Information of spin space group $P{\phantom{A}}_{\phantom{}}^{\phantom{\text{-}1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\text{-}1}{6}_3{\phantom{A}}_{\phantom{}}^{\phantom{\text{-}1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\text{-}1}\text{c}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$ ( nSSG:157.185.1.1 )

General Positions

The continuously infinite group ^∞1 is omitted.

Coordinates	Seitz symbol
a, b, c | x, y, z	{ 1 ‖ 1 | 0 }
-b, a-b, c | x, y, z	{ 1 ‖ 3+001 | 0 }
-a+b, -a, c | x, y, z	{ 1 ‖ 3-001 | 0 }
-a, -a+b, c | x, y, z	{ 1 ‖ m210 | 0 }
a-b, -b, c | x, y, z	{ 1 ‖ m120 | 0 }
b, a, c | x, y, z	{ 1 ‖ m1-10 | 0 }
a-b, a, c+1/2 | -x, -y, -z	{ -1 ‖ 6+001 | 0 0 1/2 }
-a, -b, c+1/2 | -x, -y, -z	{ -1 ‖ 2001 | 0 0 1/2 }
b, -a+b, c+1/2 | -x, -y, -z	{ -1 ‖ 6-001 | 0 0 1/2 }
-a+b, b, c+1/2 | -x, -y, -z	{ -1 ‖ m100 | 0 0 1/2 }
-b, -a, c+1/2 | -x, -y, -z	{ -1 ‖ m110 | 0 0 1/2 }
a, a-b, c+1/2 | -x, -y, -z	{ -1 ‖ m010 | 0 0 1/2 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
-b, a-b, c | -x, y, z	{ m ‖ 3+001 | 0 }
-a+b, -a, c | -x, y, z	{ m ‖ 3-001 | 0 }
-a, -a+b, c | -x, y, z	{ m ‖ m210 | 0 }
a-b, -b, c | -x, y, z	{ m ‖ m120 | 0 }
b, a, c | -x, y, z	{ m ‖ m1-10 | 0 }
a-b, a, c+1/2 | x, -y, -z	{ 2 ‖ 6+001 | 0 0 1/2 }
-a, -b, c+1/2 | x, -y, -z	{ 2 ‖ 2001 | 0 0 1/2 }
b, -a+b, c+1/2 | x, -y, -z	{ 2 ‖ 6-001 | 0 0 1/2 }
-a+b, b, c+1/2 | x, -y, -z	{ 2 ‖ m100 | 0 0 1/2 }
-b, -a, c+1/2 | x, -y, -z	{ 2 ‖ m110 | 0 0 1/2 }
a, a-b, c+1/2 | x, -y, -z	{ 2 ‖ m010 | 0 0 1/2 }

Wyckoff Positions

WP	Site symmetry	Representative
2a	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{3}.{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(0,0,c | 0,0,z)
4b	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{3}..{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(1/3,2/3,c | 0,0,z)
6c	$..{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(a,0,c | 0,0,z)
12d	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{1}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(a,b,c | 0,0,z)

Spin Brillouin Zone & k-little co-groups

Wavevector-k	Little co-group
A:(0,0,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{6}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$
Γ:(0,0,0)	${\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{6}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$
H:(1/3,1/3,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{6}{\phantom{A}}_{\phantom{}}^{\phantom{2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}2}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
K:(1/3,1/3,0)	${\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{6}{\phantom{A}}_{\phantom{}}^{\phantom{2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}2}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
L:(1/2,0,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}2}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}2}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$
M:(1/2,0,0)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}2}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}2}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$
Δ:(0,0,w)	${\phantom{A}}_{\phantom{}}^{\phantom{2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}2}\text{6}{\phantom{A}}_{\phantom{}}^{\phantom{2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}2}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
P:(1/3,1/3,w)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{3}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{1}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
Λ:(u,u,0)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
Σ:(u,0,0)	${\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}2}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
Q:(u,u,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
R:(u,0,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}2}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
U:(1/2,0,w)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}2}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}2}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
B:(u,v,0)	${\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
C:(u,u,w)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
D:(u,0,w)	${\phantom{A}}_{\phantom{}}^{\phantom{2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}2}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
E:(u,v,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }}\text{1}$
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)

2a

k-vector	k-vector-G↑	A1↑G(2)
A:(0,0,1/2)	A:(0,0,1/2)	$A_1^S(2)$
Γ:(0,0,0)	Γ:(0,0,0)	${\Gamma }_1^S(2)$
H:(1/3,1/3,1/2)	H:(1/3,1/3,1/2)	$H_1^S(2)$
K:(1/3,1/3,0)	K:(1/3,1/3,0)	$K_1^S(2)$
L:(1/2,0,1/2)	L:(1/2,0,1/2)	$L_1^S(2)$
M:(1/2,0,0)	M:(1/2,0,0)	$M_1^S(2)$
B:(u,v,0)	B:(u,v,0)	$B_1^S(2)$
C:(u,u,w)	C:(u,u,w)	$2C_1^S(1)$	Spin Splitting
D:(u,0,w)	D:(u,0,w)	$D_1^S(2)$
Δ:(0,0,w)	Δ:(0,0,w)	${\Delta }_1^S(2)$
E:(u,v,1/2)	E:(u,v,1/2)	$E_1^S(2)$
GP:(u,v,w)	GP:(u,v,w)	$2G{P}_1^S(1)$	Spin Splitting
Λ:(u,u,0)	Λ:(u,u,0)	${\Lambda }_1^S(2)$
P:(1/3,1/3,w)	P:(1/3,1/3,w)	$2P_1^S(1)$	Spin Splitting
Q:(u,u,1/2)	Q:(u,u,1/2)	$Q_1^S(2)$
R:(u,0,1/2)	R:(u,0,1/2)	$R_1^S(2)$
Σ:(u,0,0)	Σ:(u,0,0)	${\Sigma }_1^S(2)$
U:(1/2,0,w)	U:(1/2,0,w)	$U_1^S(2)$

4b

k-vector	k-vector-G↑	A1↑G(4)
A:(0,0,1/2)	A:(0,0,1/2)	$A_1^S(2)\oplus A_2^S(2)$
Γ:(0,0,0)	Γ:(0,0,0)	${\Gamma }_1^S(2)\oplus {\Gamma }_2^S(2)$
H:(1/3,1/3,1/2)	H:(1/3,1/3,1/2)	$H_3^S(4)$
K:(1/3,1/3,0)	K:(1/3,1/3,0)	$K_3^S(4)$
L:(1/2,0,1/2)	L:(1/2,0,1/2)	$L_1^S(2)\oplus L_2^S(2)$
M:(1/2,0,0)	M:(1/2,0,0)	$M_1^S(2)\oplus M_2^S(2)$
B:(u,v,0)	B:(u,v,0)	$2B_1^S(2)$
C:(u,u,w)	C:(u,u,w)	$2C_1^S(1)\oplus 2C_2^S(1)$	Spin Splitting
D:(u,0,w)	D:(u,0,w)	$2D_1^S(2)$
Δ:(0,0,w)	Δ:(0,0,w)	${\Delta }_1^S(2)\oplus {\Delta }_2^S(2)$
E:(u,v,1/2)	E:(u,v,1/2)	$2E_1^S(2)$
GP:(u,v,w)	GP:(u,v,w)	$4G{P}_1^S(1)$	Spin Splitting
Λ:(u,u,0)	Λ:(u,u,0)	${\Lambda }_1^S(2)\oplus {\Lambda }_2^S(2)$
P:(1/3,1/3,w)	P:(1/3,1/3,w)	$2P_3^S(2)$	Spin Splitting
Q:(u,u,1/2)	Q:(u,u,1/2)	$Q_1^S(2)\oplus Q_2^S(2)$
R:(u,0,1/2)	R:(u,0,1/2)	$2R_1^S(2)$
Σ:(u,0,0)	Σ:(u,0,0)	$2{\Sigma }_1^S(2)$
U:(1/2,0,w)	U:(1/2,0,w)	$U_1^S(2)\oplus U_2^S(2)$

6c

k-vector	k-vector-G↑	A'↑G(6)
A:(0,0,1/2)	A:(0,0,1/2)	$A_1^S(2)\oplus A_3^S(4)$
Γ:(0,0,0)	Γ:(0,0,0)	${\Gamma }_1^S(2)\oplus {\Gamma }_3^S(4)$
H:(1/3,1/3,1/2)	H:(1/3,1/3,1/2)	$H_1^S(2)\oplus H_3^S(4)$
K:(1/3,1/3,0)	K:(1/3,1/3,0)	$K_1^S(2)\oplus K_3^S(4)$
L:(1/2,0,1/2)	L:(1/2,0,1/2)	$2L_1^S(2)\oplus L_2^S(2)$
M:(1/2,0,0)	M:(1/2,0,0)	$2M_1^S(2)\oplus M_2^S(2)$
B:(u,v,0)	B:(u,v,0)	$3B_1^S(2)$
C:(u,u,w)	C:(u,u,w)	$4C_1^S(1)\oplus 2C_2^S(1)$	Spin Splitting
D:(u,0,w)	D:(u,0,w)	$3D_1^S(2)$
Δ:(0,0,w)	Δ:(0,0,w)	${\Delta }_1^S(2)\oplus {\Delta }_3^S(4)$
E:(u,v,1/2)	E:(u,v,1/2)	$3E_1^S(2)$
GP:(u,v,w)	GP:(u,v,w)	$6G{P}_1^S(1)$	Spin Splitting
Λ:(u,u,0)	Λ:(u,u,0)	$2{\Lambda }_1^S(2)\oplus {\Lambda }_2^S(2)$
P:(1/3,1/3,w)	P:(1/3,1/3,w)	$2P_1^S(1)\oplus 2P_3^S(2)$	Spin Splitting
Q:(u,u,1/2)	Q:(u,u,1/2)	$2Q_1^S(2)\oplus Q_2^S(2)$
R:(u,0,1/2)	R:(u,0,1/2)	$3R_1^S(2)$
Σ:(u,0,0)	Σ:(u,0,0)	$3{\Sigma }_1^S(2)$
U:(1/2,0,w)	U:(1/2,0,w)	$2U_1^S(2)\oplus U_2^S(2)$

12d

k-vector	k-vector-G↑	A↑G(12)
A:(0,0,1/2)	A:(0,0,1/2)	$A_1^S(2)\oplus A_2^S(2)\oplus 2A_3^S(4)$
Γ:(0,0,0)	Γ:(0,0,0)	${\Gamma }_1^S(2)\oplus {\Gamma }_2^S(2)\oplus 2{\Gamma }_3^S(4)$
H:(1/3,1/3,1/2)	H:(1/3,1/3,1/2)	$H_1^S(2)\oplus H_2^S(2)\oplus 2H_3^S(4)$
K:(1/3,1/3,0)	K:(1/3,1/3,0)	$K_1^S(2)\oplus K_2^S(2)\oplus 2K_3^S(4)$
L:(1/2,0,1/2)	L:(1/2,0,1/2)	$3L_1^S(2)\oplus 3L_2^S(2)$
M:(1/2,0,0)	M:(1/2,0,0)	$3M_1^S(2)\oplus 3M_2^S(2)$
B:(u,v,0)	B:(u,v,0)	$6B_1^S(2)$
C:(u,u,w)	C:(u,u,w)	$6C_1^S(1)\oplus 6C_2^S(1)$	Spin Splitting
D:(u,0,w)	D:(u,0,w)	$6D_1^S(2)$
Δ:(0,0,w)	Δ:(0,0,w)	${\Delta }_1^S(2)\oplus {\Delta }_2^S(2)\oplus 2{\Delta }_3^S(4)$
E:(u,v,1/2)	E:(u,v,1/2)	$6E_1^S(2)$
GP:(u,v,w)	GP:(u,v,w)	$12G{P}_1^S(1)$	Spin Splitting
Λ:(u,u,0)	Λ:(u,u,0)	$3{\Lambda }_1^S(2)\oplus 3{\Lambda }_2^S(2)$
P:(1/3,1/3,w)	P:(1/3,1/3,w)	$2P_1^S(1)\oplus 2P_2^S(1)\oplus 4P_3^S(2)$	Spin Splitting
Q:(u,u,1/2)	Q:(u,u,1/2)	$3Q_1^S(2)\oplus 3Q_2^S(2)$
R:(u,0,1/2)	R:(u,0,1/2)	$6R_1^S(2)$
Σ:(u,0,0)	Σ:(u,0,0)	$6{\Sigma }_1^S(2)$
U:(1/2,0,w)	U:(1/2,0,w)	$3U_1^S(2)\oplus 3U_2^S(2)$