Hahn Th. (ed.) International tables for crystallography. Vol. A. Space-group symmetry
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I 4
1
32 No. 214 CONTINUED
Symmetry of special projections
Along [001] p4mm
a
=
1
2
(a − b) b
=
1
2
(a + b)
Origin at
1
4
,0, z
Along [111] p3m1
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 +
1
4
,
1
8
Maximal non-isomorphic subgroups
I
[2] I 2
1
31(I 2
1
3, 199) (1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12)+
[3] I 4
1
12(I 4
1
22, 98) (1; 2; 3; 4; 13; 14; 15; 16)+
[3] I 4
1
12(I 4
1
22, 98) (1; 2; 3; 4; 17; 18; 19; 20)+
[3] I 4
1
12(I 4
1
22, 98) (1; 2; 3; 4; 21; 22; 23; 24)+
⎧
⎪
⎨
⎪
⎩
[4] I 132(R32, 155) (1; 5; 9; 14; 19; 24)+
[4] I 132(R32, 155) (1; 6; 12; 13; 18; 24)+
[4] I 132(R32, 155) (1; 7; 10; 13; 19; 22)+
[4] I 132(R32, 155) (1; 8; 11; 14; 18; 22)+
IIa [2] P4
1
32 (213) 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] P4
3
32 (212) 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] I 4
1
32(a
= 3a,b
= 3b,c
= 3c) (214)
Minimal non-isomorphic supergroups
I
[2] Ia
¯
3d (230)
II [4] P4
2
32(a
=
1
2
a,b
=
1
2
b,c
=
1
2
c) (208)
654
P
¯
43mT
1
d
¯
43m Cubic
No. 215 P
¯
43m
Patterson symmetry Pm
¯
3m
Origin at
¯
43m
Asymmetric unit 0 ≤ x ≤ 1; 0 ≤ y ≤
1
2
;0≤ z ≤
1
2
; y ≤ min(x,1 − x); z ≤ y
Vertices 0,0, 01,0, 0
1
2
,
1
2
,0
1
2
,
1
2
,
1
2
Symmetry operations
(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) mx,x,z (14) mx, ¯x,z (15)
¯
4
+
0,0,z;0,0, 0 (16)
¯
4
−
0,0,z;0,0,0
(17) mx,y, y (18)
¯
4
+
x,0, 0; 0,0,0 (19)
¯
4
−
x,0, 0; 0, 0,0 (20) mx,y, ¯y
(21) mx,y, x (22)
¯
4
−
0,y,0; 0,0, 0 (23) m ¯x,y,x (24)
¯
4
+
0,y,0; 0,0, 0
656
International Tables for Crystallography (2006). Vol. A, Space group 215, pp. 656–657.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 215 P
¯
43m
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 permutable
General:
24 j 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) y,x , z (14) ¯
y, ¯x,z (15) y, ¯x, ¯z (16) ¯y,x, ¯z
(17) x,z,y (18) ¯x,z, ¯y (19) ¯x, ¯z, y (20) x, ¯z, ¯y
(21) z,y, x (22) z, ¯y, ¯x (23) ¯z, y, ¯x (24) ¯z, ¯y,x
no conditions
Special: no extra conditions
12 i ..mx,x,z ¯x, ¯x,z ¯x, x, ¯zx
, ¯x, ¯zz,x, xz, ¯x, ¯x
¯z, ¯x, x ¯z,x, ¯xx, z,x ¯x,z, ¯xx, ¯z, ¯x ¯x, ¯z,x
12 h 2 .. x,
1
2
,0¯x,
1
2
,00, x,
1
2
0, ¯x,
1
2
1
2
,0, x
1
2
,0, ¯x
1
2
,x,0
1
2
, ¯x,0 x,0,
1
2
¯x,0,
1
2
0,
1
2
,x 0,
1
2
, ¯x
6 g 2 . mm x,
1
2
,
1
2
¯x,
1
2
,
1
2
1
2
,x,
1
2
1
2
, ¯x,
1
2
1
2
,
1
2
,x
1
2
,
1
2
, ¯x
6 f 2 . mm x,0,0¯x,0, 00, x,00, ¯x,00, 0,x 0,0, ¯x
4 e . 3 mx, x,x ¯x, ¯x, x ¯x,x, ¯xx, ¯x, ¯x
3 d
¯
42.m
1
2
,0, 00,
1
2
,00,0,
1
2
3 c
¯
42.m 0,
1
2
,
1
2
1
2
,0,
1
2
1
2
,
1
2
,0
1 b
¯
43m
1
2
,
1
2
,
1
2
1 a
¯
43m 0,0,0
Symmetry of special projections
Along [001] p4mm
a
= ab
= b
Origin at 0,0,z
Along [111] p31m
a
=
1
3
(2a − b− c) b
=
1
3
(−a + 2b − c)
Origin at x,x , x
Along [110] p1m1
a
=
1
2
(−a + b) b
= c
Origin at x,x , 0
Maximal non-isomorphic subgroups
I
[2] P231 (P23, 195) 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12
⎧
⎨
⎩
[3] P
¯
41m (P
¯
42m, 111) 1; 2; 3; 4; 13; 14; 15; 16
[3] P
¯
41m (P
¯
42m, 111) 1; 2; 3; 4; 17; 18; 19; 20
[3] P
¯
41m (P
¯
42m, 111) 1; 2; 3; 4; 21; 22; 23; 24
⎧
⎪
⎨
⎪
⎩
[4] P13m (R3m, 160) 1; 5; 9; 13; 17; 21
[4] P13m (R3m, 160) 1; 6; 12; 14; 20; 21
[4] P13m (R3m, 160) 1; 7; 10; 14; 17; 23
[4] P13m (R3m, 160) 1; 8; 11; 13; 20; 23
IIa none
IIb [2] F
¯
43c (a
= 2a,b
= 2b,c
= 2c) (219); [2] F
¯
43m (a
= 2a,b
= 2b,c
= 2c) (216); [4] I
¯
43m (a
= 2a,b
= 2b,c
= 2c) (217)
Maximal isomorphic subgroups of lowest index
IIc
[27] P
¯
43m (a
= 3a,b
= 3b,c
= 3c) (215)
Minimal non-isomorphic supergroups
I
[2] Pm
¯
3m (221); [2] Pn
¯
3m (224)
II [2] I
¯
43m (217); [4] F
¯
43m (216)
657
F
¯
43mT
2
d
¯
43m Cubic
No. 216 F
¯
43m
Patterson symmetry Fm
¯
3m
Origin at
¯
43m
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
4
; −
1
4
≤ z ≤
1
4
; y ≤ min(x,
1
2
− x); −y ≤ z ≤ y
Vertices 0,0, 0
1
2
,0, 0
1
4
,
1
4
,
1
4
1
4
,
1
4
,−
1
4
658
International Tables for Crystallography (2006). Vol. A, Space group 216, pp. 658–660.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 216 F
¯
43m
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) mx,x,z (14) mx, ¯x,z (15)
¯
4
+
0,0,z;0,0,0 (16)
¯
4
−
0,0,z;0,0,0
(17) mx,y,y (18)
¯
4
+
x,0,0; 0,0,0 (19)
¯
4
−
x,0,0; 0,0,0 (20) mx,y, ¯y
(21) mx,y,x (22)
¯
4
−
0,y,0; 0,0,0 (23) m ¯x,y,x (24)
¯
4
+
0,y,0; 0,0,0
For (0,
1
2
,
1
2
)+ set
(1) t(0,
1
2
,
1
2
) (2) 2(0,0,
1
2
) 0,
1
4
,z (3) 2(0,
1
2
,0) 0,y,
1
4
(4) 2 x,
1
4
,
1
4
(5) 3
+
(
1
3
,
1
3
,
1
3
) x−
1
3
,x−
1
6
,x (6) 3
+
¯x,x+
1
2
, ¯x (7) 3
+
(−
1
3
,
1
3
,
1
3
) x+
1
3
, ¯x−
1
6
, ¯x (8) 3
+
¯x, ¯x+
1
2
,x
(9) 3
−
(
1
3
,
1
3
,
1
3
) x−
1
6
,x+
1
6
,x (10) 3
−
(−
1
3
,
1
3
,
1
3
) x+
1
6
, ¯x+
1
6
, ¯x (11) 3
−
¯x+
1
2
, ¯x+
1
2
,x (12) 3
−
¯x−
1
2
,x+
1
2
, ¯x
(13) g(
1
4
,
1
4
,
1
2
) x−
1
4
,x,z (14) g(−
1
4
,
1
4
,
1
2
) x+
1
4
, ¯x,z (15)
¯
4
+
1
4
,
1
4
,z;
1
4
,
1
4
,
1
4
(16)
¯
4
−
−
1
4
,
1
4
,z; −
1
4
,
1
4
,
1
4
(17) g(0,
1
2
,
1
2
) x,y,y (18)
¯
4
+
x,
1
2
,0; 0,
1
2
,0 (19)
¯
4
−
x,0,
1
2
;0,0,
1
2
(20) mx,y+
1
2
, ¯y
(21) g(
1
4
,
1
2
,
1
4
) x−
1
4
,y,x (22)
¯
4
−
1
4
,y,
1
4
;
1
4
,
1
4
,
1
4
(23) g(−
1
4
,
1
2
,
1
4
) ¯x+
1
4
,y,x (24)
¯
4
+
−
1
4
,y,
1
4
; −
1
4
,
1
4
,
1
4
For (
1
2
,0,
1
2
)+ set
(1) t(
1
2
,0,
1
2
) (2) 2(0,0,
1
2
)
1
4
,0,z (3) 2
1
4
,y,
1
4
(4) 2(
1
2
,0,0) x,0,
1
4
(5) 3
+
(
1
3
,
1
3
,
1
3
) x+
1
6
,x−
1
6
,x (6) 3
+
(
1
3
,−
1
3
,
1
3
) ¯x+
1
6
,x+
1
6
, ¯x (7) 3
+
x+
1
2
, ¯x−
1
2
, ¯x (8) 3
+
¯x+
1
2
, ¯x+
1
2
,x
(9) 3
−
(
1
3
,
1
3
,
1
3
) x−
1
6
,x−
1
3
,x (10) 3
−
x+
1
2
, ¯x, ¯x (11) 3
−
¯x+
1
2
, ¯x,x (12) 3
−
(
1
3
,−
1
3
,
1
3
) ¯x−
1
6
,x+
1
3
, ¯x
(13) g(
1
4
,
1
4
,
1
2
) x+
1
4
,x,z (14) g(
1
4
,−
1
4
,
1
2
) x+
1
4
, ¯x,z (15)
¯
4
+
1
4
,−
1
4
,z;
1
4
,−
1
4
,
1
4
(16)
¯
4
−
1
4
,
1
4
,z;
1
4
,
1
4
,
1
4
(17) g(
1
2
,
1
4
,
1
4
) x,y−
1
4
,y (18)
¯
4
+
x,
1
4
,
1
4
;
1
4
,
1
4
,
1
4
(19)
¯
4
−
x,−
1
4
,
1
4
;
1
4
,−
1
4
,
1
4
(20) g(
1
2
,−
1
4
,
1
4
) x,y+
1
4
, ¯y
(21) g(
1
2
,0,
1
2
) x,y,x (22)
¯
4
−
1
2
,y,0;
1
2
,0,0 (23) m ¯x+
1
2
,y,x (24)
¯
4
+
0,y,
1
2
;0,0,
1
2
For (
1
2
,
1
2
,0)+ set
(1) t(
1
2
,
1
2
,0) (2) 2
1
4
,
1
4
,z (3) 2(0,
1
2
,0)
1
4
,y,0(4)2(
1
2
,0,0) x,
1
4
,0
(5) 3
+
(
1
3
,
1
3
,
1
3
) x+
1
6
,x+
1
3
,x (6) 3
+
¯x+
1
2
,x, ¯x (7) 3
+
x+
1
2
, ¯x, ¯x (8) 3
+
(
1
3
,
1
3
,−
1
3
) ¯x+
1
6
, ¯x+
1
3
,x
(9) 3
−
(
1
3
,
1
3
,
1
3
) x+
1
3
,x+
1
6
,x (10) 3
−
x, ¯x+
1
2
, ¯x (11) 3
−
(
1
3
,
1
3
,−
1
3
) ¯x+
1
3
, ¯x+
1
6
,x (12) 3
−
¯x,x+
1
2
, ¯x
(13) g(
1
2
,
1
2
,0) x,x,z (14) mx+
1
2
, ¯x,z (15)
¯
4
+
1
2
,0,z;
1
2
,0,0 (16)
¯
4
−
0,
1
2
,z;0,
1
2
,0
(17) g(
1
2
,
1
4
,
1
4
) x,y+
1
4
,y (18)
¯
4
+
x,
1
4
,−
1
4
;
1
4
,
1
4
,−
1
4
(19)
¯
4
−
x,
1
4
,
1
4
;
1
4
,
1
4
,
1
4
(20) g(
1
2
,
1
4
,−
1
4
) x,y+
1
4
, ¯y
(21) g(
1
4
,
1
2
,
1
4
) x+
1
4
,y,x (22)
¯
4
−
1
4
,y,−
1
4
;
1
4
,
1
4
,−
1
4
(23) g(
1
4
,
1
2
,−
1
4
) ¯x+
1
4
,y,x (24)
¯
4
+
1
4
,y,
1
4
;
1
4
,
1
4
,
1
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 permutable
General:
96 i 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) y,
x,z (14) ¯y, ¯x,z (15) y, ¯x, ¯z (16) ¯y,x, ¯z
(17) x,z,y (18) ¯x, z, ¯y (19) ¯x, ¯z, y (20) x, ¯z, ¯y
(21) z,y, x (22) z, ¯y, ¯x (23) ¯z, y, ¯x (24) ¯z, ¯y,x
hkl : h + k,h + l,k + l = 2n
0kl : k
,l = 2n
hhl : h + l = 2n
h00 : h = 2n
Special: no extra conditions
48 h ..mx, x,z ¯x, ¯x,z ¯x,x, ¯zx, ¯x, ¯zz,x,xz, ¯x, ¯x
¯z, ¯x, x ¯z,x, ¯xx,z,x ¯x,z, ¯xx, ¯z, ¯x ¯x, ¯z,x
24 g 2 . mm x
,
1
4
,
1
4
¯x,
3
4
,
1
4
1
4
,x,
1
4
1
4
, ¯x,
3
4
1
4
,
1
4
,x
3
4
,
1
4
, ¯x
24 f 2 . mm x, 0,0¯x, 0,00, x,00, ¯x,00,0,x 0,0, ¯x
16 e . 3 mx, x,x ¯x, ¯x,x ¯x,x, ¯xx, ¯x, ¯x
4 d
¯
43m
3
4
,
3
4
,
3
4
4 c
¯
43m
1
4
,
1
4
,
1
4
4 b
¯
43m
1
2
,
1
2
,
1
2
4 a
¯
43m 0,0,0
659
F
¯
43m No. 216 CONTINUED
Symmetry of special projections
Along [001] p4mm
a
=
1
2
ab
=
1
2
b
Origin at 0,0,z
Along [111] p31m
a
=
1
6
(2a − b− c) b
=
1
6
(−a + 2b − c)
Origin at x,x , x
Along [110] c1m1
a
=
1
2
(−a + b) b
= c
Origin at x,x , 0
Maximal non-isomorphic subgroups
I
[2] F 231(F 23, 196) (1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12)+
⎧
⎨
⎩
[3] F
¯
41m (I
¯
4m2, 119) (1; 2; 3; 4; 13; 14; 15; 16)+
[3] F
¯
41m (I
¯
4m2, 119) (1; 2; 3; 4; 17; 18; 19; 20)+
[3] F
¯
41m (I
¯
4m2, 119) (1; 2; 3; 4; 21; 22; 23; 24)+
⎧
⎪
⎨
⎪
⎩
[4] F 13m ( R3m, 160) (1; 5; 9; 13; 17; 21)+
[4] F 13m ( R3m, 160) (1; 6; 12; 14; 20; 21)+
[4] F 13m ( R3m, 160) (1; 7; 10; 14; 17; 23)+
[4] F 13m ( R3
m, 160) (1; 8; 11; 13; 20; 23)+
IIa
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
[4] P
¯
43m (215) 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24
[4] P
¯
43m (215) 1; 2; 3; 4; 13; 14; 15; 16; (9; 10; 11; 12; 17; 18; 19; 20)+(0,
1
2
,
1
2
); (5; 6; 7; 8; 21; 22; 23;
24)+(
1
2
,0,
1
2
)
[4] P
¯
43m (215) 1; 2; 3; 4; 17; 18; 19; 20; (9; 10; 11; 12; 21; 22; 23; 24)+(
1
2
,0,
1
2
); (5; 6; 7; 8; 13; 14; 15;
16)+(
1
2
,
1
2
,0)
[4] P
¯
43m (215) 1; 2; 3; 4; 21; 22; 23; 24; (5; 6; 7; 8; 17; 18; 19; 20)+(0,
1
2
,
1
2
); (9; 10; 11; 12; 13; 14; 15;
16)+(
1
2
,
1
2
,0)
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
[4] P
¯
43m (215) 1; 5; 9; 13; 17; 21; (4; 6; 11; 15; 20; 22)+(0,
1
2
,
1
2
); (3; 8; 10; 16; 18; 23)+(
1
2
,0,
1
2
); (2; 7; 12;
14; 19; 24)+(
1
2
,
1
2
,0)
[4] P
¯
43m (215) 1; 6; 12; 14; 20; 21; (4; 5; 10; 16; 17; 22)+(0,
1
2
,
1
2
); (3; 7; 11; 15; 19; 23)+(
1
2
,0,
1
2
); (2; 8; 9;
13; 18; 24)+(
1
2
,
1
2
,0)
[4] P
¯
43m (215) 1; 7; 10; 14; 17; 23; (4; 8; 12; 16; 20; 24)+(0,
1
2
,
1
2
); (3; 6; 9; 15; 18; 21)+(
1
2
,0,
1
2
); (2; 5; 11;
13; 19; 22)+(
1
2
,
1
2
,0)
[4] P
¯
43m (215) 1; 8; 11; 13; 20; 23; (4; 7; 9; 15; 17; 24)+(0,
1
2
,
1
2
); (3; 5; 12; 16; 19; 21)+(
1
2
,0,
1
2
); (2; 6; 10;
14; 18; 22)+(
1
2
,
1
2
,0)
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[27] F
¯
43m (a
= 3a,b
= 3b,c
= 3c) (216)
Minimal non-isomorphic supergroups
I
[2] Fm
¯
3m (225); [2] Fd
¯
3m (227)
II [2] P
¯
43m (a
=
1
2
a,b
=
1
2
b,c
=
1
2
c) (215)
660
I
¯
43mT
3
d
¯
43m Cubic
No. 217 I
¯
43m
Patterson symmetry Im
¯
3m
Origin at
¯
43m
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
662
International Tables for Crystallography (2006). Vol. A, Space group 217, pp. 661–663.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 217 I
¯
43m
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) mx,x,z (14) mx, ¯x,z (15)
¯
4
+
0,0,z;0,0,0 (16)
¯
4
−
0,0,z;0,0,0
(17) mx,y,y (18)
¯
4
+
x,0,0; 0,0,0 (19)
¯
4
−
x,0,0; 0,0,0 (20) mx,y, ¯y
(21) mx,y,x (22)
¯
4
−
0,y,0; 0,0,0 (23) m ¯x,y,x (24)
¯
4
+
0,y,0; 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) n(
1
2
,
1
2
,
1
2
) x,x,z (14) cx+
1
2
, ¯x,z (15)
¯
4
+
1
2
,0,z;
1
2
,0,
1
4
(16)
¯
4
−
0,
1
2
,z;0,
1
2
,
1
4
(17) n(
1
2
,
1
2
,
1
2
) x,y,y (18)
¯
4
+
x,
1
2
,0;
1
4
,
1
2
,0 (19)
¯
4
−
x,0,
1
2
;
1
4
,0,
1
2
(20) ax,y+
1
2
, ¯y
(21) n(
1
2
,
1
2
,
1
2
) x,y,x (22)
¯
4
−
1
2
,y,0;
1
2
,
1
4
,0 (23) b ¯x+
1
2
,y,x (24)
¯
4
+
0,y,
1
2
;0,
1
4
,
1
2
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 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) y,x,
z (14) ¯y, ¯x,z (15) y, ¯x, ¯z (16) ¯y,x, ¯z
(17) x,z,y (18) ¯x,z, ¯y (19) ¯x, ¯z, y (20) x, ¯z, ¯y
(21) z,y, x (22) z, ¯y, ¯x (23) ¯z, y, ¯x (24) ¯z, ¯y,x
hkl : h + k + l = 2n
0kl : k + l = 2n
hhl : l
= 2n
h00 : h = 2n
Special: no extra conditions
24 g ..mx, x,z ¯x, ¯x,z ¯x,x, ¯zx, ¯x, ¯zz,x,xz, ¯x, ¯x
¯z, ¯x, x ¯z,x, ¯xx,z,x ¯x,z, ¯xx, ¯z, ¯x ¯x, ¯z,x
24 f 2 .. x,
1
2
,0¯x,
1
2
,00, x,
1
2
0, ¯x,
1
2
1
2
,0, x
1
2
,0, ¯x
1
2
,x,0
1
2
, ¯x,0 x,0,
1
2
¯x,0,
1
2
0,
1
2
,x 0,
1
2
, ¯x
12 e 2 . mm x,0,0¯x,0, 00, x,00, ¯x,00, 0,x 0,0, ¯x
12 d
¯
4 ..
1
4
,
1
2
,0
3
4
,
1
2
,00,
1
4
,
1
2
0,
3
4
,
1
2
1
2
,0,
1
4
1
2
,0,
3
4
8 c . 3 mx, x,x ¯x, ¯x,x ¯x,x, ¯xx, ¯x, ¯x
6 b
¯
42.m 0,
1
2
,
1
2
1
2
,0,
1
2
1
2
,
1
2
,0
2 a
¯
43m 0,0,0
Symmetry of special projections
Along [001] p4mm
a
=
1
2
(a − b) b
=
1
2
(a + b)
Origin at 0,0,z
Along [111] p31m
a
=
1
3
(2a − b− c) b
=
1
3
(−a + 2b − c)
Origin at x, x,x
Along [110] p1m1
a
=
1
2
(−a + b) b
=
1
2
c
Origin at x,x, 0
(Continued on page 661)
663
CONTINUED
(from page 663)
No. 217 I
¯
43m
Maximal non-isomorphic subgroups
I
[2] I 231(I 23, 197) (1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12)+
⎧
⎨
⎩
[3] I
¯
41m (I
¯
42m, 121) (1; 2; 3; 4; 13; 14; 15; 16)+
[3] I
¯
41m (I
¯
42m, 121) (1; 2; 3; 4; 17; 18; 19; 20)+
[3] I
¯
41m (I
¯
42m, 121) (1; 2; 3; 4; 21; 22; 23; 24)+
⎧
⎪
⎨
⎪
⎩
[4] I 13m ( R3m, 160) (1; 5; 9; 13; 17; 21)+
[4] I 13m ( R3m, 160) (1; 6; 12; 14; 20; 21)+
[4] I 13m (R3m, 160) (1; 7; 10; 14; 17; 23)+
[4] I 13m (R3
m, 160) (1; 8; 11; 13; 20; 23)+
IIa [2] P
¯
43n (218) 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] P
¯
43m (215) 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] I
¯
43m (a
= 3a,b
= 3b,c
= 3c) (217)
Minimal non-isomorphic supergroups
I
[2] Im
¯
3m (229)
II [4] P
¯
43m (a
=
1
2
a,b
=
1
2
b,c
=
1
2
c) (215)
661
P
¯
43nT
4
d
¯
43m Cubic
No. 218 P
¯
43n
Patterson symmetry Pm
¯
3m
Origin at 23
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,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) n(
1
2
,
1
2
,
1
2
) x,x,z (14) cx+
1
2
, ¯x,z (15)
¯
4
+
1
2
,0, z;
1
2
,0,
1
4
(16)
¯
4
−
0,
1
2
,z;0,
1
2
,
1
4
(17) n(
1
2
,
1
2
,
1
2
) x,y, y (18)
¯
4
+
x,
1
2
,0;
1
4
,
1
2
,0 (19)
¯
4
−
x,0,
1
2
;
1
4
,0,
1
2
(20) ax,y +
1
2
, ¯y
(21) n(
1
2
,
1
2
,
1
2
) x,y, x (22)
¯
4
−
1
2
,y,0;
1
2
,
1
4
,0 (23) b ¯x +
1
2
,y,x (24)
¯
4
+
0,y,
1
2
;0,
1
4
,
1
2
664
International Tables for Crystallography (2006). Vol. A, Space group 218, pp. 664–665.
Copyright © 2006 International Union of Crystallography