Information of spin space group $P_I{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}{4}_2/{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{n}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$ ( nSSG:136.139.2.1 )

General Positions

The continuously infinite group ^∞1 is omitted.

Coordinates	Seitz symbol
a, b, c | x, y, z	{ 1 ‖ 1 | 0 }
-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 }
a+1/2, -b+1/2, -c+1/2 | x, y, z	{ 1 ‖ 2100 | 1/2 1/2 1/2 }
-a+1/2, b+1/2, -c+1/2 | x, y, z	{ 1 ‖ 2010 | 1/2 1/2 1/2 }
-a, -b, c | x, y, z	{ 1 ‖ 2001 | 0 }
b, a, -c | x, y, z	{ 1 ‖ 2110 | 0 }
-b, -a, -c | x, y, z	{ 1 ‖ 21-10 | 0 }
-a, -b, -c | x, y, z	{ 1 ‖ -1 | 0 }
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 }
-a+1/2, b+1/2, c+1/2 | x, y, z	{ 1 ‖ m100 | 1/2 1/2 1/2 }
a+1/2, -b+1/2, c+1/2 | x, y, z	{ 1 ‖ m010 | 1/2 1/2 1/2 }
a, b, -c | x, y, z	{ 1 ‖ m001 | 0 }
-b, -a, c | x, y, z	{ 1 ‖ m110 | 0 }
b, a, c | x, y, z	{ 1 ‖ m1-10 | 0 }
a+1/2, b+1/2, c+1/2 | -x, -y, -z	{ -1 ‖ 1 | 1/2 1/2 1/2 }
-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 ‖ 2100 | 0 }
-a, b, -c | -x, -y, -z	{ -1 ‖ 2010 | 0 }
-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 ‖ 2110 | 1/2 1/2 1/2 }
-b+1/2, -a+1/2, -c+1/2 | -x, -y, -z	{ -1 ‖ 21-10 | 1/2 1/2 1/2 }
-a+1/2, -b+1/2, -c+1/2 | -x, -y, -z	{ -1 ‖ -1 | 1/2 1/2 1/2 }
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 ‖ m100 | 0 }
a, -b, c | -x, -y, -z	{ -1 ‖ m010 | 0 }
a+1/2, b+1/2, -c+1/2 | -x, -y, -z	{ -1 ‖ m001 | 1/2 1/2 1/2 }
-b+1/2, -a+1/2, c+1/2 | -x, -y, -z	{ -1 ‖ m110 | 1/2 1/2 1/2 }
b+1/2, a+1/2, c+1/2 | -x, -y, -z	{ -1 ‖ m1-10 | 1/2 1/2 1/2 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
-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 }
a+1/2, -b+1/2, -c+1/2 | -x, y, z	{ m ‖ 2100 | 1/2 1/2 1/2 }
-a+1/2, b+1/2, -c+1/2 | -x, y, z	{ m ‖ 2010 | 1/2 1/2 1/2 }
-a, -b, c | -x, y, z	{ m ‖ 2001 | 0 }
b, a, -c | -x, y, z	{ m ‖ 2110 | 0 }
-b, -a, -c | -x, y, z	{ m ‖ 21-10 | 0 }
-a, -b, -c | -x, y, z	{ m ‖ -1 | 0 }
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 }
-a+1/2, b+1/2, c+1/2 | -x, y, z	{ m ‖ m100 | 1/2 1/2 1/2 }
a+1/2, -b+1/2, c+1/2 | -x, y, z	{ m ‖ m010 | 1/2 1/2 1/2 }
a, b, -c | -x, y, z	{ m ‖ m001 | 0 }
-b, -a, c | -x, y, z	{ m ‖ m110 | 0 }
b, a, c | -x, y, z	{ m ‖ m1-10 | 0 }
a+1/2, b+1/2, c+1/2 | x, -y, -z	{ 2 ‖ 1 | 1/2 1/2 1/2 }
-b, a, c | x, -y, -z	{ 2 ‖ 4+001 | 0 }
b, -a, c | x, -y, -z	{ 2 ‖ 4-001 | 0 }
a, -b, -c | x, -y, -z	{ 2 ‖ 2100 | 0 }
-a, b, -c | x, -y, -z	{ 2 ‖ 2010 | 0 }
-a+1/2, -b+1/2, c+1/2 | x, -y, -z	{ 2 ‖ 2001 | 1/2 1/2 1/2 }
b+1/2, a+1/2, -c+1/2 | x, -y, -z	{ 2 ‖ 2110 | 1/2 1/2 1/2 }
-b+1/2, -a+1/2, -c+1/2 | x, -y, -z	{ 2 ‖ 21-10 | 1/2 1/2 1/2 }
-a+1/2, -b+1/2, -c+1/2 | x, -y, -z	{ 2 ‖ -1 | 1/2 1/2 1/2 }
b, -a, -c | x, -y, -z	{ 2 ‖ -4+001 | 0 }
-b, a, -c | x, -y, -z	{ 2 ‖ -4-001 | 0 }
-a, b, c | x, -y, -z	{ 2 ‖ m100 | 0 }
a, -b, c | x, -y, -z	{ 2 ‖ m010 | 0 }
a+1/2, b+1/2, -c+1/2 | x, -y, -z	{ 2 ‖ m001 | 1/2 1/2 1/2 }
-b+1/2, -a+1/2, c+1/2 | x, -y, -z	{ 2 ‖ m110 | 1/2 1/2 1/2 }
b+1/2, a+1/2, c+1/2 | x, -y, -z	{ 2 ‖ m1-10 | 1/2 1/2 1/2 }

Wyckoff Positions

WP	Site symmetry	Representative
2a	${\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{4}/{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(0,0,0 | 0,0,0)
2b	${\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{4}/{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(0,0,1/2 | 0,0,0)
4c	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}.{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(0,1/2,0 | 0,0,0)
4d	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{-4}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(0,1/2,1/4 | 0,0,0)
4e	${\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{4}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(0,0,c | 0,0,0)
8f	$..{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}/{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(1/4,1/4,1/4 | 0,0,0)
8g	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}.{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(0,1/2,c | 0,0,0)
8h	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}.{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(a,a,0 | 0,0,z)
8i	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}.{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(a,0,0 | 0,0,0)
8j	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}.{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(a,1/2,0 | 0,0,0)
16k	$..{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{2}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(a,a+1/2,1/4 | 0,0,0)
16l	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}..{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(a,b,0 | 0,0,z)
16m	$..{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(a,a,c | 0,0,z)
16n	$.{\phantom{A}}_{\phantom{}}^{\phantom{-1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}-1}\text{m}.{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(0,b,c | 0,0,0)
32o	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{1}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }\mathrm{m}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }\mathrm{m}}\text{1}$	(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)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{4/}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }/\mathrm{mm}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }/\mathrm{mm}}\text{1}$
Γ:(0,0,0)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{4/}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }/\mathrm{mm}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }/\mathrm{mm}}\text{1}$
M:(1/2,1/2,0)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{4/}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }/\mathrm{mm}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }/\mathrm{mm}}\text{1}$
Z:(0,0,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{4/}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }/\mathrm{mm}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }/\mathrm{mm}}\text{1}$
R:(0,1/2,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }/\mathrm{mm}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }/\mathrm{mm}}\text{1}$
X:(0,1/2,0)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }/\mathrm{mm}}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }/\mathrm{mm}}\text{1}$
Λ:(0,0,w)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{4/}{\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }2}\text{1}$
V:(1/2,1/2,w)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{4/}{\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }2}\text{1}$
Δ:(0,v,0)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }2}\text{1}$
S:(u,u,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }2}\text{1}$
Σ:(u,u,0)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }2}\text{1}$
T:(u,1/2,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }2}\text{1}$
U:(0,v,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }2}\text{1}$
W:(0,1/2,w)	${\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }2}\text{1}$
Y:(u,1/2,0)	${\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }2}\text{1}$
B:(0,v,w)	${\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{2/}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }2}\text{1}$
C:(u,u,w)	${\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{2/}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }2}\text{1}$
D:(u,v,0)	${\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{2/}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }2}\text{1}$
E:(u,v,1/2)	${\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{2/}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }2}\text{1}$
F:(u,1/2,w)	${\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{2/}{\phantom{A}}_{\phantom{}}^{\phantom{1}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}1}\text{m}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }2}\text{1}$
GP:(u,v,w)	${\phantom{A}}_{\phantom{}}^{\phantom{m}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}m}\text{-1}{\phantom{A}}_{\phantom{}}^{\phantom{\mathrm{\infty }2}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}\mathrm{\infty }2}\text{1}$

Spin Brillouin Zone

Band Representations (Magnon)

8h

k-vector	A1↑G(8)
A:(1/2,1/2,1/2)	$A_1^S(4)\oplus A_4^S(4)$
Γ:(0,0,0)	${\Gamma }_1^{S,+}(2)\oplus {\Gamma }_4^{S,+}(2)\oplus {\Gamma }_5^{S,-}(4)$
M:(1/2,1/2,0)	$M_1^{S,+}{M}_4^{S,+}(4)\oplus M_5^{S,-}(4)$
Z:(0,0,1/2)	$Z_1^S(4)\oplus Z_4^S(4)$
R:(0,1/2,1/2)	$R_1^{S,+}(4)\oplus R_1^{S,-}(4)$
X:(0,1/2,0)	$2X_2^S(4)$
Λ:(0,0,w)	${\Lambda }_1^S(2)\oplus {\Lambda }_3^S(2)\oplus {\Lambda }_5^S(4)$
V:(1/2,1/2,w)	$V_1^S{V}_3^S(4)\oplus V_5^S(4)$
Δ:(0,v,0)	$2{\Delta }_1^S(2)\oplus 2{\Delta }_4^S(2)$
S:(u,u,1/2)	$2S_1^S(2)\oplus S_3^S(2)\oplus S_4^S(2)$
Σ:(u,u,0)	$3{\Sigma }_1^S(2)\oplus {\Sigma }_2^S(2)$
T:(u,1/2,1/2)	$2T_1^S(4)$
U:(0,v,1/2)	$2U_1^S(4)$
W:(0,1/2,w)	$2W_1^S(4)$
Y:(u,1/2,0)	$2Y_1^S{Y}_4^S(4)$
B:(0,v,w)	$2B_1^S(2)\oplus 2B_2^S(2)$
C:(u,u,w)	$3C_1^S(2)\oplus C_2^S(2)$
D:(u,v,0)	$4D_1^S(2)$
E:(u,v,1/2)	$2E_1^S(2)\oplus 2E_2^S(2)$
F:(u,1/2,w)	$2F_1^S{F}_2^S(4)$
GP:(u,v,w)	$4G{P}_1^S(2)$

16l

k-vector	A'↑G(16)
A:(1/2,1/2,1/2)	$A_1^S(4)\oplus A_2^S(4)\oplus A_3^S(4)\oplus A_4^S(4)$
Γ:(0,0,0)	${\Gamma }_1^{S,+}(2)\oplus {\Gamma }_2^{S,+}(2)\oplus {\Gamma }_3^{S,+}(2)\oplus {\Gamma }_4^{S,+}(2)\oplus 2{\Gamma }_5^{S,-}(4)$
M:(1/2,1/2,0)	$M_1^{S,+}{M}_4^{S,+}(4)\oplus M_2^{S,+}{M}_3^{S,+}(4)\oplus 2M_5^{S,-}(4)$
Z:(0,0,1/2)	$Z_1^S(4)\oplus Z_2^S(4)\oplus Z_3^S(4)\oplus Z_4^S(4)$
R:(0,1/2,1/2)	$2R_1^{S,+}(4)\oplus 2R_1^{S,-}(4)$
X:(0,1/2,0)	$4X_2^S(4)$
Λ:(0,0,w)	${\Lambda }_1^S(2)\oplus {\Lambda }_2^S(2)\oplus {\Lambda }_3^S(2)\oplus {\Lambda }_4^S(2)\oplus 2{\Lambda }_5^S(4)$
V:(1/2,1/2,w)	$V_1^S{V}_3^S(4)\oplus V_2^S{V}_4^S(4)\oplus 2V_5^S(4)$
Δ:(0,v,0)	$4{\Delta }_1^S(2)\oplus 4{\Delta }_4^S(2)$
S:(u,u,1/2)	$2S_1^S(2)\oplus 2S_2^S(2)\oplus 2S_3^S(2)\oplus 2S_4^S(2)$
Σ:(u,u,0)	$4{\Sigma }_1^S(2)\oplus 4{\Sigma }_2^S(2)$
T:(u,1/2,1/2)	$4T_1^S(4)$
U:(0,v,1/2)	$4U_1^S(4)$
W:(0,1/2,w)	$4W_1^S(4)$
Y:(u,1/2,0)	$4Y_1^S{Y}_4^S(4)$
B:(0,v,w)	$4B_1^S(2)\oplus 4B_2^S(2)$
C:(u,u,w)	$4C_1^S(2)\oplus 4C_2^S(2)$
D:(u,v,0)	$8D_1^S(2)$
E:(u,v,1/2)	$4E_1^S(2)\oplus 4E_2^S(2)$
F:(u,1/2,w)	$4F_1^S{F}_2^S(4)$
GP:(u,v,w)	$8G{P}_1^S(2)$

16m

k-vector	A'↑G(16)
A:(1/2,1/2,1/2)	$2A_1^S(4)\oplus A_3^S(4)\oplus A_4^S(4)$
Γ:(0,0,0)	${\Gamma }_1^{S,+}(2)\oplus {\Gamma }_2^{S,-}(2)\oplus {\Gamma }_3^{S,-}(2)\oplus {\Gamma }_4^{S,+}(2)\oplus {\Gamma }_5^{S,+}(4)\oplus {\Gamma }_5^{S,-}(4)$
M:(1/2,1/2,0)	$M_1^{S,+}{M}_4^{S,+}(4)\oplus M_2^{S,-}{M}_3^{S,-}(4)\oplus M_5^{S,+}(4)\oplus M_5^{S,-}(4)$
Z:(0,0,1/2)	$2Z_1^S(4)\oplus Z_3^S(4)\oplus Z_4^S(4)$
R:(0,1/2,1/2)	$2R_1^{S,+}(4)\oplus 2R_1^{S,-}(4)$
X:(0,1/2,0)	$2X_1^S(4)\oplus 2X_2^S(4)$
Λ:(0,0,w)	$2{\Lambda }_1^S(2)\oplus 2{\Lambda }_3^S(2)\oplus 2{\Lambda }_5^S(4)$
V:(1/2,1/2,w)	$2V_1^S{V}_3^S(4)\oplus 2V_5^S(4)$
Δ:(0,v,0)	$2{\Delta }_1^S(2)\oplus 2{\Delta }_2^S(2)\oplus 2{\Delta }_3^S(2)\oplus 2{\Delta }_4^S(2)$
S:(u,u,1/2)	$3S_1^S(2)\oplus S_2^S(2)\oplus 3S_3^S(2)\oplus S_4^S(2)$
Σ:(u,u,0)	$3{\Sigma }_1^S(2)\oplus {\Sigma }_2^S(2)\oplus 3{\Sigma }_3^S(2)\oplus {\Sigma }_4^S(2)$
T:(u,1/2,1/2)	$4T_1^S(4)$
U:(0,v,1/2)	$4U_1^S(4)$
W:(0,1/2,w)	$4W_1^S(4)$
Y:(u,1/2,0)	$2Y_1^S{Y}_4^S(4)\oplus 2Y_2^S{Y}_3^S(4)$
B:(0,v,w)	$4B_1^S(2)\oplus 4B_2^S(2)$
C:(u,u,w)	$6C_1^S(2)\oplus 2C_2^S(2)$
D:(u,v,0)	$4D_1^S(2)\oplus 4D_2^S(2)$
E:(u,v,1/2)	$4E_1^S(2)\oplus 4E_2^S(2)$
F:(u,1/2,w)	$4F_1^S{F}_2^S(4)$
GP:(u,v,w)	$8G{P}_1^S(2)$

32o

k-vector	A↑G(32)
A:(1/2,1/2,1/2)	$2A_1^S(4)\oplus 2A_2^S(4)\oplus 2A_3^S(4)\oplus 2A_4^S(4)$
Γ:(0,0,0)	${\Gamma }_1^{S,+}(2)\oplus {\Gamma }_1^{S,-}(2)\oplus {\Gamma }_2^{S,+}(2)\oplus {\Gamma }_2^{S,-}(2)\oplus {\Gamma }_3^{S,+}(2)\oplus {\Gamma }_3^{S,-}(2)\oplus {\Gamma }_4^{S,+}(2)\oplus {\Gamma }_4^{S,-}(2)\oplus 2{\Gamma }_5^{S,+}(4)\oplus 2{\Gamma }_5^{S,-}(4)$
M:(1/2,1/2,0)	$M_1^{S,+}{M}_4^{S,+}(4)\oplus M_1^{S,-}{M}_4^{S,-}(4)\oplus M_2^{S,+}{M}_3^{S,+}(4)\oplus M_2^{S,-}{M}_3^{S,-}(4)\oplus 2M_5^{S,+}(4)\oplus 2M_5^{S,-}(4)$
Z:(0,0,1/2)	$2Z_1^S(4)\oplus 2Z_2^S(4)\oplus 2Z_3^S(4)\oplus 2Z_4^S(4)$
R:(0,1/2,1/2)	$4R_1^{S,+}(4)\oplus 4R_1^{S,-}(4)$
X:(0,1/2,0)	$4X_1^S(4)\oplus 4X_2^S(4)$
Λ:(0,0,w)	$2{\Lambda }_1^S(2)\oplus 2{\Lambda }_2^S(2)\oplus 2{\Lambda }_3^S(2)\oplus 2{\Lambda }_4^S(2)\oplus 4{\Lambda }_5^S(4)$
V:(1/2,1/2,w)	$2V_1^S{V}_3^S(4)\oplus 2V_2^S{V}_4^S(4)\oplus 4V_5^S(4)$
Δ:(0,v,0)	$4{\Delta }_1^S(2)\oplus 4{\Delta }_2^S(2)\oplus 4{\Delta }_3^S(2)\oplus 4{\Delta }_4^S(2)$
S:(u,u,1/2)	$4S_1^S(2)\oplus 4S_2^S(2)\oplus 4S_3^S(2)\oplus 4S_4^S(2)$
Σ:(u,u,0)	$4{\Sigma }_1^S(2)\oplus 4{\Sigma }_2^S(2)\oplus 4{\Sigma }_3^S(2)\oplus 4{\Sigma }_4^S(2)$
T:(u,1/2,1/2)	$8T_1^S(4)$
U:(0,v,1/2)	$8U_1^S(4)$
W:(0,1/2,w)	$8W_1^S(4)$
Y:(u,1/2,0)	$4Y_1^S{Y}_4^S(4)\oplus 4Y_2^S{Y}_3^S(4)$
B:(0,v,w)	$8B_1^S(2)\oplus 8B_2^S(2)$
C:(u,u,w)	$8C_1^S(2)\oplus 8C_2^S(2)$
D:(u,v,0)	$8D_1^S(2)\oplus 8D_2^S(2)$
E:(u,v,1/2)	$8E_1^S(2)\oplus 8E_2^S(2)$
F:(u,1/2,w)	$8F_1^S{F}_2^S(4)$
GP:(u,v,w)	$16G{P}_1^S(2)$