 0 1/2 1/2 }
-c+1/2, -a, b+1/2 | x, y, z	{ 1 ‖ 3+1-1-1 | 1/2 0 1/2 }
-b+1/2, -c, a+1/2 | x, y, z	{ 1 ‖ 3--11-1 | 1/2 0 1/2 }
c+1/2, -a+1/2, -b | x, y, z	{ 1 ‖ 3+-11-1 | 1/2 1/2 0 }
b+1/2, -c+1/2, -a | x, y, z	{ 1 ‖ 3--1-11 | 1/2 1/2 0 }
-c, a+1/2, -b+1/2 | x, y, z	{ 1 ‖ 3+-1-11 | 0 1/2 1/2 }
-a+1/2, -b+1/2, -c+1/2 | -x, -y, -z	{ -1 ‖ -1 | 1/2 1/2 1/2 }
-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 }
-c+1/2, -a+1/2, -b+1/2 | -x, -y, -z	{ -1 ‖ -3+111 | 1/2 1/2 1/2 }
-b+1/2, -c+1/2, -a+1/2 | -x, -y, -z	{ -1 ‖ -3-111 | 1/2 1/2 1/2 }
b, -c+1/2, a+1/2 | -x, -y, -z	{ -1 ‖ -3-1-1-1 | 0 1/2 1/2 }
c+1/2, a, -b+1/2 | -x, -y, -z	{ -1 ‖ -3+1-1-1 | 1/2 0 1/2 }
b+1/2, c, -a+1/2 | -x, -y, -z	{ -1 ‖ -3--11-1 | 1/2 0 1/2 }
-c+1/2, a+1/2, b | -x, -y, -z	{ -1 ‖ -3+-11-1 | 1/2 1/2 0 }
-b+1/2, c+1/2, a | -x, -y, -z	{ -1 ‖ -3--1-11 | 1/2 1/2 0 }
c, -a+1/2, b+1/2 | -x, -y, -z	{ -1 ‖ -3+-1-11 | 0 1/2 1/2 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
a, -b, -c+1/2 | -x, y, z	{ m ‖ 2100 | 0 0 1/2 }
-a+1/2, b, -c | -x, y, z	{ m ‖ 2010 | 1/2 0 0 }
-a, -b+1/2, c | -x, y, z	{ m ‖ 2001 | 0 1/2 0 }
c, a, b | -x, y, z	{ m ‖ 3+111 | 0 }
b, c, a | -x, y, z	{ m ‖ 3-111 | 0 }
-b+1/2, c, -a | -x, y, z	{ m ‖ 3-1-1-1 | 1/2 0 0 }
-c, -a+1/2, b | -x, y, z	{ m ‖ 3+1-1-1 | 0 1/2 0 }
-b, -c+1/2, a | -x, y, z	{ m ‖ 3--11-1 | 0 1/2 0 }
c, -a, -b+1/2 | -x, y, z	{ m ‖ 3+-11-1 | 0 0 1/2 }
b, -c, -a+1/2 | -x, y, z	{ m ‖ 3--1-11 | 0 0 1/2 }
-c+1/2, a, -b | -x, y, z	{ m ‖ 3+-1-11 | 1/2 0 0 }
-a, -b, -c | x, -y, -z	{ 2 ‖ -1 | 0 }
-a, b, c+1/2 | x, -y, -z	{ 2 ‖ m100 | 0 0 1/2 }
a+1/2, -b, c | x, -y, -z	{ 2 ‖ m010 | 1/2 0 0 }
a, b+1/2, -c | x, -y, -z	{ 2 ‖ m001 | 0 1/2 0 }
-c, -a, -b | x, -y, -z	{ 2 ‖ -3+111 | 0 }
-b, -c, -a | x, -y, -z	{ 2 ‖ -3-111 | 0 }
b+1/2, -c, a | x, -y, -z	{ 2 ‖ -3-1-1-1 | 1/2 0 0 }
c, a+1/2, -b | x, -y, -z	{ 2 ‖ -3+1-1-1 | 0 1/2 0 }
b, c+1/2, -a | x, -y, -z	{ 2 ‖ -3--11-1 | 0 1/2 0 }
-c, a, b+1/2 | x, -y, -z	{ 2 ‖ -3+-11-1 | 0 0 1/2 }
-b, c, a+1/2 | x, -y, -z	{ 2 ‖ -3--1-11 | 0 0 1/2 }
c+1/2, -a, b | x, -y, -z	{ 2 ‖ -3+-1-11 | 1/2 0 0 }
a+1/2, b+1/2, c+1/2 | -x, y, z	{ m ‖ 1 | 1/2 1/2 1/2 }
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 }
c+1/2, a+1/2, b+1/2 | -x, y, z	{ m ‖ 3+111 | 1/2 1/2 1/2 }
b+1/2, c+1/2, a+1/2 | -x, y, z	{ m ‖ 3-111 | 1/2 1/2 1/2 }
-b, c+1/2, -a+1/2 | -x, y, z	{ m ‖ 3-1-1-1 | 0 1/2 1/2 }
-c+1/2, -a, b+1/2 | -x, y, z	{ m ‖ 3+1-1-1 | 1/2 0 1/2 }
-b+1/2, -c, a+1/2 | -x, y, z	{ m ‖ 3--11-1 | 1/2 0 1/2 }
c+1/2, -a+1/2, -b | -x, y, z	{ m ‖ 3+-11-1 | 1/2 1/2 0 }
b+1/2, -c+1/2, -a | -x, y, z	{ m ‖ 3--1-11 | 1/2 1/2 0 }
-c, a+1/2, -b+1/2 | -x, y, z	{ m ‖ 3+-1-11 | 0 1/2 1/2 }
-a+1/2, -b+1/2, -c+1/2 | x, -y, -z	{ 2 ‖ -1 | 1/2 1/2 1/2 }
-a+1/2, b+1/2, c | x, -y, -z	{ 2 ‖ m100 | 1/2 1/2 0 }
a, -b+1/2, c+1/2 | x, -y, -z	{ 2 ‖ m010 | 0 1/2 1/2 }
a+1/2, b, -c+1/2 | x, -y, -z	{ 2 ‖ m001 | 1/2 0 1/2 }
-c+1/2, -a+1/2, -b+1/2 | x, -y, -z	{ 2 ‖ -3+111 | 1/2 1/2 1/2 }
-b+1/2, -c+1/2, -a+1/2 | x, -y, -z	{ 2 ‖ -3-111 | 1/2 1/2 1/2 }
b, -c+1/2, a+1/2 | x, -y, -z	{ 2 ‖ -3-1-1-1 | 0 1/2 1/2 }
c+1/2, a, -b+1/2 | x, -y, -z	{ 2 ‖ -3+1-1-1 | 1/2 0 1/2 }
b+1/2, c, -a+1/2 | x, -y, -z	{ 2 ‖ -3--11-1 | 1/2 0 1/2 }
-c+1/2, a+1/2, b | x, -y, -z	{ 2 ‖ -3+-11-1 | 1/2 1/2 0 }
-b+1/2, c+1/2, a | x, -y, -z	{ 2 ‖ -3--1-11 | 1/2 1/2 0 }
c, -a+1/2, b+1/2 | x, -y, -z	{ 2 ‖ -3+-1-11 | 0 1/2 1/2 }

Wyckoff Positions

WP	Site symmetry	Representative
8a	$.\ce{^{-1}{-3}}.\ce{^{\infty m}{1}} $	(0,0,0 | 0,0,0)
8b	$.\ce{^{-1}{-3}}.\ce{^{\infty m}{1}} $	(1/4,1/4,1/4 | 0,0,0)
16c	$.\ce{^{1}{3}}.\ce{^{\infty m}{1}} $	(a,a,a | 0,0,z)
24d	$\ce{^{1}{2}}..\ce{^{\infty m}{1}} $	(a,0,1/4 | 0,0,z)
48e	$\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}{-3}}\ce{^{\infty m}{1}} $
H:(1,1,1)	$\ce{^{-1}{m}}\ce{^{-1}{-3}}\ce{^{\infty m}{1}} $
P:(1/2,1/2,1/2)	$\ce{^{-1}{m}}\ce{^{-1}{-3}}\ce{^{\infty}{1}} $
N:(1/2,1/2,0)	$\ce{^{1}{2/}}\ce{^{2}{m}}\ce{^{\infty m}{1}} $
Δ:(0,v,0)	$\ce{^{2}{m}}\ce{^{-1}{m}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
Λ:(u,u,u)	$\ce{^{-1}{-3}}\ce{^{\infty}{1}} $
A:(u,v,0)	$\ce{^{m}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
D:(1/2,1/2,w)	$\ce{^{1}{2/}}\ce{^{-1}{m}}\ce{^{\infty}{1}} $
G:(1+u,1-u,1)	$\ce{^{m}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
Σ:(u,u,0)	$\ce{^{m}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
C:(u,u,w)	$\ce{^{-1}{-1}}\ce{^{\infty}{1}} $
GP:(u,v,w)	$\ce{^{-1}{-1}}\ce{^{\infty}{1}} $

Spin Brillouin Zone

Band Representations (Magnon)

24d

k-vector	k-vector-G↑	A↑G(12)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{2}^{S}Γ_{3}^{S}(4)⊕Γ_{4}^{S}(6) $
H:(1,1,1)	H:(1,1,1)	$2H_{4}^{S}(6) $
N:(1/2,1/2,0)	N:(1/2,1/2,0)	$3N_{1}^{S}(2)⊕3N_{2}^{S}(2) $
P:(1/2,1/2,1/2)	P:(1/2,1/2,1/2)	$P_{1}^{S}(4)⊕P_{2}^{S}(4)⊕P_{3}^{S}(4) $
D:(1/2,1/2,w)	D:(1/2,1/2,w)	$3D_{1}^{S}(2)⊕3D_{2}^{S}(2) $
Δ:(0,v,0)	Δ:(0,v,0)	$4Δ_{1}^{S}(2)⊕2Δ_{2}^{S}(2) $
G:(1+u,1-u,1)	G:(1+u,1-u,1)	$6G_{1}^{S}(2) $
Λ:(u,u,u)	Λ:(u,u,u)	$2Λ_{1}^{S}(2)⊕2Λ_{2}^{S}(2)⊕2Λ_{3}^{S}(2) $
Σ:(u,u,0)	Σ:(u,u,0)	$6Σ_{1}^{S}(2) $
A:(u,v,0)	A:(u,v,0)	$6A_{1}^{S}(2) $
C:(u,u,w)	C:(u,u,w)	$6C_{1}^{S}(2) $
GP:(u,v,w)	GP:(u,v,w)	$6GP_{1}^{S}(2) $

48e

k-vector	k-vector-G↑	A↑G(24)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{2}^{S}Γ_{3}^{S}(4)⊕3Γ_{4}^{S}(6) $
H:(1,1,1)	H:(1,1,1)	$H_{1}^{S}(2)⊕H_{2}^{S}H_{3}^{S}(4)⊕3H_{4}^{S}(6) $
N:(1/2,1/2,0)	N:(1/2,1/2,0)	$6N_{1}^{S}(2)⊕6N_{2}^{S}(2) $
P:(1/2,1/2,1/2)	P:(1/2,1/2,1/2)	$2P_{1}^{S}(4)⊕2P_{2}^{S}(4)⊕2P_{3}^{S}(4) $
D:(1/2,1/2,w)	D:(1/2,1/2,w)	$6D_{1}^{S}(2)⊕6D_{2}^{S}(2) $
Δ:(0,v,0)	Δ:(0,v,0)	$6Δ_{1}^{S}(2)⊕6Δ_{2}^{S}(2) $
G:(1+u,1-u,1)	G:(1+u,1-u,1)	$12G_{1}^{S}(2) $
Λ:(u,u,u)	Λ:(u,u,u)	$4Λ_{1}^{S}(2)⊕4Λ_{2}^{S}(2)⊕4Λ_{3}^{S}(2) $
Σ:(u,u,0)	Σ:(u,u,0)	$12Σ_{1}^{S}(2) $
A:(u,v,0)	A:(u,v,0)	$12A_{1}^{S}(2) $
C:(u,u,w)	C:(u,u,w)	$12C_{1}^{S}(2) $
GP:(u,v,w)	GP:(u,v,w)	$12GP_{1}^{S}(2) $

16c

k-vector	k-vector-G↑	A1↑G(8)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{4}^{S}(6) $
H:(1,1,1)	H:(1,1,1)	$H_{1}^{S}(2)⊕H_{4}^{S}(6) $
N:(1/2,1/2,0)	N:(1/2,1/2,0)	$2N_{1}^{S}(2)⊕2N_{2}^{S}(2) $
P:(1/2,1/2,1/2)	P:(1/2,1/2,1/2)	$P_{1}^{S}(4)⊕P_{3}^{S}(4) $
D:(1/2,1/2,w)	D:(1/2,1/2,w)	$2D_{1}^{S}(2)⊕2D_{2}^{S}(2) $
Δ:(0,v,0)	Δ:(0,v,0)	$2Δ_{1}^{S}(2)⊕2Δ_{2}^{S}(2) $
G:(1+u,1-u,1)	G:(1+u,1-u,1)	$4G_{1}^{S}(2) $
Λ:(u,u,u)	Λ:(u,u,u)	$2Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2)⊕Λ_{3}^{S}(2) $
Σ:(u,u,0)	Σ:(u,u,0)	$4Σ_{1}^{S}(2) $
A:(u,v,0)	A:(u,v,0)	$4A_{1}^{S}(2) $
C:(u,u,w)	C:(u,u,w)	$4C_{1}^{S}(2) $
GP:(u,v,w)	GP:(u,v,w)	$4GP_{1}^{S}(2) $