Information of spin space group $I{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{-}\text{4}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{d}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$ ( nSSG:122.122.1.1 )

General Positions

The continuously infinite group ^∞1 is omitted.

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

Wyckoff Positions

WP	Site symmetry	Representative
4a	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{-4}..{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(0,0,0 |0,0,z)
4b	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{-4}..{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(0,0,1/2 |0,0,z)
8c	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2}..{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(0,0,c |0,0,z)
8d	$.{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2}.{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(a,1/4,1/8 |0,0,z)
16e	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{1}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(a,b,c |0,0,z)

Spin Brillouin Zone & k-little co-groups

Wavevector-k	Little co-group
Γ:(0,0,0)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{-4}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$
M:(1,1,1)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{-4}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$
P:(1/2,1/2,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{-4}{\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
X:(1/2,1/2,0)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$
N:(1/2,0,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$
Λ:(0,0,w)	${\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{-4}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
Δ:(u,u,0)	${\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
Q:(1/2,v,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
Σ:(u,0,0)	${\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
W:(1/2,1/2,w)	${\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
Y:(u,1-u,0)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
A:(u,u,w)	${\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
B:(u,0,w)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
C:(u,v,0)	${\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
GP:(u,v,w)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{1}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$

Spin Brillouin Zone

Band Representations (Magnon)

4a

k-vector	A↑G(2)
Γ:(0,0,0)	${\Gamma }_1^S(1)\oplus {\Gamma }_4^S(1)$
M:(1,1,1)	$M_5^S(2)$
P:(1/2,1/2,1/2)	$P_1^S(2)$
X:(1/2,1/2,0)	$X_1^S(2)$
N:(1/2,0,1/2)	$N_1^S(1)\oplus N_2^S(1)$
Λ:(0,0,w)	${\Lambda }_1^S(1)\oplus {\Lambda }_2^S(1)$
Δ:(u,u,0)	${\Delta }_1^S(1)\oplus {\Delta }_2^S(1)$
Q:(1/2,v,1/2)	$Q_1^S(1)\oplus Q_2^S(1)$
Σ:(u,0,0)	${\Sigma }_1^S(1)\oplus {\Sigma }_2^S(1)$
W:(1/2,1/2,w)	$W_1^S(2)$
Y:(u,1-u,0)	$Y_1^S{Y}_2^S(2)$
A:(u,u,w)	$A_1^S(1)\oplus A_2^S(1)$
B:(u,0,w)	$2B_1^S(1)$
C:(u,v,0)	$2C_1^S(1)$
GP:(u,v,w)	$2G{P}_1^S(1)$

4b

k-vector	A↑G(2)
Γ:(0,0,0)	${\Gamma }_1^S(1)\oplus {\Gamma }_4^S(1)$
M:(1,1,1)	$M_5^S(2)$
P:(1/2,1/2,1/2)	$P_2^S(2)$
X:(1/2,1/2,0)	$X_1^S(2)$
N:(1/2,0,1/2)	$N_1^S(1)\oplus N_2^S(1)$
Λ:(0,0,w)	${\Lambda }_1^S(1)\oplus {\Lambda }_2^S(1)$
Δ:(u,u,0)	${\Delta }_1^S(1)\oplus {\Delta }_2^S(1)$
Q:(1/2,v,1/2)	$Q_1^S(1)\oplus Q_2^S(1)$
Σ:(u,0,0)	${\Sigma }_1^S(1)\oplus {\Sigma }_2^S(1)$
W:(1/2,1/2,w)	$W_1^S(2)$
Y:(u,1-u,0)	$Y_1^S{Y}_2^S(2)$
A:(u,u,w)	$A_1^S(1)\oplus A_2^S(1)$
B:(u,0,w)	$2B_1^S(1)$
C:(u,v,0)	$2C_1^S(1)$
GP:(u,v,w)	$2G{P}_1^S(1)$

8c

k-vector	A↑G(4)
Γ:(0,0,0)	${\Gamma }_1^S(1)\oplus {\Gamma }_2^S(1)\oplus {\Gamma }_3^S(1)\oplus {\Gamma }_4^S(1)$
M:(1,1,1)	$2M_5^S(2)$
P:(1/2,1/2,1/2)	$P_1^S(2)\oplus P_2^S(2)$
X:(1/2,1/2,0)	$2X_1^S(2)$
N:(1/2,0,1/2)	$2N_1^S(1)\oplus 2N_2^S(1)$
Λ:(0,0,w)	$2{\Lambda }_1^S(1)\oplus 2{\Lambda }_2^S(1)$
Δ:(u,u,0)	$2{\Delta }_1^S(1)\oplus 2{\Delta }_2^S(1)$
Q:(1/2,v,1/2)	$2Q_1^S(1)\oplus 2Q_2^S(1)$
Σ:(u,0,0)	$2{\Sigma }_1^S(1)\oplus 2{\Sigma }_2^S(1)$
W:(1/2,1/2,w)	$2W_1^S(2)$
Y:(u,1-u,0)	$2Y_1^S{Y}_2^S(2)$
A:(u,u,w)	$2A_1^S(1)\oplus 2A_2^S(1)$
B:(u,0,w)	$4B_1^S(1)$
C:(u,v,0)	$4C_1^S(1)$
GP:(u,v,w)	$4G{P}_1^S(1)$

8d

k-vector	A↑G(4)
Γ:(0,0,0)	${\Gamma }_1^S(1)\oplus {\Gamma }_2^S(1)\oplus {\Gamma }_5^S(2)$
M:(1,1,1)	$M_1^S{M}_2^S(2)\oplus M_5^S(2)$
P:(1/2,1/2,1/2)	$P_1^S(2)\oplus P_2^S(2)$
X:(1/2,1/2,0)	$2X_1^S(2)$
N:(1/2,0,1/2)	$2N_1^S(1)\oplus 2N_2^S(1)$
Λ:(0,0,w)	${\Lambda }_1^S(1)\oplus {\Lambda }_2^S(1)\oplus {\Lambda }_3^S{\Lambda }_4^S(2)$
Δ:(u,u,0)	$2{\Delta }_1^S(1)\oplus 2{\Delta }_2^S(1)$
Q:(1/2,v,1/2)	$2Q_1^S(1)\oplus 2Q_2^S(1)$
Σ:(u,0,0)	$3{\Sigma }_1^S(1)\oplus {\Sigma }_2^S(1)$
W:(1/2,1/2,w)	$2W_1^S(2)$
Y:(u,1-u,0)	$2Y_1^S{Y}_2^S(2)$
A:(u,u,w)	$2A_1^S(1)\oplus 2A_2^S(1)$
B:(u,0,w)	$4B_1^S(1)$
C:(u,v,0)	$4C_1^S(1)$
GP:(u,v,w)	$4G{P}_1^S(1)$

16e

k-vector	A↑G(8)
Γ:(0,0,0)	${\Gamma }_1^S(1)\oplus {\Gamma }_2^S(1)\oplus {\Gamma }_3^S(1)\oplus {\Gamma }_4^S(1)\oplus 2{\Gamma }_5^S(2)$
M:(1,1,1)	$M_1^S{M}_2^S(2)\oplus M_3^S{M}_4^S(2)\oplus 2M_5^S(2)$
P:(1/2,1/2,1/2)	$2P_1^S(2)\oplus 2P_2^S(2)$
X:(1/2,1/2,0)	$4X_1^S(2)$
N:(1/2,0,1/2)	$4N_1^S(1)\oplus 4N_2^S(1)$
Λ:(0,0,w)	$2{\Lambda }_1^S(1)\oplus 2{\Lambda }_2^S(1)\oplus 2{\Lambda }_3^S{\Lambda }_4^S(2)$
Δ:(u,u,0)	$4{\Delta }_1^S(1)\oplus 4{\Delta }_2^S(1)$
Q:(1/2,v,1/2)	$4Q_1^S(1)\oplus 4Q_2^S(1)$
Σ:(u,0,0)	$4{\Sigma }_1^S(1)\oplus 4{\Sigma }_2^S(1)$
W:(1/2,1/2,w)	$4W_1^S(2)$
Y:(u,1-u,0)	$4Y_1^S{Y}_2^S(2)$
A:(u,u,w)	$4A_1^S(1)\oplus 4A_2^S(1)$
B:(u,0,w)	$8B_1^S(1)$
C:(u,v,0)	$8C_1^S(1)$
GP:(u,v,w)	$8G{P}_1^S(1)$