Information of spin space group $P_c{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{4}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$ ( nSSG:75.75.2.1 )

General Positions

The continuously infinite group ^∞1 is omitted.

Coordinates	Seitz symbol
a, b, c | x, y, z	{ 1 ‖ 1 | 0 }
-b, a, c | x, y, z	{ 1 ‖ 4+001 | 0 }
b, -a, c | x, y, z	{ 1 ‖ 4-001 | 0 }
-a, -b, c | x, y, z	{ 1 ‖ 2001 | 0 }
a, b, c+1/2 | -x, -y, -z	{ -1 ‖ 1 | 0 0 1/2 }
-b, a, c+1/2 | -x, -y, -z	{ -1 ‖ 4+001 | 0 0 1/2 }
b, -a, c+1/2 | -x, -y, -z	{ -1 ‖ 4-001 | 0 0 1/2 }
-a, -b, c+1/2 | -x, -y, -z	{ -1 ‖ 2001 | 0 0 1/2 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
-b, a, c | -x, y, z	{ m ‖ 4+001 | 0 }
b, -a, c | -x, y, z	{ m ‖ 4-001 | 0 }
-a, -b, c | -x, y, z	{ m ‖ 2001 | 0 }
a, b, c+1/2 | x, -y, -z	{ 2 ‖ 1 | 0 0 1/2 }
-b, a, c+1/2 | x, -y, -z	{ 2 ‖ 4+001 | 0 0 1/2 }
b, -a, c+1/2 | x, -y, -z	{ 2 ‖ 4-001 | 0 0 1/2 }
-a, -b, c+1/2 | x, -y, -z	{ 2 ‖ 2001 | 0 0 1/2 }

Wyckoff Positions

WP	Site symmetry	Representative
2a	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{4}..{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(0,0,c | 0,0,z)
2b	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{4}..{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(1/2,1/2,c | 0,0,z)
4c	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2}..{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(0,1/2,c | 0,0,z)
8d	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{1}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(a,b,c | 0,0,z)

Spin Brillouin Zone & k-little co-groups

Wavevector-k	Little co-group
A:(1/2,1/2,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{4}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }/\mathrm{mm}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }/\mathrm{mm}}\text{1}$
Γ:(0,0,0)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{4}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }/\mathrm{mm}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }/\mathrm{mm}}\text{1}$
M:(1/2,1/2,0)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{4}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }/\mathrm{mm}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }/\mathrm{mm}}\text{1}$
Z:(0,0,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{4}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }/\mathrm{mm}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }/\mathrm{mm}}\text{1}$
R:(0,1/2,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }/\mathrm{mm}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }/\mathrm{mm}}\text{1}$
X:(0,1/2,0)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }/\mathrm{mm}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }/\mathrm{mm}}\text{1}$
Λ:(0,0,w)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{4}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }2}\text{1}$
V:(1/2,1/2,w)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{4}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }2}\text{1}$
D:(u,v,0)	${\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }2}\text{1}$
Δ:(0,v,0)	${\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }2}\text{1}$
E:(u,v,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }2}\text{1}$
S:(u,u,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }2}\text{1}$
Σ:(u,u,0)	${\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }2}\text{1}$
T:(u,1/2,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }2}\text{1}$
U:(0,v,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }2}\text{1}$
W:(0,1/2,w)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }2}\text{1}$
Y:(u,1/2,0)	${\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }2}\text{1}$
B:(0,v,w)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{1}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }2}\text{1}$
C:(u,u,w)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{1}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }2}\text{1}$
F:(u,1/2,w)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{1}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }2}\text{1}$
GP:(u,v,w)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{1}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }2}\text{1}$

Spin Brillouin Zone

4/mP

Band Representations (Magnon)

2a

k-vector	A↑G(2)
A:(1/2,1/2,1/2)	$A_1^S(2)$
Γ:(0,0,0)	${\Gamma }_1^S(2)$
M:(1/2,1/2,0)	$M_1^S(2)$
Z:(0,0,1/2)	$Z_1^S(2)$
R:(0,1/2,1/2)	$R_1^S(2)$
X:(0,1/2,0)	$X_1^S(2)$
Λ:(0,0,w)	${\Lambda }_1^S(2)$
V:(1/2,1/2,w)	$V_1^S(2)$
D:(u,v,0)	$D_1^S(2)$
Δ:(0,v,0)	${\Delta }_1^S(2)$
E:(u,v,1/2)	$E_1^S(2)$
S:(u,u,1/2)	$S_1^S(2)$
Σ:(u,u,0)	${\Sigma }_1^S(2)$
T:(u,1/2,1/2)	$T_1^S(2)$
U:(0,v,1/2)	$U_1^S(2)$
W:(0,1/2,w)	$W_1^S(2)$
Y:(u,1/2,0)	$Y_1^S(2)$
B:(0,v,w)	$B_1^S(2)$
C:(u,u,w)	$C_1^S(2)$
F:(u,1/2,w)	$F_1^S(2)$
GP:(u,v,w)	$G{P}_1^S(2)$

2b

k-vector	A↑G(2)
A:(1/2,1/2,1/2)	$A_2^S(2)$
Γ:(0,0,0)	${\Gamma }_1^S(2)$
M:(1/2,1/2,0)	$M_2^S(2)$
Z:(0,0,1/2)	$Z_1^S(2)$
R:(0,1/2,1/2)	$R_2^S(2)$
X:(0,1/2,0)	$X_2^S(2)$
Λ:(0,0,w)	${\Lambda }_1^S(2)$
V:(1/2,1/2,w)	$V_2^S(2)$
D:(u,v,0)	$D_1^S(2)$
Δ:(0,v,0)	${\Delta }_1^S(2)$
E:(u,v,1/2)	$E_1^S(2)$
S:(u,u,1/2)	$S_1^S(2)$
Σ:(u,u,0)	${\Sigma }_1^S(2)$
T:(u,1/2,1/2)	$T_1^S(2)$
U:(0,v,1/2)	$U_1^S(2)$
W:(0,1/2,w)	$W_2^S(2)$
Y:(u,1/2,0)	$Y_1^S(2)$
B:(0,v,w)	$B_1^S(2)$
C:(u,u,w)	$C_1^S(2)$
F:(u,1/2,w)	$F_1^S(2)$
GP:(u,v,w)	$G{P}_1^S(2)$

4c

k-vector	A↑G(4)
A:(1/2,1/2,1/2)	$A_3^S{A}_4^S(4)$
Γ:(0,0,0)	${\Gamma }_1^S(2)\oplus {\Gamma }_2^S(2)$
M:(1/2,1/2,0)	$M_3^S{M}_4^S(4)$
Z:(0,0,1/2)	$Z_1^S(2)\oplus Z_2^S(2)$
R:(0,1/2,1/2)	$R_1^S(2)\oplus R_2^S(2)$
X:(0,1/2,0)	$X_1^S(2)\oplus X_2^S(2)$
Λ:(0,0,w)	${\Lambda }_1^S(2)\oplus {\Lambda }_2^S(2)$
V:(1/2,1/2,w)	$V_3^S(2)\oplus V_4^S(2)$
D:(u,v,0)	$2D_1^S(2)$
Δ:(0,v,0)	$2{\Delta }_1^S(2)$
E:(u,v,1/2)	$2E_1^S(2)$
S:(u,u,1/2)	$2S_1^S(2)$
Σ:(u,u,0)	$2{\Sigma }_1^S(2)$
T:(u,1/2,1/2)	$2T_1^S(2)$
U:(0,v,1/2)	$2U_1^S(2)$
W:(0,1/2,w)	$W_1^S(2)\oplus W_2^S(2)$
Y:(u,1/2,0)	$2Y_1^S(2)$
B:(0,v,w)	$2B_1^S(2)$
C:(u,u,w)	$2C_1^S(2)$
F:(u,1/2,w)	$2F_1^S(2)$
GP:(u,v,w)	$2G{P}_1^S(2)$

8d

k-vector	A↑G(8)
A:(1/2,1/2,1/2)	$A_1^S(2)\oplus A_2^S(2)\oplus A_3^S{A}_4^S(4)$
Γ:(0,0,0)	${\Gamma }_1^S(2)\oplus {\Gamma }_2^S(2)\oplus {\Gamma }_3^S{\Gamma }_4^S(4)$
M:(1/2,1/2,0)	$M_1^S(2)\oplus M_2^S(2)\oplus M_3^S{M}_4^S(4)$
Z:(0,0,1/2)	$Z_1^S(2)\oplus Z_2^S(2)\oplus Z_3^S{Z}_4^S(4)$
R:(0,1/2,1/2)	$2R_1^S(2)\oplus 2R_2^S(2)$
X:(0,1/2,0)	$2X_1^S(2)\oplus 2X_2^S(2)$
Λ:(0,0,w)	${\Lambda }_1^S(2)\oplus {\Lambda }_2^S(2)\oplus {\Lambda }_3^S(2)\oplus {\Lambda }_4^S(2)$
V:(1/2,1/2,w)	$V_1^S(2)\oplus V_2^S(2)\oplus V_3^S(2)\oplus V_4^S(2)$
D:(u,v,0)	$4D_1^S(2)$
Δ:(0,v,0)	$4{\Delta }_1^S(2)$
E:(u,v,1/2)	$4E_1^S(2)$
S:(u,u,1/2)	$4S_1^S(2)$
Σ:(u,u,0)	$4{\Sigma }_1^S(2)$
T:(u,1/2,1/2)	$4T_1^S(2)$
U:(0,v,1/2)	$4U_1^S(2)$
W:(0,1/2,w)	$2W_1^S(2)\oplus 2W_2^S(2)$
Y:(u,1/2,0)	$4Y_1^S(2)$
B:(0,v,w)	$4B_1^S(2)$
C:(u,u,w)	$4C_1^S(2)$
F:(u,1/2,w)	$4F_1^S(2)$
GP:(u,v,w)	$4G{P}_1^S(2)$