Hahn Th. (ed.) International tables for crystallography. Vol. A. Space-group symmetry
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F 222 D
7
2
222 Orthorhombic
No. 22 F 222
Patterson symmetry F mmm
Origin at 222
Asymmetric unit 0 ≤ x ≤
1
4
;0≤ y ≤
1
4
;0≤ z ≤ 1
Symmetry operations
For (0,0,0)+ set
(1) 1 (2) 2 0, 0,z (3) 2 0,y,0(4)2x,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
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
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
Maximal isomorphic subgroups of lowest index
IIc
[3] F 222(a
= 3a or b
= 3b or c
= 3c) (22)
Minimal non-isomorphic supergroups
I
[2] F mmm(69); [2] Fddd(70); [2] I 422 (97); [2] I 4
1
22 (98); [2] I
¯
4m2 (119); [2] I
¯
4c2 (120); [3] F 23 (196)
II [2] P222(a
=
1
2
a,b
=
1
2
b,c
=
1
2
c) (16)
212
International Tables for Crystallography (2006). Vol. A, Space group 22, pp. 212–213.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 22 F 222
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)
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
General:
16 k 1(1)x, y,z (2) ¯x, ¯y,z (3) ¯x,y, ¯z (4) x, ¯y, ¯zhkl: h + k,h + l,k + l = 2n
0kl : k,l = 2n
h0l : h, l = 2n
hk0: h, k = 2n
h00 : h = 2n
0k0: k = 2n
00l : l = 2n
Special: no extra conditions
8 j
2 .. x,
1
4
,
1
4
¯x,
3
4
,
1
4
8 i . 2 .
1
4
,y,
1
4
3
4
, ¯y,
1
4
8 h ..2
1
4
,
1
4
,z
3
4
,
1
4
, ¯z
8 g ..20, 0,z 0,0, ¯z
8 f . 2 . 0, y,00, ¯y,0
8 e 2 .. x,0, 0¯x,0,0
4 d 222
1
4
,
1
4
,
3
4
4 c 222
1
4
,
1
4
,
1
4
4 b 222 0,0,
1
2
4 a 222 0,0,0
Symmetry of special projections
Along [001] p2mm
a
=
1
2
ab
=
1
2
b
Origin at 0,0,z
Along [100] p2mm
a
=
1
2
bb
=
1
2
c
Origin at x,0,0
Along [010] p2mm
a
=
1
2
cb
=
1
2
a
Origin at 0,y, 0
Maximal non-isomorphic subgroups
I
[2] F 112(C2, 5) (1; 2)+
[2] F 121(C2, 5) (1; 3)+
[2] F 211(C2, 5) (1; 4)+
IIa [2] A222 (C222, 21) 1; 2; 3; 4; (1; 2; 3; 4)+(0,
1
2
,
1
2
)
[2] A222 (C222, 21) 1; 4; (1; 4)+(0,
1
2
,
1
2
); (2; 3)+(
1
2
,0,
1
2
); (2; 3)+(
1
2
,
1
2
,0)
[2] B222 (C222, 21) 1; 2; 3; 4; (1; 2; 3; 4)+(
1
2
,0,
1
2
)
[2] B222 (C222, 21) 1; 3; (1; 3)+(
1
2
,0,
1
2
); (2; 4)+(
1
2
,
1
2
,0); (2; 4)+(0,
1
2
,
1
2
)
[2] C 222 (21) 1; 2; 3; 4; (1; 2; 3; 4)+(
1
2
,
1
2
,0)
[2] C 222 (21) 1; 2; (1; 2)+(
1
2
,
1
2
,0); (3; 4)+(0,
1
2
,
1
2
); (3; 4)+(
1
2
,0,
1
2
)
[2] A2
1
22(C222
1
, 20) 1; 2; (1; 2)+(0,
1
2
,
1
2
); (3; 4)+(
1
2
,0,
1
2
); (3; 4)+(
1
2
,
1
2
,0)
[2] A2
1
22(C222
1
, 20) 1; 3; (1; 3)+(0,
1
2
,
1
2
); (2; 4)+(
1
2
,0,
1
2
); (2; 4)+(
1
2
,
1
2
,0)
[2] B22
1
2(C 222
1
, 20) 1; 2; (1; 2)+(
1
2
,0,
1
2
); (3; 4)+(
1
2
,
1
2
,0); (3; 4)+(0,
1
2
,
1
2
)
[2] B22
1
2(C 222
1
, 20) 1; 4; (1; 4)+(
1
2
,0,
1
2
); (2; 3)+(
1
2
,
1
2
,0); (2; 3)+(0,
1
2
,
1
2
)
[2] C 222
1
(20) 1; 3; (1; 3)+(
1
2
,
1
2
,0); (2; 4)+(0,
1
2
,
1
2
); (2; 4)+(
1
2
,0,
1
2
)
[2] C 222
1
(20) 1; 4; (1; 4)+(
1
2
,
1
2
,0); (2; 3)+(0,
1
2
,
1
2
); (2; 3)+(
1
2
,0,
1
2
)
IIb none
(Continued on preceding page)
213
I 222 D
8
2
222 Orthorhombic
No. 23 I 222
Patterson symmetry Immm
Origin at 222
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
2
;0≤ z ≤
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
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
214
International Tables for Crystallography (2006). Vol. A, Space group 23, pp. 214–215.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 23 I 222
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); t(
1
2
,
1
2
,
1
2
); (2); (3)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates
(0,0, 0)+ (
1
2
,
1
2
,
1
2
)+
Reflection conditions
General:
8 k 1(1)x,y,z (2) ¯x, ¯y,z (3) ¯x,y, ¯z (4) x, ¯y, ¯zhkl: h + k + l = 2n
0kl : k + l = 2n
h0l : h + l = 2n
hk0: h + k = 2n
h00 : h = 2n
0k0: k = 2n
00l : l = 2n
Special: no extra conditions
4 j ..20,
1
2
,z 0,
1
2
, ¯z
4 i ..20,0, z 0,0, ¯z
4 h . 2 .
1
2
,y,0
1
2
, ¯y, 0
4 g . 2 . 0, y,00, ¯y, 0
4 f 2 .. x,0,
1
2
¯x,0,
1
2
4 e 2 .. x,0,0¯x, 0,0
2 d 222 0,
1
2
,0
2 c 222 0,0,
1
2
2 b 222
1
2
,0, 0
2 a 222 0,0,0
Symmetry of special projections
Along [001] c2mm
a
= ab
= b
Origin at 0,0,z
Along [100] c2mm
a
= bb
= c
Origin at x, 0,0
Along [010] c2mm
a
= cb
= a
Origin at 0,y, 0
Maximal non-isomorphic subgroups
I
[2] I 112(C2, 5) (1; 2)+
[2] I 121(C2, 5) (1; 3)+
[2] I 211(C2, 5) (1; 4)+
IIa [2] P2
1
2
1
2 (18) 1; 2; (3; 4)+(
1
2
,
1
2
,
1
2
)
[2] P2
1
22
1
(P2
1
2
1
2, 18) 1; 3; (2; 4)+(
1
2
,
1
2
,
1
2
)
[2] P22
1
2
1
(P2
1
2
1
2, 18) 1; 4; (2; 3)+(
1
2
,
1
2
,
1
2
)
[2] P222 (16) 1; 2; 3; 4
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] I 222(a
= 3a or b
= 3b or c
= 3c) (23)
Minimal non-isomorphic supergroups
I
[2] Immm (71); [2] Ibam (72); [2] I 422 (97); [2] I
¯
42m (121); [3] I 23 (197)
II [2] A222(a
=
1
2
a)(C 222, 21); [2] B222 (b
=
1
2
b)(C222, 21); [2] C 222(c
=
1
2
c) (21)
215
I 2
1
2
1
2
1
D
9
2
222 Orthorhombic
No. 24 I 2
1
2
1
2
1
Patterson symmetry Immm
Origin at midpoint of three non-intersecting pairs of parallel 2 axes
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
2
;0≤ z ≤
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
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
216
International Tables for Crystallography (2006). Vol. A, Space group 24, pp. 216–217.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 24 I 2
1
2
1
2
1
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); t(
1
2
,
1
2
,
1
2
); (2); (3)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates
(0,0, 0)+ (
1
2
,
1
2
,
1
2
)+
Reflection conditions
General:
8 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
, ¯zhkl: h + k + l = 2n
0kl : k + l = 2n
h0l : h + l = 2n
hk0: h + k = 2n
h00 : h = 2n
0k0: k = 2n
00l : l = 2n
Special: as above, plus
4 c ..20,
1
4
,z 0,
3
4
, ¯z +
1
2
hk0: h = 2n
4 b . 2 .
1
4
,y,0
1
4
, ¯y,
1
2
h0l : h = 2n
4 a 2 .. x, 0,
1
4
¯x +
1
2
,0,
3
4
0kl : k = 2n
Symmetry of special projections
Along [001] c2mm
a
= ab
= b
Origin at
1
4
,0, z
Along [100] c2mm
a
= bb
= c
Origin at x,
1
4
,0
Along [010] c2mm
a
= cb
= a
Origin at 0,y,
1
4
Maximal non-isomorphic subgroups
I
[2] I 112
1
(C 2, 5) (1; 2)+
[2] I 12
1
1(C 2, 5) (1; 3)+
[2] I 2
1
11(C 2, 5) (1; 4)+
IIa [2] P2
1
2
1
2
1
(19) 1; 2; 3; 4
[2] P222
1
(17) 1; 2; (3; 4)+(
1
2
,
1
2
,
1
2
)
[2] P22
1
2(P222
1
, 17) 1; 3; (2; 4)+(
1
2
,
1
2
,
1
2
)
[2] P2
1
22(P222
1
, 17) 1; 4; (2; 3)+(
1
2
,
1
2
,
1
2
)
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] I 2
1
2
1
2
1
(a
= 3a or b
= 3b or c
= 3c) (24)
Minimal non-isomorphic supergroups
I
[2] Ibca(73); [2] Imma (74); [2] I 4
1
22 (98); [2] I
¯
42d (122); [3] I 2
1
3 (199)
II [2] A222(a
=
1
2
a)(C 222, 21); [2] B222 (b
=
1
2
b)(C222, 21); [2] C 222(c
=
1
2
c) (21)
217
Pmm2 C
1
2v
mm2 Orthorhombic
No. 25 Pmm2
Patterson symmetry Pmmm
Origin on mm2
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
2
;0≤ z ≤ 1
Symmetry operations
(1) 1 (2) 2 0,0,z (3) mx,0, z (4) m 0,y,z
218
International Tables for Crystallography (2006). Vol. A, Space group 25, pp. 218–219.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 25 Pmm2
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
General:
4 i 1(1)x,y,z (2) ¯x, ¯y,z (3) x, ¯y,z (4) ¯x,y,z no conditions
Special: no extra conditions
2 hm..
1
2
,y,z
1
2
, ¯y, z
2 gm.. 0, y,z 0, ¯y,z
2 f . m . x,
1
2
,z ¯x,
1
2
,z
2 e . m . x,0,z ¯x, 0,z
1 dmm2
1
2
,
1
2
,z
1 cmm2
1
2
,0, z
1 bmm20,
1
2
,z
1 amm20, 0,z
Symmetry of special projections
Along [001] p2mm
a
= ab
= b
Origin at 0,0,z
Along [100] p1m1
a
= bb
= c
Origin at x,0,0
Along [010] p11m
a
= cb
= a
Origin at 0,y, 0
Maximal non-isomorphic subgroups
I
[2] P1m1(Pm,6) 1; 3
[2] Pm11(Pm,6) 1; 4
[2] P112 (P2, 3) 1; 2
IIa none
IIb [2] Pma2(a
= 2a) (28); [2] Pbm2(b
= 2b)(Pma2, 28); [2] Pcc2(c
= 2c) (27); [2] Pmc2
1
(c
= 2c) (26);
[2] Pcm2
1
(c
= 2c)(Pmc2
1
, 26); [2] Aem2(b
= 2b,c
= 2c) (39); [2] Amm2(b
= 2b,c
= 2c) (38);
[2] Bme2(a
= 2a,c
= 2c)(Aem2, 39); [2] Bmm2(a
= 2a,c
= 2c)(Amm2, 38); [2] Cmm2(a
= 2a,b
= 2b) (35);
[2] Fmm2(a
= 2a,b
= 2b,c
= 2c) (42)
Maximal isomorphic subgroups of lowest index
IIc
[2] Pmm2(a
= 2a or b
= 2b) (25); [2] Pmm2(c
= 2c) (25)
Minimal non-isomorphic supergroups
I
[2] Pmmm (47); [2] Pmma (51); [2] Pmmn (59); [2] P4mm (99); [2] P4
2
mc(105); [2] P
¯
4m2 (115)
II [2] Cmm2 (35); [2] Amm2 (38); [2] Bmm2(Amm2, 38); [2] Imm2 (44)
219
Pmc2
1
C
2
2v
mm2 Orthorhombic
No. 26 Pmc2
1
Patterson symmetry Pmmm
Origin on mc2
1
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
2
;0≤ z ≤ 1
Symmetry operations
(1) 1 (2) 2(0, 0,
1
2
) 0,0,z (3) cx,0,z (4) m 0,y, z
220
International Tables for Crystallography (2006). Vol. A, Space group 26, pp. 220–221.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 26 Pmc2
1
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
General:
4 c 1(1)x,y,z (2) ¯x, ¯y,z +
1
2
(3) x, ¯y, z +
1
2
(4) ¯x,y,zh0l : l = 2n
00l : l = 2n
Special: no extra conditions
2 bm..
1
2
,y,z
1
2
, ¯y, z +
1
2
2 am.. 0, y,z 0, ¯y,z +
1
2
Symmetry of special projections
Along [001] p2mm
a
= ab
= b
Origin at 0,0,z
Along [100] p1g1
a
= bb
= c
Origin at x,0,0
Along [010] p11m
a
=
1
2
cb
= a
Origin at 0,y, 0
Maximal non-isomorphic subgroups
I
[2] P1c1(Pc,7) 1; 3
[2] Pm11(Pm,6) 1; 4
[2] P112
1
(P2
1
,4) 1; 2
IIa none
IIb [2] Pmn2
1
(a
= 2a) (31); [2] Pbc2
1
(b
= 2b)(Pca2
1
, 29); [2]Cmc2
1
(a
= 2a,b
= 2b) (36)
Maximal isomorphic subgroups of lowest index
IIc
[2] Pmc2
1
(a
= 2a) (26); [2] Pmc2
1
(b
= 2b) (26); [3] Pmc2
1
(c
= 3c) (26)
Minimal non-isomorphic supergroups
I
[2] Pmma (51); [2] Pbam (55); [2] Pbcm (57); [2] Pnma (62)
II [2] Cmc2
1
(36); [2] Amm2 (38); [2] Bme2(Aem2, 39); [2] Ima2 (46); [2] Pmm2(c
=
1
2
c) (25)
221