Hahn Th. (ed.) International tables for crystallography. Vol. A. Space-group symmetry
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CONTINUED
(from pages 625 and 627)
No. 203 Fd
¯
3
ORIGIN CHOICE 1
Symmetry of special projections
Along [001] c2mm
a
=
1
2
ab
=
1
2
b
Origin at 0,0,z
Along [111] p6
a
=
1
6
(2a − b− c) b
=
1
6
(−a + 2b − c)
Origin at x,x , x
Along [110] c2mm
a
=
1
2
(−a + b) b
= c
Origin at x,x ,
1
8
ORIGIN CHOICE 2
Symmetry of special projections
Along [001] c2mm
a
=
1
2
ab
=
1
2
b
Origin at
1
8
,
1
8
,z
Along [111] p6
a
=
1
6
(2a − b− c) b
=
1
6
(−a + 2b − c)
Origin at x,x, x
Along [110] c2mm
a
=
1
2
(−a + b) b
= c
Origin at x,x , 0
ORIGIN CHOICES 1 AND 2
Maximal non-isomorphic subgroups
I
[2] F 23 (196) (1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12)+
[3] Fd1(Fddd, 70) (1; 2; 3; 4; 13; 14; 15; 16)+
⎧
⎪
⎪
⎨
⎪
⎪
⎩
[4] F 1
¯
3(R
¯
3, 148) (1; 5; 9; 13; 17; 21)+
[4] F 1
¯
3(R
¯
3, 148) (1; 6; 12; 13; 18; 24)+
[4] F 1
¯
3(R
¯
3, 148) (1; 7; 10; 13; 19; 22)+
[4] F 1
¯
3(R
¯
3, 148) (1; 8; 11; 13; 20; 23)+
IIa none
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[27] Fd
¯
3(a
= 3a,b
= 3b,c
= 3c) (203)
Minimal non-isomorphic supergroups
I
[2] Fd
¯
3m (227); [2] Fd
¯
3c (228)
II [2] Pn
¯
3(a
=
1
2
a,b
=
1
2
b,c
=
1
2
c) (201)
623
Fd
¯
3 T
4
h
m
¯
3 Cubic
No. 203 F 2/d
¯
3
Patterson symmetry Fm
¯
3
ORIGIN CHOICE 2
Origin at centre (
¯
3),at
1
8
,
1
8
,
1
8
from 23
Asymmetric unit −
1
8
≤ x ≤
3
8
; −
1
8
≤ y ≤
1
8
; −
3
8
≤ z ≤
1
8
; y ≤ min(x,
1
4
− x); −y −
1
4
≤ z ≤ y
Vertices −
1
8
,−
1
8
,−
1
8
3
8
,−
1
8
,−
1
8
1
8
,
1
8
,
1
8
1
8
,
1
8
,−
3
8
626
CONTINUED No. 203 Fd
¯
3
Symmetry operations
For (0,0,0)+ set
(1) 1 (2) 2
3
8
,
3
8
,z (3) 2
3
8
,y,
3
8
(4) 2 x,
3
8
,
3
8
(5) 3
+
x,x,x (6) 3
+
¯x,x+
3
4
, ¯x (7) 3
+
x+
3
4
, ¯x, ¯x (8) 3
+
¯x+
3
4
, ¯x+
3
4
,x
(9) 3
−
x,x,x (10) 3
−
x+
3
4
, ¯x, ¯x (11) 3
−
¯x+
3
4
, ¯x+
3
4
,x (12) 3
−
¯x,x+
3
4
, ¯x
(13)
¯
10,0,0 (14) d(
1
4
,
1
4
,0) x,y,0 (15) d(
1
4
,0,
1
4
) x,0,z (16) d(0 ,
1
4
,
1
4
) 0,y,z
(17)
¯
3
+
x,x,x;0,0,0 (18)
¯
3
+
¯x−
1
2
,x+
1
4
, ¯x; −
1
4
,0,
1
4
(19)
¯
3
+
x−
1
4
, ¯x+
1
2
, ¯x;0,
1
4
,−
1
4
(20)
¯
3
+
¯x+
1
4
, ¯x−
1
4
,x;
1
4
,−
1
4
,0
(21)
¯
3
−
x,x,x;0,0,0 (22)
¯
3
−
x+
1
4
, ¯x−
1
2
, ¯x;0,−
1
4
,
1
4
(23)
¯
3
−
¯x−
1
4
, ¯x+
1
4
,x; −
1
4
,
1
4
,0 (24)
¯
3
−
¯x+
1
2
,x−
1
4
, ¯x;
1
4
,0,−
1
4
For (0,
1
2
,
1
2
)+ set
(1) t(0,
1
2
,
1
2
) (2) 2(0,0,
1
2
)
3
8
,
1
8
,z (3) 2(0,
1
2
,0)
3
8
,y,
1
8
(4) 2 x,
1
8
,
1
8
(5) 3
+
(
1
3
,
1
3
,
1
3
) x−
1
3
,x−
1
6
,x (6) 3
+
¯x,x+
1
4
, ¯x (7) 3
+
x+
3
4
, ¯x−
1
2
, ¯x (8) 3
+
(
1
3
,
1
3
,−
1
3
) ¯x+
5
12
, ¯x+
7
12
,x
(9) 3
−
(
1
3
,
1
3
,
1
3
) x−
1
6
,x+
1
6
,x (10) 3
−
x+
1
4
, ¯x+
1
2
, ¯x (11) 3
−
¯x+
1
4
, ¯x+
1
4
,x (12) 3
−
(
1
3
,−
1
3
,
1
3
) ¯x−
1
6
,x+
7
12
, ¯x
(13)
¯
10,
1
4
,
1
4
(14) d(
1
4
,
3
4
,0) x,y,
1
4
(15) d(
1
4
,0,
3
4
) x,
1
4
,z (16) d(0,
3
4
,
3
4
) 0,y,z
(17)
¯
3
+
x,x+
1
2
,x;0,
1
2
,0 (18)
¯
3
+
¯x−
3
2
,x+
3
4
, ¯x; −
3
4
,0,
3
4
(19)
¯
3
+
x−
1
4
, ¯x+1, ¯x;0,
3
4
,−
1
4
(20)
¯
3
+
¯x+
5
4
, ¯x+
1
4
,x;
3
4
,−
1
4
,
1
2
(21)
¯
3
−
x−
1
2
,x−
1
2
,x;0,0,
1
2
(22)
¯
3
−
x+
3
4
, ¯x−1, ¯x;0,−
1
4
,
3
4
(23)
¯
3
−
¯x−
3
4
, ¯x+
3
4
,x; −
3
4
,
3
4
,0 (24)
¯
3
−
¯x+1,x+
1
4
, ¯x;
3
4
,
1
2
,−
1
4
For (
1
2
,0,
1
2
)+ set
(1) t(
1
2
,0,
1
2
) (2) 2(0,0,
1
2
)
1
8
,
3
8
,z (3) 2
1
8
,y,
1
8
(4) 2(
1
2
,0,0) x,
3
8
,
1
8
(5) 3
+
(
1
3
,
1
3
,
1
3
) x+
1
6
,x−
1
6
,x (6) 3
+
¯x+
1
2
,x+
1
4
, ¯x (7) 3
+
(−
1
3
,
1
3
,
1
3
) x+
7
12
, ¯x−
1
6
, ¯x (8) 3
+
¯x+
1
4
, ¯x+
1
4
,x
(9) 3
−
(
1
3
,
1
3
,
1
3
) x−
1
6
,x−
1
3
,x (10) 3
−
x+
1
4
, ¯x, ¯x (11) 3
−
(
1
3
,
1
3
,−
1
3
) ¯x+
7
12
, ¯x+
5
12
,x (12) 3
−
¯x−
1
2
,x+
3
4
, ¯x
(13)
¯
1
1
4
,0,
1
4
(14) d(
3
4
,
1
4
,0) x,y,
1
4
(15) d(
3
4
,0,
3
4
) x,0,z (16) d(0 ,
1
4
,
3
4
)
1
4
,y,z
(17)
¯
3
+
x−
1
2
,x−
1
2
,x;0,0,
1
2
(18)
¯
3
+
¯x−1,x+
3
4
, ¯x; −
1
4
,0,
3
4
(19)
¯
3
+
x+
1
4
, ¯x+1, ¯x;
1
2
,
3
4
,−
1
4
(20)
¯
3
+
¯x+
3
4
, ¯x−
3
4
,x;
3
4
,−
3
4
,0
(21)
¯
3
−
x+
1
2
,x,x;
1
2
,0,0 (22)
¯
3
−
x+
3
4
, ¯x−
3
2
, ¯x;0,−
3
4
,
3
4
(23)
¯
3
−
¯x+
1
4
, ¯x+
5
4
,x; −
1
4
,
3
4
,
1
2
(24)
¯
3
−
¯x+1,x−
1
4
, ¯x;
3
4
,0,−
1
4
For (
1
2
,
1
2
,0)+ set
(1) t(
1
2
,
1
2
,0) (2) 2
1
8
,
1
8
,z (3) 2(0,
1
2
,0)
1
8
,y,
3
8
(4) 2(
1
2
,0,0) x,
1
8
,
3
8
(5) 3
+
(
1
3
,
1
3
,
1
3
) x+
1
6
,x+
1
3
,x (6) 3
+
(
1
3
,−
1
3
,
1
3
) ¯x+
1
6
,x+
5
12
, ¯x (7) 3
+
x+
1
4
, ¯x, ¯x (8) 3
+
¯x+
1
4
, ¯x+
3
4
,x
(9) 3
−
(
1
3
,
1
3
,
1
3
) x+
1
3
,x+
1
6
,x (10) 3
−
(−
1
3
,
1
3
,
1
3
) x+
5
12
, ¯x+
1
6
, ¯x (11) 3
−
¯x+
3
4
, ¯x+
1
4
,x (12) 3
−
¯x,x+
1
4
, ¯x
(13)
¯
1
1
4
,
1
4
,0 (14) d(
3
4
,
3
4
,0) x,y,0 (15) d(
3
4
,0,
1
4
) x,
1
4
,z (16) d(0,
3
4
,
1
4
)
1
4
,y,z
(17)
¯
3
+
x+
1
2
,x,x;
1
2
,0,0 (18)
¯
3
+
¯x−1,x+
5
4
, ¯x; −
1
4
,
1
2
,
3
4
(19)
¯
3
+
x−
3
4
, ¯x+
3
2
, ¯x;0,
3
4
,−
3
4
(20)
¯
3
+
¯x+
3
4
, ¯x−
1
4
,x;
3
4
,−
1
4
,0
(21)
¯
3
−
x,x+
1
2
,x;0,
1
2
,0 (22)
¯
3
−
x+
5
4
, ¯x−1, ¯x;
1
2
,−
1
4
,
3
4
(23)
¯
3
−
¯x−
1
4
, ¯x+
3
4
,x; −
1
4
,
3
4
,0 (24)
¯
3
−
¯x+
3
2
,x−
3
4
, ¯x;
3
4
,0,−
3
4
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); t(0,
1
2
,
1
2
); t(
1
2
,0,
1
2
); (2); (3); (5); (13)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates
(0,0, 0)+ (0,
1
2
,
1
2
)+ (
1
2
,0,
1
2
)+ (
1
2
,
1
2
,0)+
Reflection conditions
h,k,l cyclically permutable
General:
96 g 1(1)x,y, z (2) ¯x +
3
4
, ¯y +
3
4
,z (3) ¯x +
3
4
,y, ¯z +
3
4
(4) x, ¯y +
3
4
, ¯z +
3
4
(5) z,x,y (6) z, ¯x +
3
4
, ¯y +
3
4
(7) ¯z +
3
4
, ¯x +
3
4
,y (8) ¯z +
3
4
,x, ¯y +
3
4
(9) y,z,x (10) ¯y +
3
4
,z, ¯x +
3
4
(11) y, ¯z +
3
4
, ¯x +
3
4
(12) ¯y +
3
4
, ¯z +
3
4
,x
(13) ¯x, ¯y, ¯z (14) x +
1
4
,y +
1
4
, ¯z (15) x +
1
4
, ¯y, z +
1
4
(16) ¯x,y +
1
4
,z +
1
4
(17) ¯z, ¯x, ¯y (18) ¯z, x +
1
4
,y +
1
4
(19) z +
1
4
,x +
1
4
, ¯y (20) z+
1
4
, ¯x,y +
1
4
(21) ¯y, ¯z, ¯x (22) y +
1
4
, ¯z, x +
1
4
(23) ¯y,z +
1
4
,x +
1
4
(24) y +
1
4
,z +
1
4
, ¯x
hkl : h + k,h + l,k + l = 2n
0kl : k + l = 4n, k,l = 2n
hhl : h + l = 2n
h00 : h = 4n
Special: as above, plus
48 f 2 .. x,
1
8
,
1
8
¯x +
3
4
,
5
8
,
1
8
1
8
,x,
1
8
1
8
, ¯x +
3
4
,
5
8
1
8
,
1
8
,x
5
8
,
1
8
, ¯x +
3
4
¯x,
7
8
,
7
8
x +
1
4
,
3
8
,
7
8
7
8
, ¯x,
7
8
7
8
,x +
1
4
,
3
8
7
8
,
7
8
, ¯x
3
8
,
7
8
,x +
1
4
hkl : h = 2n + 1
or h + k + l = 4n
32 e . 3 . x, x,x ¯x +
3
4
, ¯x +
3
4
,x ¯x +
3
4
,x, ¯x +
3
4
x, ¯x +
3
4
, ¯x +
3
4
¯x, ¯x, ¯xx+
1
4
,x +
1
4
, ¯xx+
1
4
, ¯x, x +
1
4
¯x,x +
1
4
,x +
1
4
no extra conditions
16 d .
¯
3 .
1
2
,
1
2
,
1
2
1
4
,
1
4
,
1
2
1
4
,
1
2
,
1
4
1
2
,
1
4
,
1
4
16 c .
¯
3 . 0,0,0
3
4
,
3
4
,0
3
4
,0,
3
4
0,
3
4
,
3
4
hkl : h = 2n + 1
or h, k, l = 4n + 2
or h, k, l = 4n
8 b 23.
5
8
,
5
8
,
5
8
3
8
,
3
8
,
3
8
8 a 23.
1
8
,
1
8
,
1
8
7
8
,
7
8
,
7
8
hkl : h = 2n + 1
or h + k + l = 4n
(Continued on page 623)
627
CONTINUED
(from pages 625 and 627)
No. 203 Fd
¯
3
ORIGIN CHOICE 1
Symmetry of special projections
Along [001] c2mm
a
=
1
2
ab
=
1
2
b
Origin at 0,0,z
Along [111] p6
a
=
1
6
(2a − b− c) b
=
1
6
(−a + 2b − c)
Origin at x,x , x
Along [110] c2mm
a
=
1
2
(−a + b) b
= c
Origin at x,x ,
1
8
ORIGIN CHOICE 2
Symmetry of special projections
Along [001] c2mm
a
=
1
2
ab
=
1
2
b
Origin at
1
8
,
1
8
,z
Along [111] p6
a
=
1
6
(2a − b− c) b
=
1
6
(−a + 2b − c)
Origin at x,x, x
Along [110] c2mm
a
=
1
2
(−a + b) b
= c
Origin at x,x , 0
ORIGIN CHOICES 1 AND 2
Maximal non-isomorphic subgroups
I
[2] F 23 (196) (1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12)+
[3] Fd1(Fddd, 70) (1; 2; 3; 4; 13; 14; 15; 16)+
⎧
⎪
⎪
⎨
⎪
⎪
⎩
[4] F 1
¯
3(R
¯
3, 148) (1; 5; 9; 13; 17; 21)+
[4] F 1
¯
3(R
¯
3, 148) (1; 6; 12; 13; 18; 24)+
[4] F 1
¯
3(R
¯
3, 148) (1; 7; 10; 13; 19; 22)+
[4] F 1
¯
3(R
¯
3, 148) (1; 8; 11; 13; 20; 23)+
IIa none
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[27] Fd
¯
3(a
= 3a,b
= 3b,c
= 3c) (203)
Minimal non-isomorphic supergroups
I
[2] Fd
¯
3m (227); [2] Fd
¯
3c (228)
II [2] Pn
¯
3(a
=
1
2
a,b
=
1
2
b,c
=
1
2
c) (201)
623
Im
¯
3 T
5
h
m
¯
3 Cubic
No. 204 I 2/m
¯
3
Patterson symmetry Im
¯
3
Origin at centre (m
¯
3)
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
2
;0≤ z ≤
1
2
; y ≤ x; z ≤ y
Vertices 0,0, 0
1
2
,0, 0
1
2
,
1
2
,0
1
2
,
1
2
,
1
2
Symmetry operations
For (0,0,0)+ set
(1) 1 (2) 2 0,0,z (3) 2 0,y,0(4)2x,0,0
(5) 3
+
x,x,x (6) 3
+
¯x,x, ¯x (7) 3
+
x, ¯x, ¯x (8) 3
+
¯x, ¯x,x
(9) 3
−
x,x,x (10) 3
−
x, ¯x, ¯x (11) 3
−
¯x, ¯x,x (12) 3
−
¯x,x, ¯x
(13)
¯
10,0,0 (14) mx,y,0 (15) mx,0,z (16) m 0,y,z
(17)
¯
3
+
x,x,x;0,0,0 (18)
¯
3
+
¯x,x, ¯x;0,0,0 (19)
¯
3
+
x, ¯x, ¯x;0,0,0 (20)
¯
3
+
¯x, ¯x,x;0,0,0
(21)
¯
3
−
x,x,x;0,0,0 (22)
¯
3
−
x, ¯x, ¯x;0,0,0 (23)
¯
3
−
¯x, ¯x,x ;0,0,0 (24)
¯
3
−
¯x,x, ¯x;0,0,0
For (
1
2
,
1
2
,
1
2
)+ set
(1) t(
1
2
,
1
2
,
1
2
) (2) 2(0,0,
1
2
)
1
4
,
1
4
,z (3) 2(0,
1
2
,0)
1
4
,y,
1
4
(4) 2(
1
2
,0,0) x,
1
4
,
1
4
(5) 3
+
(
1
2
,
1
2
,
1
2
) x,x,x (6) 3
+
(
1
6
,−
1
6
,
1
6
) ¯x+
1
3
,x+
1
3
, ¯x (7) 3
+
(−
1
6
,
1
6
,
1
6
) x+
2
3
, ¯x−
1
3
, ¯x (8) 3
+
(
1
6
,
1
6
,−
1
6
) ¯x+
1
3
, ¯x+
2
3
,x
(9) 3
−
(
1
2
,
1
2
,
1
2
) x,x,x (10) 3
−
(−
1
6
,
1
6
,
1
6
) x+
1
3
, ¯x+
1
3
, ¯x (11) 3
−
(
1
6
,
1
6
,−
1
6
) ¯x+
2
3
, ¯x+
1
3
,x (12) 3
−
(
1
6
,−
1
6
,
1
6
) ¯x−
1
3
,x+
2
3
, ¯x
(13)
¯
1
1
4
,
1
4
,
1
4
(14) n(
1
2
,
1
2
,0) x,y,
1
4
(15) n(
1
2
,0,
1
2
) x,
1
4
,z (16) n(0,
1
2
,
1
2
)
1
4
,y,z
(17)
¯
3
+
x,x,x;
1
4
,
1
4
,
1
4
(18)
¯
3
+
¯x−1,x +1, ¯x; −
1
4
,
1
4
,
3
4
(19)
¯
3
+
x, ¯x+1, ¯x;
1
4
,
3
4
,−
1
4
(20)
¯
3
+
¯x+1, ¯x,x;
3
4
,−
1
4
,
1
4
(21)
¯
3
−
x,x,x;
1
4
,
1
4
,
1
4
(22)
¯
3
−
x+1, ¯x−1, ¯x;
1
4
,−
1
4
,
3
4
(23)
¯
3
−
¯x, ¯x+1,x; −
1
4
,
3
4
,
1
4
(24)
¯
3
−
¯x+1,x, ¯x;
3
4
,
1
4
,−
1
4
628
International Tables for Crystallography (2006). Vol. A, Space group 204, pp. 628–629.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 204 Im
¯
3
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); t(
1
2
,
1
2
,
1
2
); (2); (3); (5); (13)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates
(0,0, 0)+ (
1
2
,
1
2
,
1
2
)+
Reflection conditions
h,k,l cyclically permutable
General:
48 h 1(1)x,y, z (2) ¯x, ¯y,z (3) ¯x,y, ¯z (4) x, ¯y, ¯z
(5) z,x,y (6) z, ¯x, ¯y (7) ¯z, ¯x,y (8) ¯z,x, ¯y
(9) y,z,x (10) ¯y,z, ¯x (11) y, ¯z, ¯x (12) ¯y, ¯z,x
(13) ¯x, ¯y
, ¯z (14) x,y, ¯z (15) x, ¯y,z (16) ¯x,y, z
(17) ¯z, ¯x, ¯y (18) ¯z, x,y (19) z,x, ¯y (20) z, ¯x,y
(21) ¯y, ¯z, ¯x (22) y, ¯z,x (23) ¯y,z,x (24) y,z, ¯x
hkl : h + k + l = 2n
0kl : k + l = 2 n
hhl : l
= 2n
h00 : h = 2n
Special: as above, plus
24 gm.. 0, y,z 0, ¯y,z 0,y, ¯z 0, ¯y, ¯zz,0, yz,0, ¯y
¯z,0,y ¯z,0, ¯yy,z,0¯y , z,0 y, ¯z,0¯y, ¯z,0
no extra conditions
16 f . 3 . x,x,x ¯x, ¯x,x ¯x,
x, ¯xx, ¯x, ¯x
¯x, ¯x, ¯xx,x, ¯xx, ¯x,x ¯x,x, x
no extra conditions
12 emm2.. x,0,
1
2
¯x,0,
1
2
1
2
,x,0
1
2
, ¯x, 00,
1
2
,x 0,
1
2
, ¯x no extra conditions
12 dmm2.. x,0 , 0¯x,0,00,x,00, ¯x,00,0,x 0,0, ¯x no extra conditions
8 c .
¯
3 .
1
4
,
1
4
,
1
4
3
4
,
3
4
,
1
4
3
4
,
1
4
,
3
4
1
4
,
3
4
,
3
4
hkl : k,l = 2n
6 bmmm.. 0,
1
2
,
1
2
1
2
,0,
1
2
1
2
,
1
2
,0 no extra conditions
2 am
¯
3 . 0,0,0 no extra conditions
Symmetry of special projections
Along [001] c2mm
a
= ab
= b
Origin at 0,0,z
Along [111] p6
a
=
1
3
(2a − b− c) b
=
1
3
(−a + 2b − c)
Origin at x,x , x
Along [110] p2mm
a
=
1
2
(−a + b) b
=
1
2
c
Origin at x,x , 0
Maximal non-isomorphic subgroups
I
[2] I 23 (197) (1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12)+
[3] Im1(Immm, 71) (1; 2; 3; 4; 13; 14; 15; 16)+
⎧
⎪
⎪
⎨
⎪
⎪
⎩
[4] I 1
¯
3(R
¯
3, 148) (1; 5; 9; 13; 17; 21)+
[4] I 1
¯
3(R
¯
3, 148) (1; 6; 12; 13; 18; 24)+
[4] I 1
¯
3(R
¯
3, 148) (1; 7; 10; 13; 19; 22)+
[4] I 1
¯
3(R
¯
3, 148) (1; 8; 11; 13; 20; 23)+
IIa [2] Pn
¯
3 (201) 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; (13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24)+(
1
2
,
1
2
,
1
2
)
[2] Pm
¯
3 (200) 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[27] Im
¯
3(a
= 3a,b
= 3b,c
= 3c) (204)
Minimal non-isomorphic supergroups
I
[2] Im
¯
3m (229)
II [4] Pm
¯
3(a
=
1
2
a,b
=
1
2
b,c
=
1
2
c) (200)
629
Pa
¯
3 T
6
h
m
¯
3 Cubic
No. 205 P2
1
/a
¯
3 Patterson symmetry Pm
¯
3
Origin at centre (
¯
3)
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
2
;0≤ z ≤
1
2
; z ≤ min(x,y)
Vertices 0,0, 0
1
2
,0, 0
1
2
,
1
2
,00,
1
2
,0
1
2
,
1
2
,
1
2
Symmetry operations
(1) 1 (2) 2(0,0,
1
2
)
1
4
,0,z (3) 2(0,
1
2
,0) 0,y,
1
4
(4) 2(
1
2
,0,0) x,
1
4
,0
(5) 3
+
x,x,x (6) 3
+
¯x+
1
2
,x, ¯x (7) 3
+
x+
1
2
, ¯x−
1
2
, ¯x (8) 3
+
¯x, ¯x+
1
2
,x
(9) 3
−
x,x,x (10) 3
−
(−
1
3
,
1
3
,
1
3
) x+
1
6
, ¯x+
1
6
, ¯x (11) 3
−
(
1
3
,
1
3
,−
1
3
) ¯x+
1
3
, ¯x+
1
6
,x (12) 3
−
(
1
3
,−
1
3
,
1
3
) ¯x−
1
6
,x+
1
3
, ¯x
(13)
¯
10,0,0 (14) ax,y,
1
4
(15) cx,
1
4
,z (16) b
1
4
,y,z
(17)
¯
3
+
x,x,x;0,0,0 (18)
¯
3
+
¯x−
1
2
,x+1, ¯x;0,
1
2
,
1
2
(19)
¯
3
+
x+
1
2
, ¯x+
1
2
, ¯x;
1
2
,
1
2
,0 (20)
¯
3
+
¯x+1, ¯x+
1
2
,x;
1
2
,0,
1
2
(21)
¯
3
−
x,x,x;0,0,0 (22)
¯
3
−
x+
1
2
, ¯x−
1
2
, ¯x;0,0,
1
2
(23)
¯
3
−
¯x, ¯x+
1
2
,x;0,
1
2
,0 (24)
¯
3
−
¯x+
1
2
,x, ¯x;
1
2
,0,0
630
International Tables for Crystallography (2006). Vol. A, Space group 205, pp. 630–631.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 205 Pa
¯
3
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3); (5); (13)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
h,k,l cyclically permutable
General:
24 d 1(1)x,y,z (2) ¯x +
1
2
, ¯y, z +
1
2
(3) ¯x,y +
1
2
, ¯z +
1
2
(4) x +
1
2
, ¯y +
1
2
, ¯z
(5) z,x,y (6) z +
1
2
, ¯x +
1
2
, ¯y (7) ¯z +
1
2
, ¯x, y +
1
2
(8) ¯z,x +
1
2
, ¯y +
1
2
(9) y,z,x (10) ¯y,z +
1
2
, ¯x +
1
2
(11) y +
1
2
, ¯z +
1
2
, ¯x (12) ¯y+
1
2
, ¯z, x +
1
2
(13) ¯x, ¯y, ¯z (14) x +
1
2
,y, ¯z +
1
2
(15) x, ¯y +
1
2
,z +
1
2
(16) ¯x +
1
2
,y +
1
2
,z
(17) ¯z, ¯x, ¯y (18) ¯z +
1
2
,x +
1
2
,y (19) z +
1
2
,x, ¯y +
1
2
(20) z, ¯x +
1
2
,y +
1
2
(21) ¯y, ¯z, ¯x (22) y, ¯z +
1
2
,x +
1
2
(23) ¯y +
1
2
,z +
1
2
,x (24) y+
1
2
,z, ¯x +
1
2
0kl : k = 2n
h00 : h = 2n
Special: as above, plus
8 c . 3 . x,x, x ¯x +
1
2
, ¯x, x +
1
2
¯x,x +
1
2
, ¯x +
1
2
x +
1
2
, ¯x +
1
2
, ¯x
¯x, ¯x, ¯xx+
1
2
,x, ¯x +
1
2
x, ¯x +
1
2
,x +
1
2
¯x+
1
2
,x +
1
2
,x
no extra conditions
4 b .
¯
3 .
1
2
,
1
2
,
1
2
0,
1
2
,0
1
2
,0, 00,0,
1
2
hkl : h + k,h + l,k + l = 2n
4 a .
¯
3 . 0,0,0
1
2
,0,
1
2
0,
1
2
,
1
2
1
2
,
1
2
,0 hkl : h + k,h + l,k + l = 2n
Symmetry of special projections
Along [001] p2gm
a
=
1
2
ab
= b
Origin at 0,0,z
Along [111] p6
a
=
1
3
(2a − b− c) b
=
1
3
(−a + 2b − c)
Origin at x, x,x
Along [110] p2gg
a
=
1
2
(−a + b) b
= c
Origin at x, x,0
Maximal non-isomorphic subgroups
I
[2] P2
1
3 (198) 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12
[3] Pa1(Pbca, 61) 1; 2; 3; 4; 13; 14; 15; 16
⎧
⎪
⎪
⎨
⎪
⎪
⎩
[4] P1
¯
3(R
¯
3, 148) 1; 5; 9; 13; 17; 21
[4] P1
¯
3(R
¯
3, 148) 1; 6; 12; 13; 18; 24
[4] P1
¯
3(R
¯
3, 148) 1; 7; 10; 13; 19; 22
[4] P1
¯
3(R
¯
3, 148) 1; 8; 11; 13; 20; 23
IIa none
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[27] Pa
¯
3(a
= 3a,b
= 3b,c
= 3c) (205)
Minimal non-isomorphic supergroups
I
none
II [2] Ia
¯
3 (206); [4] Fm
¯
3 (202)
631
Ia
¯
3 T
7
h
m
¯
3 Cubic
No. 206 I 2
1
/a
¯
3 Patterson symmetry Im
¯
3
Origin at centre (
¯
3)
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
2
;0≤ z ≤
1
4
; z ≤ min(x,
1
2
− x,y,
1
2
− y)
Vertices 0,0, 0
1
2
,0, 0
1
2
,
1
2
,00,
1
2
,0
1
4
,
1
4
,
1
4
Symmetry operations
For (0,0,0)+ set
(1) 1 (2) 2(0,0,
1
2
)
1
4
,0,z (3) 2(0,
1
2
,0) 0,y,
1
4
(4) 2(
1
2
,0,0) x,
1
4
,0
(5) 3
+
x,x,x (6) 3
+
¯x+
1
2
,x, ¯x (7) 3
+
x+
1
2
, ¯x−
1
2
, ¯x (8) 3
+
¯x, ¯x+
1
2
,x
(9) 3
−
x,x,x (10) 3
−
(−
1
3
,
1
3
,
1
3
) x+
1
6
, ¯x+
1
6
, ¯x (11) 3
−
(
1
3
,
1
3
,−
1
3
) ¯x+
1
3
, ¯x+
1
6
,x (12) 3
−
(
1
3
,−
1
3
,
1
3
) ¯x−
1
6
,x+
1
3
, ¯x
(13)
¯
10,0,0 (14) ax,y,
1
4
(15) cx,
1
4
,z (16) b
1
4
,y,z
(17)
¯
3
+
x,x,x;0,0,0 (18)
¯
3
+
¯x−
1
2
,x+1, ¯x;0,
1
2
,
1
2
(19)
¯
3
+
x+
1
2
, ¯x+
1
2
, ¯x;
1
2
,
1
2
,0 (20)
¯
3
+
¯x+1, ¯x+
1
2
,x;
1
2
,0,
1
2
(21)
¯
3
−
x,x,x;0,0,0 (22)
¯
3
−
x+
1
2
, ¯x−
1
2
, ¯x;0,0,
1
2
(23)
¯
3
−
¯x, ¯x+
1
2
,x;0,
1
2
,0 (24)
¯
3
−
¯x+
1
2
,x, ¯x;
1
2
,0,0
For (
1
2
,
1
2
,
1
2
)+ set
(1) t(
1
2
,
1
2
,
1
2
) (2) 2 0,
1
4
,z (3) 2
1
4
,y,0(4)2x,0,
1
4
(5) 3
+
(
1
2
,
1
2
,
1
2
) x,x,x (6) 3
+
(
1
6
,−
1
6
,
1
6
) ¯x−
1
6
,x+
1
3
, ¯x (7) 3
+
(−
1
6
,
1
6
,
1
6
) x+
1
6
, ¯x+
1
6
, ¯x (8) 3
+
(
1
6
,
1
6
,−
1
6
) ¯x+
1
3
, ¯x+
1
6
,x
(9) 3
−
(
1
2
,
1
2
,
1
2
) x,x,x (10) 3
−
(
1
6
,−
1
6
,−
1
6
) x+
1
6
, ¯x+
1
6
, ¯x (11) 3
−
(−
1
6
,−
1
6
,
1
6
) ¯x+
1
3
, ¯x+
1
6
,x (12) 3
−
(−
1
6
,
1
6
,−
1
6
) ¯x−
1
6
,x+
1
3
, ¯x
(13)
¯
1
1
4
,
1
4
,
1
4
(14) bx,y,0 (15) ax,0,z (16) c 0,y,z
(17)
¯
3
+
x,x,x;
1
4
,
1
4
,
1
4
(18)
¯
3
+
¯x−
1
2
,x, ¯x; −
1
4
,−
1
4
,
1
4
(19)
¯
3
+
x−
1
2
, ¯x+
1
2
, ¯x; −
1
4
,
1
4
,−
1
4
(20)
¯
3
+
¯x, ¯x−
1
2
,x;
1
4
,−
1
4
,−
1
4
(21)
¯
3
−
x,x,x;
1
4
,
1
4
,
1
4
(22)
¯
3
−
x+
1
2
, ¯x−
1
2
, ¯x;
1
4
,−
1
4
,
1
4
(23)
¯
3
−
¯x, ¯x+
1
2
,x; −
1
4
,
1
4
,
1
4
(24)
¯
3
−
¯x+
1
2
,x, ¯x;
1
4
,
1
4
,−
1
4
632
International Tables for Crystallography (2006). Vol. A, Space group 206, pp. 632–633.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 206 Ia
¯
3
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); t(
1
2
,
1
2
,
1
2
); (2); (3); (5); (13)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates
(0,0, 0)+ (
1
2
,
1
2
,
1
2
)+
Reflection conditions
h,k,l cyclically permutable
General:
48 e 1(1)x,y, z (2) ¯x +
1
2
, ¯y, z +
1
2
(3) ¯x,y +
1
2
, ¯z +
1
2
(4) x +
1
2
, ¯y +
1
2
, ¯z
(5) z,x,y (6) z +
1
2
, ¯x +
1
2
, ¯y (7) ¯z +
1
2
, ¯x, y +
1
2
(8) ¯z,x +
1
2
, ¯y +
1
2
(9) y,z,x (10) ¯y,z +
1
2
, ¯x +
1
2
(11) y +
1
2
, ¯z +
1
2
, ¯x (12) ¯y+
1
2
, ¯z, x +
1
2
(13) ¯x, ¯y, ¯z (14) x +
1
2
,y, ¯z +
1
2
(15) x, ¯y +
1
2
,z +
1
2
(16) ¯x +
1
2
,y +
1
2
,z
(17) ¯z, ¯x, ¯y (18) ¯z +
1
2
,x +
1
2
,y (19) z +
1
2
,x, ¯y +
1
2
(20) z, ¯x +
1
2
,y +
1
2
(21) ¯y, ¯z, ¯x (22) y, ¯z +
1
2
,x +
1
2
(23) ¯y +
1
2
,z +
1
2
,x (24) y+
1
2
,z, ¯x +
1
2
hkl : h + k + l = 2n
0kl : k,l = 2n
hhl : l = 2n
h00 : h = 2n
Special: as above, plus
24 d 2 .. x,0,
1
4
¯x+
1
2
,0,
3
4
1
4
,x,0
3
4
, ¯x +
1
2
,00,
1
4
,x 0,
3
4
, ¯x +
1
2
¯x,0,
3
4
x +
1
2
,0,
1
4
3
4
, ¯x,0
1
4
,x +
1
2
,00,
3
4
, ¯x 0,
1
4
,x +
1
2
no extra conditions
16 c . 3 . x,x, x ¯x+
1
2
, ¯x, x +
1
2
¯x,x +
1
2
, ¯x +
1
2
x +
1
2
, ¯x +
1
2
, ¯x
¯x, ¯x, ¯xx+
1
2
,x, ¯x +
1
2
x, ¯x +
1
2
,x +
1
2
¯x+
1
2
,x +
1
2
,x
no extra conditions
8 b .
¯
3 .
1
4
,
1
4
,
1
4
1
4
,
3
4
,
3
4
3
4
,
3
4
,
1
4
3
4
,
1
4
,
3
4
hkl : k,l = 2n
8 a .
¯
3 . 0,0,0
1
2
,0,
1
2
0,
1
2
,
1
2
1
2
,
1
2
,0 hkl : k,l = 2n
Symmetry of special projections
Along [001] p2mm
a
=
1
2
ab
=
1
2
b
Origin at 0,0,z
Along [111] p6
a
=
1
3
(2a − b− c) b
=
1
3
(−a + 2b − c)
Origin at x, x,x
Along [110] p2mg
a
=
1
2
(−a + b) b
=
1
2
c
Origin at x,x , 0
Maximal non-isomorphic subgroups
I
[2] I 2
1
3 (199) (1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12)+
[3] Ia1(Ibca, 73) (1; 2; 3; 4; 13; 14; 15; 16)+
⎧
⎪
⎪
⎨
⎪
⎪
⎩
[4] I 1
¯
3(R
¯
3, 148) (1; 5; 9; 13; 17; 21)+
[4] I 1
¯
3(R
¯
3, 148) (1; 6; 12; 13; 18; 24)+
[4] I 1
¯
3(R
¯
3, 148) (1; 7; 10; 13; 19; 22)+
[4] I 1
¯
3(R
¯
3, 148) (1; 8; 11; 13; 20; 23)+
IIa [2] Pa
¯
3 (205) 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24
[2] Pa
¯
3 (205) 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; (13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24)+(
1
2
,
1
2
,
1
2
)
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[27] Ia
¯
3(a
= 3a,b
= 3b,c
= 3c) (206)
Minimal non-isomorphic supergroups
I
[2] Ia
¯
3d (230)
II [4] Pm
¯
3(a
=
1
2
a,b
=
1
2
b,c
=
1
2
c) (200)
633