Hahn Th. (ed.) International tables for crystallography. Vol. A. Space-group symmetry
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F 23 T
2
23 Cubic
No. 196 F 23
Patterson symmetry Fm
¯
3
Origin at 23
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
2
; −
1
4
≤ z ≤
1
4
; y ≤ x;max(x−
1
2
,−y) ≤ z ≤ min(
1
2
− x,y)
Vertices 0,0, 0
1
2
,0, 0
1
2
,
1
2
,0
1
4
,
1
4
,
1
4
1
4
,
1
4
,−
1
4
604
International Tables for Crystallography (2006). Vol. A, Space group 196, pp. 604–606.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 196 F 23
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
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
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
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
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)
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:
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
hkl : h
+ k,h + l,k + l = 2n
0kl : k,l = 2n
hhl : h + l = 2n
h00 : h = 2n
Special: no extra conditions
24 g 2 .. 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 .. x,0,0¯x,0,00,x,00, ¯x,00,0,x 0,0, ¯x
16 e . 3 . x, x,x ¯x, ¯x,x ¯x,x, ¯xx, ¯x, ¯x
4 d 23.
3
4
,
3
4
,
3
4
4 c 23.
1
4
,
1
4
,
1
4
4 b 23.
1
2
,
1
2
,
1
2
4 a 23. 0,0, 0
Symmetry of special projections
Along [001] p2mm
a
=
1
2
ab
=
1
2
b
Origin at 0,0,z
Along [111] p3
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
605
F 23 No. 196 CONTINUED
Maximal non-isomorphic subgroups
I
[3] F 21 (F 222, 22) (1; 2; 3; 4)+
⎧
⎪
⎨
⎪
⎩
[4] F 13 (R3, 146) (1; 5; 9)+
[4] F 13 (R3, 146) (1; 6; 12)+
[4] F 13 (R3, 146) (1; 7; 10)+
[4] F 13 (R3, 146) (1; 8; 11)+
IIa
⎧
⎪
⎨
⎪
⎩
[4] P2
1
3 (198) 1; 5; 9; (2; 7; 12)+(0,
1
2
,
1
2
); (4; 6; 11)+(
1
2
,0,
1
2
); (3; 8; 10)+(
1
2
,
1
2
,0)
[4] P2
1
3 (198) 1; 7; 10; (2; 5; 11)+(0,
1
2
,
1
2
); (4; 8; 12)+(
1
2
,0,
1
2
); (3; 6; 9)+(
1
2
,
1
2
,0)
[4] P2
1
3 (198) 1; 8; 11; (2; 6; 10)+(0,
1
2
,
1
2
); (4; 7; 9)+(
1
2
,0,
1
2
); (3; 5; 12)+(
1
2
,
1
2
,0)
[4] P2
1
3 (198) 1; 6; 12; (2; 8; 9)+(0,
1
2
,
1
2
); (4; 5; 10)+(
1
2
,0,
1
2
); (3; 7; 11)+(
1
2
,
1
2
,0)
⎧
⎪
⎨
⎪
⎩
[4] P2
1
3 (198) 1; 5; 9; (3; 8; 10)+(0,
1
2
,
1
2
); (2; 7; 12)+(
1
2
,0,
1
2
); (4; 6; 11)+(
1
2
,
1
2
,0)
[4] P2
1
3 (198) 1; 7; 10; (3; 6; 9)+(0,
1
2
,
1
2
); (2; 5; 11)+(
1
2
,0,
1
2
); (4; 8; 12)+(
1
2
,
1
2
,0)
[4] P2
1
3 (198) 1; 8; 11; (3; 5; 12)+(0,
1
2
,
1
2
); (2; 6; 10)+(
1
2
,0,
1
2
); (4; 7; 9)+(
1
2
,
1
2
,0)
[4] P2
1
3 (198) 1; 6; 12; (3; 7; 11)+(0,
1
2
,
1
2
); (2; 8; 9)+(
1
2
,0,
1
2
); (4; 5; 10)+(
1
2
,
1
2
,0)
⎧
⎪
⎨
⎪
⎩
[4] P23 (195) 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12
[4] P23 (195) 1; 2; 3; 4; (5; 6; 7; 8)+(0,
1
2
,
1
2
); (9; 10; 11; 12)+(
1
2
,
1
2
,0)
[4] P23 (195) 1; 2; 3; 4; (5; 6; 7; 8)+(
1
2
,0,
1
2
); (9; 10; 11; 12)+(0,
1
2
,
1
2
)
[4] P23 (195) 1; 2; 3; 4; (5; 6; 7; 8)+(
1
2
,
1
2
,0); (9; 10; 11; 12)+(
1
2
,0,
1
2
)
⎧
⎪
⎨
⎪
⎩
[4] P23 (195) 1; 5; 9; (4; 6; 11)+(0,
1
2
,
1
2
); (3; 8; 10)+(
1
2
,0,
1
2
); (2; 7; 12)+(
1
2
,
1
2
,0)
[4] P23 (195) 1; 7; 10; (4; 8; 12)+(0,
1
2
,
1
2
); (3; 6; 9)+(
1
2
,0,
1
2
); (2; 5; 11)+(
1
2
,
1
2
,0)
[4] P23 (195) 1; 8; 11; (4; 7; 9)+(0,
1
2
,
1
2
); (3; 5; 12)+(
1
2
,0,
1
2
); (2; 6; 10)+(
1
2
,
1
2
,0)
[4] P23 (195) 1; 6; 12; (4; 5; 10)+(0,
1
2
,
1
2
); (3; 7; 11)+(
1
2
,0,
1
2
); (2; 8; 9)+(
1
2
,
1
2
,0)
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[27] F 23(a
= 3a,b
= 3b,c
= 3c) (196)
Minimal non-isomorphic supergroups
I
[2] Fm
¯
3 (202); [2] Fd
¯
3 (203); [2] F 432 (209); [2] F 4
1
32 (210); [2] F
¯
43m (216); [2] F
¯
43c (219)
II [2] P23(a
=
1
2
c,b
=
1
2
b,c
=
1
2
c) (195)
606
I 23 T
3
23 Cubic
No. 197 I 23
Patterson symmetry Im
¯
3
Origin at 23
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
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
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
608
International Tables for Crystallography (2006). Vol. A, Space group 197, pp. 608–609.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 197 I 23
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)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates
(0,0, 0)+ (
1
2
,
1
2
,
1
2
)+
Reflection conditions
h,k,l cyclically permutable
General:
24 f 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
hkl : h + k
+ l = 2n
0kl : k + l = 2n
hhl : l = 2n
h00 : h = 2n
Special: no extra conditions
12 e 2 .. x,
1
2
,0¯x,
1
2
,00, x,
1
2
0, ¯x,
1
2
1
2
,0, x
1
2
,0, ¯x
12 d 2 .. x,0,0¯x,0, 00,x,00, ¯x,00,0,x 0,0, ¯x
8 c . 3 . x,x,x ¯x, ¯x,x ¯x,x, ¯xx, ¯x, ¯x
6 b 222.. 0,
1
2
,
1
2
1
2
,0,
1
2
1
2
,
1
2
,0
2 a 23. 0,0,0
Symmetry of special projections
Along [001] c2mm
a
= ab
= b
Origin at 0,0,z
Along [111] p3
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
Maximal non-isomorphic subgroups
I
[3] I 21 (I 222, 23) (1; 2; 3; 4)+
⎧
⎪
⎨
⎪
⎩
[4] I 13 (R3, 146) (1; 5; 9)+
[4] I 13 (R3, 146) (1; 6; 12)+
[4] I 13 (R3, 146) (1; 7; 10)+
[4] I 13 (R3, 146) (1; 8; 11)+
IIa [2] P23 (195) 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[27] I 23(a
= 3a,b
= 3b,c
= 3c) (197)
Minimal non-isomorphic supergroups
I
[2] Im
¯
3 (204); [2] I 432 (211); [2] I
¯
43m (217)
II [4] P23(a
=
1
2
a,b
=
1
2
b,c
=
1
2
c) (195)
609
P2
1
3 T
4
23 Cubic
No. 198 P2
1
3 Patterson symmetry Pm
¯
3
Origin on 3[111] at midpoint of three non-intersecting pairs of parallel 2
1
axes
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
2
; −
1
2
≤ z ≤
1
2
;max(x −
1
2
,−y) ≤ 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
0,
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
610
International Tables for Crystallography (2006). Vol. A, Space group 198, pp. 610–611.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 198 P2
1
3
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3); (5)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
h,k,l cyclically permutable
General:
12 b 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
h00 : h = 2n
Special: no extra conditions
4 a . 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
Symmetry of special projections
Along [001] p2gg
a
= ab
= b
Origin at
1
4
,0, z
Along [111] p3
a
=
1
3
(2a − b− c) b
=
1
3
(−a + 2b − c)
Origin at x,x, x
Along [110] p1g1
a
=
1
2
(−a + b) b
= c
Origin at x +
1
4
,x,0
Maximal non-isomorphic subgroups
I
[3] P2
1
1(P2
1
2
1
2
1
, 19) 1; 2; 3; 4
⎧
⎪
⎨
⎪
⎩
[4] P13 (R3, 146) 1; 5; 9
[4] P13 (R3, 146) 1; 6; 12
[4] P13 (R3, 146) 1; 7; 10
[4] P13 (R3, 146) 1; 8; 11
IIa none
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[27] P2
1
3(a
= 3a,b
= 3b,c
= 3c) (198)
Minimal non-isomorphic supergroups
I
[2] Pa
¯
3 (205); [2] P4
3
32 (212); [2] P4
1
32 (213)
II [2] I 2
1
3 (199); [4] F 23 (196)
611
I 2
1
3 T
5
23 Cubic
No. 199 I 2
1
3 Patterson symmetry Im
¯
3
Origin on 3[111] at midpoint of three non-intersecting pairs of parallel 2 axes and of three
non-intersecting pairs of parallel 2
1
axes
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
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
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
612
International Tables for Crystallography (2006). Vol. A, Space group 199, pp. 612–613.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 199 I 2
1
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)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates
(0,0, 0)+ (
1
2
,
1
2
,
1
2
)+
Reflection conditions
h,k,l cyclically permutable
General:
24 c 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
hkl : h + k + l = 2n
0kl : k + l = 2n
hhl : l = 2n
h00 : h = 2n
Special: no extra conditions
12 b 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
8 a . 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
Symmetry of special projections
Along [001] c2mm
a
= ab
= b
Origin at
1
4
,0, z
Along [111] p3
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 +
1
4
,0
Maximal non-isomorphic subgroups
I
[3] I 2
1
1(I 2
1
2
1
2
1
, 24) (1; 2; 3; 4)+
⎧
⎪
⎨
⎪
⎩
[4] I 13 (R3, 146) (1; 5; 9)+
[4] I 13 (R3, 146) (1; 6; 12)+
[4] I 13 (R3, 146) (1; 7; 10)+
[4] I 13 (R3, 146) (1; 8; 11)+
IIa [2] P2
1
3 (198) 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[27] I 2
1
3(a
= 3a,b
= 3b,c
= 3c) (199)
Minimal non-isomorphic supergroups
I
[2] Ia
¯
3 (206); [2] I 4
1
32 (214); [2] I
¯
43d (220)
II [4] P23(a
=
1
2
a,b
=
1
2
b,c
=
1
2
c) (195)
613
Pm
¯
3 T
1
h
m
¯
3 Cubic
No. 200 P2/m
¯
3
Patterson symmetry Pm
¯
3
Origin at centre (m
¯
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,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
Minimal non-isomorphic supergroups
I
[2] Pm
¯
3m (221); [2] Pm
¯
3n (223)
II [2] Im
¯
3 (204); [4] Fm
¯
3 (202)
614
International Tables for Crystallography (2006). Vol. A, Space group 200, pp. 614–615.
Copyright © 2006 International Union of Crystallography