Information of spin space group $P{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}{6}_3/{\phantom{A}}_{\phantom{}}^{\phantom{\text{-}1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\text{-}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$ ( nSSG:173.176.1.1 )

General Positions

The continuously infinite group ^∞1 is omitted.

Coordinates	Seitz symbol
a, b, c | x, y, z	{ 1 ‖ 1 | 0 }
a-b, a, c+1/2 | x, y, z	{ 1 ‖ 6+001 | 0 0 1/2 }
-b, a-b, c | x, y, z	{ 1 ‖ 3+001 | 0 }
-a, -b, c+1/2 | x, y, z	{ 1 ‖ 2001 | 0 0 1/2 }
-a+b, -a, c | x, y, z	{ 1 ‖ 3-001 | 0 }
b, -a+b, c+1/2 | x, y, z	{ 1 ‖ 6-001 | 0 0 1/2 }
-a, -b, -c | -x, -y, -z	{ -1 ‖ -1 | 0 }
-a+b, -a, -c+1/2 | -x, -y, -z	{ -1 ‖ -6+001 | 0 0 1/2 }
b, -a+b, -c | -x, -y, -z	{ -1 ‖ -3+001 | 0 }
a, b, -c+1/2 | -x, -y, -z	{ -1 ‖ m001 | 0 0 1/2 }
a-b, a, -c | -x, -y, -z	{ -1 ‖ -3-001 | 0 }
-b, a-b, -c+1/2 | -x, -y, -z	{ -1 ‖ -6-001 | 0 0 1/2 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
a-b, a, c+1/2 | -x, y, z	{ m ‖ 6+001 | 0 0 1/2 }
-b, a-b, c | -x, y, z	{ m ‖ 3+001 | 0 }
-a, -b, c+1/2 | -x, y, z	{ m ‖ 2001 | 0 0 1/2 }
-a+b, -a, c | -x, y, z	{ m ‖ 3-001 | 0 }
b, -a+b, c+1/2 | -x, y, z	{ m ‖ 6-001 | 0 0 1/2 }
-a, -b, -c | x, -y, -z	{ 2 ‖ -1 | 0 }
-a+b, -a, -c+1/2 | x, -y, -z	{ 2 ‖ -6+001 | 0 0 1/2 }
b, -a+b, -c | x, -y, -z	{ 2 ‖ -3+001 | 0 }
a, b, -c+1/2 | x, -y, -z	{ 2 ‖ m001 | 0 0 1/2 }
a-b, a, -c | x, -y, -z	{ 2 ‖ -3-001 | 0 }
-b, a-b, -c+1/2 | x, -y, -z	{ 2 ‖ -6-001 | 0 0 1/2 }

Wyckoff Positions

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

Spin Brillouin Zone & k-little co-groups

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

6/mP

Band Representations (Magnon)

4e

k-vector	k-vector-G↑	A1↑G(4)
Γ:(0,0,0)	Γ:(0,0,0)	${\Gamma }_1^S(2)\oplus {\Gamma }_2^S(2)$
A:(0,0,1/2)	A:(0,0,1/2)	$A_1^S{A}_2^S(4)$
H:(1/3,1/3,1/2)	H:(1/3,1/3,1/2)	$H_1^S{H}_1^S(4)$
K:(1/3,1/3,0)	K:(1/3,1/3,0)	$2K_1^S(2)$
L:(1/2,0,1/2)	L:(1/2,0,1/2)	$L_1^S{L}_2^S(4)$
M:(1/2,0,0)	M:(1/2,0,0)	$M_1^S(2)\oplus M_2^S(2)$
Δ:(0,0,w)	Δ:(0,0,w)	${\Delta }_1^S(2)\oplus {\Delta }_2^S(2)$
Λ:(u,u,0)	Λ:(u,u,0)	$2{\Lambda }_1^S(2)$
P:(1/3,1/3,w)	P:(1/3,1/3,w)	$2P_1^S(2)$
Q:(u,u,1/2)	Q:(u,u,1/2)	$Q_1^S{Q}_1^S(4)$
R:(u,0,1/2)	R:(u,0,1/2)	$R_1^S{R}_1^S(4)$
Σ:(u,0,0)	Σ:(u,0,0)	$2{\Sigma }_1^S(2)$
U:(1/2,0,w)	U:(1/2,0,w)	$U_1^S(2)\oplus U_2^S(2)$
B:(u,v,0)	B:(u,v,0)	$2B_1^S(2)$
C:(u,u,w)	C:(u,u,w)	$2C_1^S(2)$
D:(u,0,w)	D:(u,0,w)	$2D_1^S(2)$
E:(u,v,1/2)	E:(u,v,1/2)	$E_1^S{E}_1^S(4)$
GP:(u,v,w)	GP:(u,v,w)	$2G{P}_1^S(2)$

4f

k-vector	k-vector-G↑	A1↑G(4)
Γ:(0,0,0)	Γ:(0,0,0)	${\Gamma }_1^S(2)\oplus {\Gamma }_2^S(2)$
A:(0,0,1/2)	A:(0,0,1/2)	$A_1^S{A}_2^S(4)$
H:(1/3,1/3,1/2)	H:(1/3,1/3,1/2)	$H_2^S{H}_3^S(4)$
K:(1/3,1/3,0)	K:(1/3,1/3,0)	$K_2^S{K}_3^S(4)$
L:(1/2,0,1/2)	L:(1/2,0,1/2)	$L_1^S{L}_2^S(4)$
M:(1/2,0,0)	M:(1/2,0,0)	$M_1^S(2)\oplus M_2^S(2)$
Δ:(0,0,w)	Δ:(0,0,w)	${\Delta }_1^S(2)\oplus {\Delta }_2^S(2)$
Λ:(u,u,0)	Λ:(u,u,0)	$2{\Lambda }_1^S(2)$
P:(1/3,1/3,w)	P:(1/3,1/3,w)	$P_2^S(2)\oplus P_3^S(2)$
Q:(u,u,1/2)	Q:(u,u,1/2)	$Q_1^S{Q}_1^S(4)$
R:(u,0,1/2)	R:(u,0,1/2)	$R_1^S{R}_1^S(4)$
Σ:(u,0,0)	Σ:(u,0,0)	$2{\Sigma }_1^S(2)$
U:(1/2,0,w)	U:(1/2,0,w)	$U_1^S(2)\oplus U_2^S(2)$
B:(u,v,0)	B:(u,v,0)	$2B_1^S(2)$
C:(u,u,w)	C:(u,u,w)	$2C_1^S(2)$
D:(u,0,w)	D:(u,0,w)	$2D_1^S(2)$
E:(u,v,1/2)	E:(u,v,1/2)	$E_1^S{E}_1^S(4)$
GP:(u,v,w)	GP:(u,v,w)	$2G{P}_1^S(2)$

12i

k-vector	k-vector-G↑	A↑G(12)
Γ:(0,0,0)	Γ:(0,0,0)	${\Gamma }_1^S(2)\oplus {\Gamma }_2^S(2)\oplus {\Gamma }_3^S{\Gamma }_5^S(4)\oplus {\Gamma }_4^S{\Gamma }_6^S(4)$
A:(0,0,1/2)	A:(0,0,1/2)	$A_1^S{A}_2^S(4)\oplus A_3^S{A}_6^S(4)\oplus A_4^S{A}_5^S(4)$
H:(1/3,1/3,1/2)	H:(1/3,1/3,1/2)	$H_1^S{H}_1^S(4)\oplus 2H_2^S{H}_3^S(4)$
K:(1/3,1/3,0)	K:(1/3,1/3,0)	$2K_1^S(2)\oplus 2K_2^S{K}_3^S(4)$
L:(1/2,0,1/2)	L:(1/2,0,1/2)	$3L_1^S{L}_2^S(4)$
M:(1/2,0,0)	M:(1/2,0,0)	$3M_1^S(2)\oplus 3M_2^S(2)$
Δ:(0,0,w)	Δ:(0,0,w)	${\Delta }_1^S(2)\oplus {\Delta }_2^S(2)\oplus {\Delta }_3^S(2)\oplus {\Delta }_4^S(2)\oplus {\Delta }_5^S(2)\oplus {\Delta }_6^S(2)$
Λ:(u,u,0)	Λ:(u,u,0)	$6{\Lambda }_1^S(2)$
P:(1/3,1/3,w)	P:(1/3,1/3,w)	$2P_1^S(2)\oplus 2P_2^S(2)\oplus 2P_3^S(2)$
Q:(u,u,1/2)	Q:(u,u,1/2)	$3Q_1^S{Q}_1^S(4)$
R:(u,0,1/2)	R:(u,0,1/2)	$3R_1^S{R}_1^S(4)$
Σ:(u,0,0)	Σ:(u,0,0)	$6{\Sigma }_1^S(2)$
U:(1/2,0,w)	U:(1/2,0,w)	$3U_1^S(2)\oplus 3U_2^S(2)$
B:(u,v,0)	B:(u,v,0)	$6B_1^S(3)$
C:(u,u,w)	C:(u,u,w)	$6C_1^S(3)$
D:(u,0,w)	D:(u,0,w)	$6D_1^S(3)$
E:(u,v,1/2)	E:(u,v,1/2)	$3E_1^S{E}_1^S(4)$
GP:(u,v,w)	GP:(u,v,w)	$6G{P}_1^S(2)$