Information of spin space group $P\ce{^{\text{-}1}{4/}}\ce{^{1}{m}}\ce{^{1}{c}}\ce{^{\text{-}1}{c}}\ce{^{\infty m}{1}}$ ( nSSG:49.124.1.1 )

General Positions

The continuously infinite group ^∞1 is omitted.

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

Wyckoff Positions

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

Spin Brillouin Zone

Band Representations (Magnon)

4e

k-vector	k-vector-G↑	Ag↑G(4)
$Decomposable $
$(Decomposable $
$) $
\Indecomposable	Indecomposable	$ $
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S,+}(2)⊕Γ_{2}^{S,+}(2) $
A:(1/2,1/2,1/2)	R:(1/2,1/2,1/2)	$(A)R_{2}^{S}(4) $
M:(1/2,1/2,0)	S:(1/2,1/2,0)	$(M)S_{3}^{S,-}(2)⊕(M)S_{4}^{S,-}(2) $
R:(0,1/2,1/2)	T:(0,1/2,1/2)	$(R)T_{2}^{S}(4) $
RA:(-1/2,0,1/2)	U:(1/2,0,1/2)	$(RA)U_{1}^{S}(4) $
X:(0,1/2,0)	Y:(0,1/2,0)	$(X)Y_{3}^{S,-}(2)⊕(X)Y_{4}^{S,-}(2) $
XA:(-1/2,0,0)	X:(1/2,0,0)	$(XA)X_{1}^{S,+}(2)⊕(XA)X_{2}^{S,+}(2) $
Z:(0,0,1/2)	Z:(0,0,1/2)	$Z_{1}^{S}(4) $
Δ:(0,v,0)	Δ:(0,v,0)	$Δ_{1}^{S}(2)⊕Δ_{4}^{S}(2) $
Λ:(0,0,w)	Λ:(0,0,w)	$Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2) $
S:(u,u,1/2)	W:(u,u,1/2)	$(S)W_{1}^{S}(2)⊕(S)W_{2}^{S}(2) $
Σ:(u,u,0)	V:(u,u,0)	$2(Σ)V_{1}^{S}(2) $
T:(u,1/2,1/2)	E:(u,1/2,1/2)	$(T)E_{1}^{S}(4) $
U:(0,v,1/2)	B:(0,v,1/2)	$(U)B_{1}^{S}(4) $
V:(1/2,1/2,w)	Q:(1/2,1/2,w)	$(V)Q_{3}^{S}(2)⊕(V)Q_{4}^{S}(2) $
W:(0,1/2,w)	H:(0,1/2,w)	$(W)H_{3}^{S}(2)⊕(W)H_{4}^{S}(2) $
Y:(u,1/2,0)	C:(u,1/2,0)	$(Y)C_{1}^{S}(2)⊕(Y)C_{4}^{S}(2) $
B:(0,v,w)	K:(0,v,w)	$(B)K_{1}^{S}(2)⊕(B)K_{2}^{S}(2) $
C:(u,u,w)	GP:(u,u,w)	$2(C)GP_{1}^{S}(2) $
D:(u,v,0)	V:(u,v,0)	$2(D)V_{1}^{S}(2) $
E:(u,v,1/2)	W:(u,v,1/2)	$(E)W_{1}^{S}(2)⊕(E)W_{2}^{S}(2) $
F:(u,1/2,w)	N:(u,1/2,w)	$(F)N_{1}^{S}(2)⊕(F)N_{2}^{S}(2) $
GP:(u,v,w)	GP:(u,v,w)	$2GP_{1}^{S}(2) $
BA:(-v,0,w)	M:(-v,0,-w)	$(BA)M_{1}^{S}(2)⊕(BA)M_{2}^{S}(2) $
DU:(-v,0,0)	Σ:(-v,0,0)	$(DU)Σ_{1}^{S}(2)⊕(DU)Σ_{4}^{S}(2) $
FA:(-1/2,-u,w)	L:(-1/2,u,-w)	$(FA)L_{1}^{S}(2)⊕(FA)L_{2}^{S}(2) $
TA:(-1/2,-u,1/2)	P:(-1/2,u,-1/2)	$(TA)P_{1}^{S}(4) $
UA:(-v,0,1/2)	A:(-v,0,-1/2)	$(UA)A_{1}^{S}(4) $
WA:(-1/2,0,w)	G:(-1/2,0,-w)	$(WA)G_{1}^{S}(2)⊕(WA)G_{2}^{S}(2) $
YA:(-1/2,-u,0)	D:(-1/2,u,0)	$(YA)D_{1}^{S}(2)⊕(YA)D_{4}^{S}(2) $

4f

k-vector	k-vector-G↑	A↑G(4)
$Decomposable $
$(Decomposable $
$) $
\Indecomposable	Indecomposable	$ $
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S,+}(2)⊕Γ_{1}^{S,-}(2) $
A:(1/2,1/2,1/2)	R:(1/2,1/2,1/2)	$(A)R_{2}^{S}(4) $
M:(1/2,1/2,0)	S:(1/2,1/2,0)	$(M)S_{4}^{S,+}(2)⊕(M)S_{4}^{S,-}(2) $
R:(0,1/2,1/2)	T:(0,1/2,1/2)	$(R)T_{2}^{S}(4) $
RA:(-1/2,0,1/2)	U:(1/2,0,1/2)	$(RA)U_{1}^{S}(4) $
X:(0,1/2,0)	Y:(0,1/2,0)	$(X)Y_{4}^{S,+}(2)⊕(X)Y_{4}^{S,-}(2) $
XA:(-1/2,0,0)	X:(1/2,0,0)	$(XA)X_{1}^{S,+}(2)⊕(XA)X_{1}^{S,-}(2) $
Z:(0,0,1/2)	Z:(0,0,1/2)	$Z_{1}^{S}(4) $
Δ:(0,v,0)	Δ:(0,v,0)	$Δ_{1}^{S}(2)⊕Δ_{2}^{S}(2) $
Λ:(0,0,w)	Λ:(0,0,w)	$Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2) $
S:(u,u,1/2)	W:(u,u,1/2)	$(S)W_{1}^{S}(2)⊕(S)W_{2}^{S}(2) $
Σ:(u,u,0)	V:(u,u,0)	$(Σ)V_{1}^{S}(2)⊕(Σ)V_{2}^{S}(2) $
T:(u,1/2,1/2)	E:(u,1/2,1/2)	$(T)E_{1}^{S}(4) $
U:(0,v,1/2)	B:(0,v,1/2)	$(U)B_{1}^{S}(4) $
V:(1/2,1/2,w)	Q:(1/2,1/2,w)	$(V)Q_{3}^{S}(2)⊕(V)Q_{4}^{S}(2) $
W:(0,1/2,w)	H:(0,1/2,w)	$(W)H_{3}^{S}(2)⊕(W)H_{4}^{S}(2) $
Y:(u,1/2,0)	C:(u,1/2,0)	$(Y)C_{3}^{S}(2)⊕(Y)C_{4}^{S}(2) $
B:(0,v,w)	K:(0,v,w)	$(B)K_{1}^{S}(2)⊕(B)K_{2}^{S}(2) $
C:(u,u,w)	GP:(u,u,w)	$2(C)GP_{1}^{S}(2) $
D:(u,v,0)	V:(u,v,0)	$(D)V_{1}^{S}(2)⊕(D)V_{2}^{S}(2) $
E:(u,v,1/2)	W:(u,v,1/2)	$(E)W_{1}^{S}(2)⊕(E)W_{2}^{S}(2) $
F:(u,1/2,w)	N:(u,1/2,w)	$(F)N_{1}^{S}(2)⊕(F)N_{2}^{S}(2) $
GP:(u,v,w)	GP:(u,v,w)	$2GP_{1}^{S}(2) $
BA:(-v,0,w)	M:(-v,0,-w)	$(BA)M_{1}^{S}(2)⊕(BA)M_{2}^{S}(2) $
DU:(-v,0,0)	Σ:(-v,0,0)	$(DU)Σ_{1}^{S}(2)⊕(DU)Σ_{2}^{S}(2) $
FA:(-1/2,-u,w)	L:(-1/2,u,-w)	$(FA)L_{1}^{S}(2)⊕(FA)L_{2}^{S}(2) $
TA:(-1/2,-u,1/2)	P:(-1/2,u,-1/2)	$(TA)P_{1}^{S}(4) $
UA:(-v,0,1/2)	A:(-v,0,-1/2)	$(UA)A_{1}^{S}(4) $
WA:(-1/2,0,w)	G:(-1/2,0,-w)	$(WA)G_{1}^{S}(2)⊕(WA)G_{2}^{S}(2) $
YA:(-1/2,-u,0)	D:(-1/2,u,0)	$(YA)D_{1}^{S}(2)⊕(YA)D_{2}^{S}(2) $

8k

k-vector	k-vector-G↑	A↑G(8)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S,+}(2)⊕Γ_{1}^{S,-}(2)⊕Γ_{3}^{S,+}(2)⊕Γ_{3}^{S,-}(2) $
A:(1/2,1/2,1/2)	R:(1/2,1/2,1/2)	$(A)R_{1}^{S}(4)⊕(A)R_{2}^{S}(4) $
M:(1/2,1/2,0)	S:(1/2,1/2,0)	$(M)S_{1}^{S,+}(2)⊕(M)S_{1}^{S,-}(2)⊕(M)S_{3}^{S,+}(2)⊕(M)S_{3}^{S,-}(2) $
R:(0,1/2,1/2)	T:(0,1/2,1/2)	$(R)T_{1}^{S}(4)⊕(R)T_{2}^{S}(4) $
RA:(-1/2,0,1/2)	U:(1/2,0,1/2)	$(RA)U_{1}^{S}(4)⊕(RA)U_{2}^{S}(4) $
X:(0,1/2,0)	Y:(0,1/2,0)	$(X)Y_{1}^{S,+}(2)⊕(X)Y_{1}^{S,-}(2)⊕(X)Y_{3}^{S,+}(2)⊕(X)Y_{3}^{S,-}(2) $
XA:(-1/2,0,0)	X:(1/2,0,0)	$(XA)X_{1}^{S,+}(2)⊕(XA)X_{1}^{S,-}(2)⊕(XA)X_{3}^{S,+}(2)⊕(XA)X_{3}^{S,-}(2) $
Z:(0,0,1/2)	Z:(0,0,1/2)	$Z_{1}^{S}(4)⊕Z_{2}^{S}(4) $
Δ:(0,v,0)	Δ:(0,v,0)	$Δ_{1}^{S}(2)⊕Δ_{2}^{S}(2)⊕Δ_{3}^{S}(2)⊕Δ_{4}^{S}(2) $
Λ:(0,0,w)	Λ:(0,0,w)	$Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2)⊕Λ_{3}^{S}(2)⊕Λ_{4}^{S}(2) $
S:(u,u,1/2)	W:(u,u,1/2)	$2(S)W_{1}^{S}(2)⊕2(S)W_{2}^{S}(2) $
Σ:(u,u,0)	V:(u,u,0)	$2(Σ)V_{1}^{S}(2)⊕2(Σ)V_{2}^{S}(2) $
T:(u,1/2,1/2)	E:(u,1/2,1/2)	$2(T)E_{1}^{S}(4) $
U:(0,v,1/2)	B:(0,v,1/2)	$2(U)B_{1}^{S}(4) $
V:(1/2,1/2,w)	Q:(1/2,1/2,w)	$(V)Q_{1}^{S}(2)⊕(V)Q_{2}^{S}(2)⊕(V)Q_{3}^{S}(2)⊕(V)Q_{4}^{S}(2) $
W:(0,1/2,w)	H:(0,1/2,w)	$(W)H_{1}^{S}(2)⊕(W)H_{2}^{S}(2)⊕(W)H_{3}^{S}(2)⊕(W)H_{4}^{S}(2) $
Y:(u,1/2,0)	C:(u,1/2,0)	$2(Y)C_{1}^{S}(2)⊕2(Y)C_{2}^{S}(2) $
B:(0,v,w)	K:(0,v,w)	$2(B)K_{1}^{S}(2)⊕2(B)K_{2}^{S}(2) $
C:(u,u,w)	GP:(u,u,w)	$4(C)GP_{1}^{S}(2) $
D:(u,v,0)	V:(u,v,0)	$2(D)V_{1}^{S}(2)⊕2(D)V_{2}^{S}(2) $
E:(u,v,1/2)	W:(u,v,1/2)	$2(E)W_{1}^{S}(2)⊕2(E)W_{2}^{S}(2) $
F:(u,1/2,w)	N:(u,1/2,w)	$2(F)N_{1}^{S}(2)⊕2(F)N_{2}^{S}(2) $
GP:(u,v,w)	GP:(u,v,w)	$4GP_{1}^{S}(2) $
BA:(-v,0,w)	M:(-v,0,-w)	$2(BA)M_{1}^{S}(2)⊕2(BA)M_{2}^{S}(2) $
DU:(-v,0,0)	Σ:(-v,0,0)	$2(DU)Σ_{1}^{S}(2)⊕2(DU)Σ_{2}^{S}(2) $
FA:(-1/2,-u,w)	L:(-1/2,u,-w)	$2(FA)L_{1}^{S}(2)⊕2(FA)L_{2}^{S}(2) $
TA:(-1/2,-u,1/2)	P:(-1/2,u,-1/2)	$2(TA)P_{1}^{S}(4) $
UA:(-v,0,1/2)	A:(-v,0,-1/2)	$2(UA)A_{1}^{S}(4) $
WA:(-1/2,0,w)	G:(-1/2,0,-w)	$(WA)G_{1}^{S}(2)⊕(WA)G_{2}^{S}(2)⊕(WA)G_{3}^{S}(2)⊕(WA)G_{4}^{S}(2) $
YA:(-1/2,-u,0)	D:(-1/2,u,0)	$(YA)D_{1}^{S}(2)⊕(YA)D_{2}^{S}(2)⊕(YA)D_{3}^{S}(2)⊕(YA)D_{4}^{S}(2) $

8l

k-vector	k-vector-G↑	A↑G(8)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S,+}(2)⊕Γ_{1}^{S,-}(2)⊕Γ_{3}^{S,+}(2)⊕Γ_{3}^{S,-}(2) $
A:(1/2,1/2,1/2)	R:(1/2,1/2,1/2)	$(A)R_{1}^{S}(4)⊕(A)R_{2}^{S}(4) $
M:(1/2,1/2,0)	S:(1/2,1/2,0)	$(M)S_{2}^{S,+}(2)⊕(M)S_{2}^{S,-}(2)⊕(M)S_{4}^{S,+}(2)⊕(M)S_{4}^{S,-}(2) $
R:(0,1/2,1/2)	T:(0,1/2,1/2)	$(R)T_{1}^{S}(4)⊕(R)T_{2}^{S}(4) $
RA:(-1/2,0,1/2)	U:(1/2,0,1/2)	$(RA)U_{1}^{S}(4)⊕(RA)U_{2}^{S}(4) $
X:(0,1/2,0)	Y:(0,1/2,0)	$(X)Y_{2}^{S,+}(2)⊕(X)Y_{2}^{S,-}(2)⊕(X)Y_{4}^{S,+}(2)⊕(X)Y_{4}^{S,-}(2) $
XA:(-1/2,0,0)	X:(1/2,0,0)	$(XA)X_{1}^{S,+}(2)⊕(XA)X_{1}^{S,-}(2)⊕(XA)X_{3}^{S,+}(2)⊕(XA)X_{3}^{S,-}(2) $
Z:(0,0,1/2)	Z:(0,0,1/2)	$Z_{1}^{S}(4)⊕Z_{2}^{S}(4) $
Δ:(0,v,0)	Δ:(0,v,0)	$Δ_{1}^{S}(2)⊕Δ_{2}^{S}(2)⊕Δ_{3}^{S}(2)⊕Δ_{4}^{S}(2) $
Λ:(0,0,w)	Λ:(0,0,w)	$Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2)⊕Λ_{3}^{S}(2)⊕Λ_{4}^{S}(2) $
S:(u,u,1/2)	W:(u,u,1/2)	$2(S)W_{1}^{S}(2)⊕2(S)W_{2}^{S}(2) $
Σ:(u,u,0)	V:(u,u,0)	$2(Σ)V_{1}^{S}(2)⊕2(Σ)V_{2}^{S}(2) $
T:(u,1/2,1/2)	E:(u,1/2,1/2)	$2(T)E_{1}^{S}(4) $
U:(0,v,1/2)	B:(0,v,1/2)	$2(U)B_{1}^{S}(4) $
V:(1/2,1/2,w)	Q:(1/2,1/2,w)	$(V)Q_{1}^{S}(2)⊕(V)Q_{2}^{S}(2)⊕(V)Q_{3}^{S}(2)⊕(V)Q_{4}^{S}(2) $
W:(0,1/2,w)	H:(0,1/2,w)	$(W)H_{1}^{S}(2)⊕(W)H_{2}^{S}(2)⊕(W)H_{3}^{S}(2)⊕(W)H_{4}^{S}(2) $
Y:(u,1/2,0)	C:(u,1/2,0)	$2(Y)C_{3}^{S}(2)⊕2(Y)C_{4}^{S}(2) $
B:(0,v,w)	K:(0,v,w)	$2(B)K_{1}^{S}(2)⊕2(B)K_{2}^{S}(2) $
C:(u,u,w)	GP:(u,u,w)	$4(C)GP_{1}^{S}(2) $
D:(u,v,0)	V:(u,v,0)	$2(D)V_{1}^{S}(2)⊕2(D)V_{2}^{S}(2) $
E:(u,v,1/2)	W:(u,v,1/2)	$2(E)W_{1}^{S}(2)⊕2(E)W_{2}^{S}(2) $
F:(u,1/2,w)	N:(u,1/2,w)	$2(F)N_{1}^{S}(2)⊕2(F)N_{2}^{S}(2) $
GP:(u,v,w)	GP:(u,v,w)	$4GP_{1}^{S}(2) $
BA:(-v,0,w)	M:(-v,0,-w)	$2(BA)M_{1}^{S}(2)⊕2(BA)M_{2}^{S}(2) $
DU:(-v,0,0)	Σ:(-v,0,0)	$2(DU)Σ_{1}^{S}(2)⊕2(DU)Σ_{2}^{S}(2) $
FA:(-1/2,-u,w)	L:(-1/2,u,-w)	$2(FA)L_{1}^{S}(2)⊕2(FA)L_{2}^{S}(2) $
TA:(-1/2,-u,1/2)	P:(-1/2,u,-1/2)	$2(TA)P_{1}^{S}(4) $
UA:(-v,0,1/2)	A:(-v,0,-1/2)	$2(UA)A_{1}^{S}(4) $
WA:(-1/2,0,w)	G:(-1/2,0,-w)	$(WA)G_{1}^{S}(2)⊕(WA)G_{2}^{S}(2)⊕(WA)G_{3}^{S}(2)⊕(WA)G_{4}^{S}(2) $
YA:(-1/2,-u,0)	D:(-1/2,u,0)	$(YA)D_{1}^{S}(2)⊕(YA)D_{2}^{S}(2)⊕(YA)D_{3}^{S}(2)⊕(YA)D_{4}^{S}(2) $

8i

k-vector	k-vector-G↑	A↑G(8)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S,+}(2)⊕Γ_{1}^{S,-}(2)⊕Γ_{2}^{S,+}(2)⊕Γ_{2}^{S,-}(2) $
A:(1/2,1/2,1/2)	R:(1/2,1/2,1/2)	$2(A)R_{2}^{S}(4) $
M:(1/2,1/2,0)	S:(1/2,1/2,0)	$(M)S_{3}^{S,+}(2)⊕(M)S_{3}^{S,-}(2)⊕(M)S_{4}^{S,+}(2)⊕(M)S_{4}^{S,-}(2) $
R:(0,1/2,1/2)	T:(0,1/2,1/2)	$2(R)T_{2}^{S}(4) $
RA:(-1/2,0,1/2)	U:(1/2,0,1/2)	$2(RA)U_{1}^{S}(4) $
X:(0,1/2,0)	Y:(0,1/2,0)	$(X)Y_{3}^{S,+}(2)⊕(X)Y_{3}^{S,-}(2)⊕(X)Y_{4}^{S,+}(2)⊕(X)Y_{4}^{S,-}(2) $
XA:(-1/2,0,0)	X:(1/2,0,0)	$(XA)X_{1}^{S,+}(2)⊕(XA)X_{1}^{S,-}(2)⊕(XA)X_{2}^{S,+}(2)⊕(XA)X_{2}^{S,-}(2) $
Z:(0,0,1/2)	Z:(0,0,1/2)	$2Z_{1}^{S}(4) $
Δ:(0,v,0)	Δ:(0,v,0)	$Δ_{1}^{S}(2)⊕Δ_{2}^{S}(2)⊕Δ_{3}^{S}(2)⊕Δ_{4}^{S}(2) $
Λ:(0,0,w)	Λ:(0,0,w)	$2Λ_{1}^{S}(2)⊕2Λ_{2}^{S}(2) $
S:(u,u,1/2)	W:(u,u,1/2)	$2(S)W_{1}^{S}(2)⊕2(S)W_{2}^{S}(2) $
Σ:(u,u,0)	V:(u,u,0)	$2(Σ)V_{1}^{S}(2)⊕2(Σ)V_{2}^{S}(2) $
T:(u,1/2,1/2)	E:(u,1/2,1/2)	$2(T)E_{1}^{S}(4) $
U:(0,v,1/2)	B:(0,v,1/2)	$2(U)B_{1}^{S}(4) $
V:(1/2,1/2,w)	Q:(1/2,1/2,w)	$2(V)Q_{3}^{S}(2)⊕2(V)Q_{4}^{S}(2) $
W:(0,1/2,w)	H:(0,1/2,w)	$2(W)H_{3}^{S}(2)⊕2(W)H_{4}^{S}(2) $
Y:(u,1/2,0)	C:(u,1/2,0)	$(Y)C_{1}^{S}(2)⊕(Y)C_{2}^{S}(2)⊕(Y)C_{3}^{S}(2)⊕(Y)C_{4}^{S}(2) $
B:(0,v,w)	K:(0,v,w)	$2(B)K_{1}^{S}(2)⊕2(B)K_{2}^{S}(2) $
C:(u,u,w)	GP:(u,u,w)	$4(C)GP_{1}^{S}(2) $
D:(u,v,0)	V:(u,v,0)	$2(D)V_{1}^{S}(2)⊕2(D)V_{2}^{S}(2) $
E:(u,v,1/2)	W:(u,v,1/2)	$2(E)W_{1}^{S}(2)⊕2(E)W_{2}^{S}(2) $
F:(u,1/2,w)	N:(u,1/2,w)	$2(F)N_{1}^{S}(2)⊕2(F)N_{2}^{S}(2) $
GP:(u,v,w)	GP:(u,v,w)	$4GP_{1}^{S}(2) $
BA:(-v,0,w)	M:(-v,0,-w)	$2(BA)M_{1}^{S}(2)⊕2(BA)M_{2}^{S}(2) $
DU:(-v,0,0)	Σ:(-v,0,0)	$(DU)Σ_{1}^{S}(2)⊕(DU)Σ_{2}^{S}(2)⊕(DU)Σ_{3}^{S}(2)⊕(DU)Σ_{4}^{S}(2) $
FA:(-1/2,-u,w)	L:(-1/2,u,-w)	$2(FA)L_{1}^{S}(2)⊕2(FA)L_{2}^{S}(2) $
TA:(-1/2,-u,1/2)	P:(-1/2,u,-1/2)	$2(TA)P_{1}^{S}(4) $
UA:(-v,0,1/2)	A:(-v,0,-1/2)	$2(UA)A_{1}^{S}(4) $
WA:(-1/2,0,w)	G:(-1/2,0,-w)	$2(WA)G_{1}^{S}(2)⊕2(WA)G_{2}^{S}(2) $
YA:(-1/2,-u,0)	D:(-1/2,u,0)	$(YA)D_{1}^{S}(2)⊕(YA)D_{2}^{S}(2)⊕(YA)D_{3}^{S}(2)⊕(YA)D_{4}^{S}(2) $

8m

k-vector	k-vector-G↑	A'↑G(8)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S,+}(2)⊕Γ_{2}^{S,+}(2)⊕Γ_{3}^{S,-}(2)⊕Γ_{4}^{S,-}(2) $
A:(1/2,1/2,1/2)	R:(1/2,1/2,1/2)	$(A)R_{1}^{S}(4)⊕(A)R_{2}^{S}(4) $
M:(1/2,1/2,0)	S:(1/2,1/2,0)	$(M)S_{1}^{S,+}(2)⊕(M)S_{2}^{S,+}(2)⊕(M)S_{3}^{S,-}(2)⊕(M)S_{4}^{S,-}(2) $
R:(0,1/2,1/2)	T:(0,1/2,1/2)	$(R)T_{1}^{S}(4)⊕(R)T_{2}^{S}(4) $
RA:(-1/2,0,1/2)	U:(1/2,0,1/2)	$(RA)U_{1}^{S}(4)⊕(RA)U_{2}^{S}(4) $
X:(0,1/2,0)	Y:(0,1/2,0)	$(X)Y_{1}^{S,+}(2)⊕(X)Y_{2}^{S,+}(2)⊕(X)Y_{3}^{S,-}(2)⊕(X)Y_{4}^{S,-}(2) $
XA:(-1/2,0,0)	X:(1/2,0,0)	$(XA)X_{1}^{S,+}(2)⊕(XA)X_{2}^{S,+}(2)⊕(XA)X_{3}^{S,-}(2)⊕(XA)X_{4}^{S,-}(2) $
Z:(0,0,1/2)	Z:(0,0,1/2)	$Z_{1}^{S}(4)⊕Z_{2}^{S}(4) $
Δ:(0,v,0)	Δ:(0,v,0)	$2Δ_{1}^{S}(2)⊕2Δ_{4}^{S}(2) $
Λ:(0,0,w)	Λ:(0,0,w)	$Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2)⊕Λ_{3}^{S}(2)⊕Λ_{4}^{S}(2) $
S:(u,u,1/2)	W:(u,u,1/2)	$2(S)W_{1}^{S}(2)⊕2(S)W_{2}^{S}(2) $
Σ:(u,u,0)	V:(u,u,0)	$4(Σ)V_{1}^{S}(2) $
T:(u,1/2,1/2)	E:(u,1/2,1/2)	$2(T)E_{1}^{S}(4) $
U:(0,v,1/2)	B:(0,v,1/2)	$2(U)B_{1}^{S}(4) $
V:(1/2,1/2,w)	Q:(1/2,1/2,w)	$(V)Q_{1}^{S}(2)⊕(V)Q_{2}^{S}(2)⊕(V)Q_{3}^{S}(2)⊕(V)Q_{4}^{S}(2) $
W:(0,1/2,w)	H:(0,1/2,w)	$(W)H_{1}^{S}(2)⊕(W)H_{2}^{S}(2)⊕(W)H_{3}^{S}(2)⊕(W)H_{4}^{S}(2) $
Y:(u,1/2,0)	C:(u,1/2,0)	$2(Y)C_{1}^{S}(2)⊕2(Y)C_{4}^{S}(2) $
B:(0,v,w)	K:(0,v,w)	$2(B)K_{1}^{S}(2)⊕2(B)K_{2}^{S}(2) $
C:(u,u,w)	GP:(u,u,w)	$4(C)GP_{1}^{S}(2) $
D:(u,v,0)	V:(u,v,0)	$4(D)V_{1}^{S}(2) $
E:(u,v,1/2)	W:(u,v,1/2)	$2(E)W_{1}^{S}(2)⊕2(E)W_{2}^{S}(2) $
F:(u,1/2,w)	N:(u,1/2,w)	$2(F)N_{1}^{S}(2)⊕2(F)N_{2}^{S}(2) $
GP:(u,v,w)	GP:(u,v,w)	$4GP_{1}^{S}(2) $
BA:(-v,0,w)	M:(-v,0,-w)	$2(BA)M_{1}^{S}(2)⊕2(BA)M_{2}^{S}(2) $
DU:(-v,0,0)	Σ:(-v,0,0)	$2(DU)Σ_{1}^{S}(2)⊕2(DU)Σ_{4}^{S}(2) $
FA:(-1/2,-u,w)	L:(-1/2,u,-w)	$2(FA)L_{1}^{S}(2)⊕2(FA)L_{2}^{S}(2) $
TA:(-1/2,-u,1/2)	P:(-1/2,u,-1/2)	$2(TA)P_{1}^{S}(4) $
UA:(-v,0,1/2)	A:(-v,0,-1/2)	$2(UA)A_{1}^{S}(4) $
WA:(-1/2,0,w)	G:(-1/2,0,-w)	$(WA)G_{1}^{S}(2)⊕(WA)G_{2}^{S}(2)⊕(WA)G_{3}^{S}(2)⊕(WA)G_{4}^{S}(2) $
YA:(-1/2,-u,0)	D:(-1/2,u,0)	$2(YA)D_{1}^{S}(2)⊕2(YA)D_{4}^{S}(2) $

16n

k-vector	k-vector-G↑	A↑G(16)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S,+}(2)⊕Γ_{1}^{S,-}(2)⊕Γ_{2}^{S,+}(2)⊕Γ_{2}^{S,-}(2)⊕Γ_{3}^{S,+}(2)⊕Γ_{3}^{S,-}(2)⊕Γ_{4}^{S,+}(2)⊕Γ_{4}^{S,-}(2) $
A:(1/2,1/2,1/2)	R:(1/2,1/2,1/2)	$2(A)R_{1}^{S}(4)⊕2(A)R_{2}^{S}(4) $
M:(1/2,1/2,0)	S:(1/2,1/2,0)	$(M)S_{1}^{S,+}(2)⊕(M)S_{1}^{S,-}(2)⊕(M)S_{2}^{S,+}(2)⊕(M)S_{2}^{S,-}(2)⊕(M)S_{3}^{S,+}(2)⊕(M)S_{3}^{S,-}(2)⊕(M)S_{4}^{S,+}(2)⊕(M)S_{4}^{S,-}(2) $
R:(0,1/2,1/2)	T:(0,1/2,1/2)	$2(R)T_{1}^{S}(4)⊕2(R)T_{2}^{S}(4) $
RA:(-1/2,0,1/2)	U:(1/2,0,1/2)	$2(RA)U_{1}^{S}(4)⊕2(RA)U_{2}^{S}(4) $
X:(0,1/2,0)	Y:(0,1/2,0)	$(X)Y_{1}^{S,+}(2)⊕(X)Y_{1}^{S,-}(2)⊕(X)Y_{2}^{S,+}(2)⊕(X)Y_{2}^{S,-}(2)⊕(X)Y_{3}^{S,+}(2)⊕(X)Y_{3}^{S,-}(2)⊕(X)Y_{4}^{S,+}(2)⊕(X)Y_{4}^{S,-}(2) $
XA:(-1/2,0,0)	X:(1/2,0,0)	$(XA)X_{1}^{S,+}(2)⊕(XA)X_{1}^{S,-}(2)⊕(XA)X_{2}^{S,+}(2)⊕(XA)X_{2}^{S,-}(2)⊕(XA)X_{3}^{S,+}(2)⊕(XA)X_{3}^{S,-}(2)⊕(XA)X_{4}^{S,+}(2)⊕(XA)X_{4}^{S,-}(2) $
Z:(0,0,1/2)	Z:(0,0,1/2)	$2Z_{1}^{S}(4)⊕2Z_{2}^{S}(4) $
Δ:(0,v,0)	Δ:(0,v,0)	$2Δ_{1}^{S}(2)⊕2Δ_{2}^{S}(2)⊕2Δ_{3}^{S}(2)⊕2Δ_{4}^{S}(2) $
Λ:(0,0,w)	Λ:(0,0,w)	$2Λ_{1}^{S}(2)⊕2Λ_{2}^{S}(2)⊕2Λ_{3}^{S}(2)⊕2Λ_{4}^{S}(2) $
S:(u,u,1/2)	W:(u,u,1/2)	$4(S)W_{1}^{S}(2)⊕4(S)W_{2}^{S}(2) $
Σ:(u,u,0)	V:(u,u,0)	$4(Σ)V_{1}^{S}(2)⊕4(Σ)V_{2}^{S}(2) $
T:(u,1/2,1/2)	E:(u,1/2,1/2)	$4(T)E_{1}^{S}(4) $
U:(0,v,1/2)	B:(0,v,1/2)	$4(U)B_{1}^{S}(4) $
V:(1/2,1/2,w)	Q:(1/2,1/2,w)	$2(V)Q_{1}^{S}(2)⊕2(V)Q_{2}^{S}(2)⊕2(V)Q_{3}^{S}(2)⊕2(V)Q_{4}^{S}(2) $
W:(0,1/2,w)	H:(0,1/2,w)	$2(W)H_{1}^{S}(2)⊕2(W)H_{2}^{S}(2)⊕2(W)H_{3}^{S}(2)⊕2(W)H_{4}^{S}(2) $
Y:(u,1/2,0)	C:(u,1/2,0)	$2(Y)C_{1}^{S}(2)⊕2(Y)C_{2}^{S}(2)⊕2(Y)C_{3}^{S}(2)⊕2(Y)C_{4}^{S}(2) $
B:(0,v,w)	K:(0,v,w)	$4(B)K_{1}^{S}(2)⊕4(B)K_{2}^{S}(2) $
C:(u,u,w)	GP:(u,u,w)	$8(C)GP_{1}^{S}(2) $
D:(u,v,0)	V:(u,v,0)	$4(D)V_{1}^{S}(2)⊕4(D)V_{2}^{S}(2) $
E:(u,v,1/2)	W:(u,v,1/2)	$4(E)W_{1}^{S}(2)⊕4(E)W_{2}^{S}(2) $
F:(u,1/2,w)	N:(u,1/2,w)	$4(F)N_{1}^{S}(2)⊕4(F)N_{2}^{S}(2) $
GP:(u,v,w)	GP:(u,v,w)	$8GP_{1}^{S}(2) $
BA:(-v,0,w)	M:(-v,0,-w)	$4(BA)M_{1}^{S}(2)⊕4(BA)M_{2}^{S}(2) $
DU:(-v,0,0)	Σ:(-v,0,0)	$2(DU)Σ_{1}^{S}(2)⊕2(DU)Σ_{2}^{S}(2)⊕2(DU)Σ_{3}^{S}(2)⊕2(DU)Σ_{4}^{S}(2) $
FA:(-1/2,-u,w)	L:(-1/2,u,-w)	$4(FA)L_{1}^{S}(2)⊕4(FA)L_{2}^{S}(2) $
TA:(-1/2,-u,1/2)	P:(-1/2,u,-1/2)	$4(TA)P_{1}^{S}(4) $
UA:(-v,0,1/2)	A:(-v,0,-1/2)	$4(UA)A_{1}^{S}(4) $
WA:(-1/2,0,w)	G:(-1/2,0,-w)	$2(WA)G_{1}^{S}(2)⊕2(WA)G_{2}^{S}(2)⊕2(WA)G_{3}^{S}(2)⊕2(WA)G_{4}^{S}(2) $
YA:(-1/2,-u,0)	D:(-1/2,u,0)	$2(YA)D_{1}^{S}(2)⊕2(YA)D_{2}^{S}(2)⊕2(YA)D_{3}^{S}(2)⊕2(YA)D_{4}^{S}(2) $