Hahn Th. (ed.) International tables for crystallography. Vol. A. Space-group symmetry
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Immm D
25
2h
mmm Orthorhombic
No. 71 I 2 /m 2/m 2/m
Patterson symmetry Immm
Origin at centre (mmm)
Asymmetric unit 0 ≤ x ≤
1
4
;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
(5)
¯
10, 0,0(6)mx,y,0(7)mx,0,z (8) m 0,y, z
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)
¯
1
1
4
,
1
4
,
1
4
(6) n(
1
2
,
1
2
,0) x,y,
1
4
(7) n(
1
2
,0,
1
2
) x,
1
4
,z (8) n(0,
1
2
,
1
2
)
1
4
,y,z
Maximal isomorphic subgroups of lowest index
IIc
[3] Immm (a
= 3a or b
= 3b or c
= 3c) (71)
Minimal non-isomorphic supergroups
I
[2] I 4/mmm(139); [3] Im
¯
3 (204)
II [2] Ammm (a
=
1
2
a)(Cmmm, 65); [2] Bmmm (b
=
1
2
b)(Cmmm, 65); [2] Cmmm (c
=
1
2
c) (65)
324
International Tables for Crystallography (2006). Vol. A, Space group 71, pp. 324–325.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 71 Immm
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
General:
16 o 1(1)x,y,z (2) ¯x, ¯y,z (3) ¯x,y, ¯z (4) x, ¯y, ¯z
(5) ¯x, ¯y, ¯z (6) x, y, ¯z (7) x, ¯y, z (8) ¯x,y,z
hkl : h + k + l = 2n
0kl : k + l = 2 n
h0l : h + l = 2n
hk0: h + k = 2n
h00 :
h = 2n
0k0: k = 2n
00l : l = 2n
Special: as above, plus
8 n ..mx, y,0¯x, ¯y, 0¯x,y,0 x, ¯y,0 no extra conditions
8 m . m . x,0,z ¯x, 0,z ¯x,0, ¯zx,0, ¯z no extra conditions
8 lm.. 0,y, z 0, ¯y,z 0,y, ¯z 0, ¯y, ¯z no extra conditions
8 k
¯
1
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
4 jmm2
1
2
,0, z
1
2
,0, ¯z no extra conditions
4 imm20,0, z 0,0, ¯z no extra conditions
4 hm2 m 0,y,
1
2
0, ¯y,
1
2
no extra conditions
4 gm2 m 0,y,00, ¯y,0 no extra conditions
4 f 2 mm x,
1
2
,0¯x,
1
2
,0 no extra conditions
4 e 2 mm x, 0,0¯x,0, 0 no extra conditions
2 dmmm
1
2
,0,
1
2
no extra conditions
2 cmmm
1
2
,
1
2
,0 no extra conditions
2 bmmm 0,
1
2
,
1
2
no extra conditions
2 ammm 0,0,0 no extra conditions
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] Imm2 (44) (1; 2; 7; 8)+
[2] Im2m (Imm2, 44) (1; 3; 6; 8)+
[2] I 2mm (Imm2, 44) (1; 4; 6; 7)+
[2] I 222(23) (1; 2; 3; 4)+
[2] I 112/m (C2/m, 12) (1; 2; 5; 6)+
[2] I 12/m1(C 2 /m, 12) (1; 3; 5; 7)+
[2] I 2/m11(C2/m, 12) (1; 4; 5; 8)+
IIa [2] Pmmn (59) 1; 2; 7; 8; (3; 4; 5; 6)+(
1
2
,
1
2
,
1
2
)
[2] Pmnm(Pmmn, 59) 1; 3; 6; 8; (2; 4; 5; 7)+(
1
2
,
1
2
,
1
2
)
[2] Pnmm (Pmmn, 59) 1; 4; 6; 7; (2; 3; 5; 8)+(
1
2
,
1
2
,
1
2
)
[2] Pnnm(58) 1; 2; 5; 6; (3; 4; 7; 8)+(
1
2
,
1
2
,
1
2
)
[2] Pnmn(Pnnm, 58) 1; 3; 5; 7; (2; 4; 6; 8)+(
1
2
,
1
2
,
1
2
)
[2] Pmnn(Pnnm, 58) 1; 4; 5; 8; (2; 3; 6; 7)+(
1
2
,
1
2
,
1
2
)
[2] Pnnn(48) 1; 2; 3; 4; (5; 6; 7; 8)+(
1
2
,
1
2
,
1
2
)
[2] Pmmm (47) 1; 2; 3; 4; 5; 6; 7; 8
IIb none
(Continued on preceding page)
325
Ibam D
26
2h
mmm Orthorhombic
No. 72 I 2 /b 2/a 2/m
Patterson symmetry Immm
Origin at centre (2/m) at cc2/m
Asymmetric unit 0 ≤ x ≤
1
4
;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,
1
2
,0)
1
4
,y,0(4)2(
1
2
,0, 0) x,
1
4
,0
(5)
¯
10, 0,0(6)mx,y,0(7)ax,
1
4
,z (8) b
1
4
,y,z
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, y,
1
4
(4) 2 x,0,
1
4
(5)
¯
1
1
4
,
1
4
,
1
4
(6) n(
1
2
,
1
2
,0) x,y,
1
4
(7) cx,0,z (8) c 0,y,z
Maximal isomorphic subgroups of lowest index
IIc
[3] Ibam (a
= 3a or b
= 3b) (72); [3] Ibam (c
= 3c) (72)
Minimal non-isomorphic supergroups
I
[2] I 4/mcm(140)
II [2] Cmmm (c
=
1
2
c) (65); [2] Aemm (a
=
1
2
a)(Cmme, 67); [2] Bmem (b
=
1
2
b)(Cmme, 67)
326
International Tables for Crystallography (2006). Vol. A, Space group 72, pp. 326–327.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 72 Ibam
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
General:
16 k 1(1)x, y,z (2) ¯x, ¯y, z (3) ¯x +
1
2
,y +
1
2
, ¯z (4) x +
1
2
, ¯y +
1
2
, ¯z
(5) ¯x, ¯y, ¯z (6) x, y, ¯z (7) x +
1
2
, ¯y +
1
2
,z (8) ¯x +
1
2
,y +
1
2
,z
hkl : 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
8 j ..mx, y,0¯x, ¯y, 0¯x+
1
2
,y +
1
2
,0 x +
1
2
, ¯y +
1
2
,0 no extra conditions
8 i ..20,
1
2
,z
1
2
,0, ¯z 0,
1
2
, ¯z
1
2
,0, zhkl: l = 2n
8 h ..20,0, z
1
2
,
1
2
, ¯z 0,0, ¯z
1
2
,
1
2
,zhkl: l = 2n
8 g . 2 . 0,y,
1
4
0, ¯y,
1
4
0, ¯y,
3
4
0,y,
3
4
hkl : l = 2n
8 f 2 .. x,0 ,
1
4
¯x,0,
1
4
¯x,0,
3
4
x,0,
3
4
hkl : l = 2n
8 e
¯
1
1
4
,
1
4
,
1
4
3
4
,
3
4
,
1
4
1
4
,
3
4
,
3
4
3
4
,
1
4
,
3
4
hkl : k,l = 2n
4 d ..2/m
1
2
,0, 00,
1
2
,0 hkl : l = 2n
4 c ..2/m 0,0,0
1
2
,
1
2
,0 hkl : l = 2n
4 b 222
1
2
,0,
1
4
1
2
,0,
3
4
hkl : l = 2n
4 a 222 0,0,
1
4
0,0,
3
4
hkl : l = 2n
Symmetry of special projections
Along [001] c2mm
a
= ab
= 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] Ib2m (Ima2, 46) (1; 3; 6; 8)+
[2] I 2am(Ima2, 46) (1; 4; 6; 7)+
[2] Iba2 (45) (1; 2; 7; 8)+
[2] I 222(23) (1; 2; 3; 4)+
[2] I 12/a1(C2/c , 15) (1; 3; 5; 7)+
[2] I 2/b11(C 2/c, 15) (1; 4; 5; 8)+
[2] I 112/m (C2/m, 12) (1; 2; 5; 6)+
IIa [2] Pcan( Pbcn, 60) 1; 3; 5; 7; (2; 4; 6; 8)+(
1
2
,
1
2
,
1
2
)
[2] Pbcn (60) 1; 4; 5; 8; (2; 3; 6; 7)+(
1
2
,
1
2
,
1
2
)
[2] Pbcm (57) 1; 3; 6; 8; (2; 4; 5; 7)+(
1
2
,
1
2
,
1
2
)
[2] Pcam(Pbcm, 57) 1; 4; 6; 7; (2; 3; 5; 8)+(
1
2
,
1
2
,
1
2
)
[2] Pccn (56) 1; 2; 3; 4; (5; 6; 7; 8)+(
1
2
,
1
2
,
1
2
)
[2] Pbam(55) 1; 2; 3; 4; 5; 6; 7; 8
[2] Pban(50) 1; 2; 7; 8; (3; 4; 5; 6)+(
1
2
,
1
2
,
1
2
)
[2] Pccm (49) 1; 2; 5; 6; (3; 4; 7; 8)+(
1
2
,
1
2
,
1
2
)
IIb none
(Continued on preceding page)
327
Ibca D
27
2h
mmm Orthorhombic
No. 73 I 2
1
/b 2
1
/c 2
1
/a Patterson symmetry Immm
Origin at
¯
1atcab
Asymmetric unit 0 ≤ x ≤
1
4
;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
(5)
¯
10, 0,0(6)ax,y,
1
4
(7) cx,
1
4
,z (8) b
1
4
,y,z
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)
¯
1
1
4
,
1
4
,
1
4
(6) bx,y, 0(7)ax,0,z (8) c 0,y,z
328
International Tables for Crystallography (2006). Vol. A, Space group 73, pp. 328–329.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 73 Ibca
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
General:
16 f 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) ¯x, ¯y, ¯z (6) x +
1
2
,y, ¯z +
1
2
(7) x, ¯y +
1
2
,z +
1
2
(8) ¯x+
1
2
,y +
1
2
,z
hkl : 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
8 e ..20,
1
4
,z 0,
3
4
, ¯z +
1
2
0,
3
4
, ¯z 0,
1
4
,z +
1
2
hkl : l = 2n
8 d . 2 .
1
4
,y,0
1
4
, ¯y,
1
2
3
4
, ¯y, 0
3
4
,y,
1
2
hkl : k = 2n
8 c 2 .. x,0,
1
4
¯x+
1
2
,0,
3
4
¯x,0,
3
4
x +
1
2
,0,
1
4
hkl : h = 2n
8 b
¯
1
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
¯
10, 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 [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] Ibc2(Iba2, 45) (1; 2; 7; 8)+
[2] Ib2a (Iba2, 45) (1; 3; 6; 8)+
[2] I 2ca (Iba2, 45) (1; 4; 6; 7)+
[2] I 2
1
2
1
2
1
(24) (1; 2; 3; 4)+
[2] I 112/a (C 2/c, 15) (1; 2; 5; 6)+
[2] I 12/c1(C 2/c, 15) (1; 3; 5; 7)+
[2] I 2/b11(C 2/c, 15) (1; 4; 5; 8)+
IIa [2] Pbca (61) 1; 2; 3; 4; 5; 6; 7; 8
[2] Pcab (Pbca, 61) 1; 2; 3; 4; (5; 6; 7; 8)+(
1
2
,
1
2
,
1
2
)
[2] Pcaa (Pcca, 54) 1; 2; 5; 6; (3; 4; 7; 8)+(
1
2
,
1
2
,
1
2
)
[2] Pccb (Pcca, 54) 1; 3; 5; 7; (2; 4; 6; 8)+(
1
2
,
1
2
,
1
2
)
[2] Pbab(Pcca, 54) 1; 4; 5; 8; (2; 3; 6; 7)+(
1
2
,
1
2
,
1
2
)
[2] Pbcb (Pcca, 54) 1; 2; 7; 8; (3; 4; 5; 6)+(
1
2
,
1
2
,
1
2
)
[2] Pbaa(Pcca, 54) 1; 3; 6; 8; (2; 4; 5; 7)+(
1
2
,
1
2
,
1
2
)
[2] Pcca (54) 1; 4; 6; 7; (2; 3; 5; 8)+(
1
2
,
1
2
,
1
2
)
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] Ibca(a
= 3a or b
= 3b or c
= 3c) (73)
Minimal non-isomorphic supergroups
I
[2] I 4
1
/acd (142); [3] Ia
¯
3 (206)
II [2] Aemm (a
=
1
2
a)(Cmme, 67); [2] Bmem (b
=
1
2
b)(Cmme, 67); [2] Cmme (c
=
1
2
c) (67)
329
Imma D
28
2h
mmm Orthorhombic
No. 74 I 2
1
/m 2
1
/m 2
1
/a Patterson symmetry Immm
Origin at centre (2/m) at 2/m2
1
/nb
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,
1
4
,z (3) 2(0,
1
2
,0) 0, y,0(4)2x,0,0
(5)
¯
10, 0,0(6)bx,y, 0(7)mx,
1
4
,z (8) m 0,y, z
For (
1
2
,
1
2
,
1
2
)+ set
(1) t(
1
2
,
1
2
,
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,
1
4
,
1
4
(5)
¯
1
1
4
,
1
4
,
1
4
(6) ax,y,
1
4
(7) n(
1
2
,0,
1
2
) x,0, z (8) n(0,
1
2
,
1
2
)
1
4
,y,z
Maximal isomorphic subgroups of lowest index
IIc
[3] Imma (a
= 3a or b
= 3b) (74); [3] Imma (c
= 3c) (74)
Minimal non-isomorphic supergroups
I
[2] I 4
1
/amd (141)
II [2] Ammm (a
=
1
2
a)(Cmmm, 65); [2] Bmmm (b
=
1
2
b)(Cmmm, 65); [2] Cmme (c
=
1
2
c) (67)
330
International Tables for Crystallography (2006). Vol. A, Space group 74, pp. 330–331.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 74 Imma
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
General:
16 j 1(1)x,y,z (2) ¯x, ¯y +
1
2
,z (3) ¯x,y +
1
2
, ¯z (4) x, ¯y, ¯z
(5) ¯x, ¯y, ¯z (6) x, y +
1
2
, ¯z (7) x, ¯y +
1
2
,z (8) ¯x,y, z
hkl : 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
8 i . m . x,
1
4
,z ¯x,
1
4
,z ¯x,
3
4
, ¯zx,
3
4
, ¯z no extra conditions
8 hm.. 0,y,z 0, ¯y +
1
2
,z 0,y +
1
2
, ¯z 0, ¯y, ¯z no extra conditions
8 g . 2 .
1
4
,y,
1
4
3
4
, ¯y +
1
2
,
1
4
3
4
, ¯y,
3
4
1
4
,y +
1
2
,
3
4
hkl : h = 2n
8 f 2 .. x,0 , 0¯x,
1
2
,0¯x,0,0 x,
1
2
,0 hkl : k = 2n
4 emm20,
1
4
,z 0,
3
4
, ¯z no extra conditions
4 d . 2/m .
1
4
,
1
4
,
3
4
3
4
,
1
4
,
3
4
hkl : h = 2n
4 c . 2/m .
1
4
,
1
4
,
1
4
3
4
,
1
4
,
1
4
hkl : h = 2n
4 b 2/ m .. 0, 0,
1
2
0,
1
2
,
1
2
hkl : h = 2n
4 a 2/ m .. 0, 0,00,
1
2
,0 hkl : h = 2n
Symmetry of special projections
Along [001] p2mm
a
=
1
2
ab
=
1
2
b
Origin at 0,0,z
Along [100] c2mm
a
= bb
= c
Origin at x,
1
4
,
1
4
Along [010] c2mm
a
= cb
= a
Origin at 0,y, 0
Maximal non-isomorphic subgroups
I
[2] Im2b (Ima2, 46) (1; 3; 6; 8)+
[2] I 2mb (Ima2, 46) (1; 4; 6; 7)+
[2] Imm2 (44) (1; 2; 7; 8)+
[2] I 2
1
2
1
2
1
(24) (1; 2; 3; 4)+
[2] I 112/b (C 2/c, 15) (1; 2; 5; 6)+
[2] I 12/m1(C 2 /m, 12) (1; 3; 5; 7)+
[2] I 2/m11(C2/m, 12) (1; 4; 5; 8)+
IIa [2] Pnma (62) 1; 3; 5; 7; (2; 4; 6; 8)+(
1
2
,
1
2
,
1
2
)
[2] Pmnb (Pnma, 62) 1; 3; 6; 8; (2; 4; 5; 7)+(
1
2
,
1
2
,
1
2
)
[2] Pnmb (Pmna, 53) 1; 4; 6; 7; (2; 3; 5; 8)+(
1
2
,
1
2
,
1
2
)
[2] Pmna (53) 1; 4; 5; 8; (2; 3; 6; 7)+(
1
2
,
1
2
,
1
2
)
[2] Pnna(52) 1; 2; 3; 4; (5; 6; 7; 8)+(
1
2
,
1
2
,
1
2
)
[2] Pnnb(Pnna, 52) 1; 2; 5; 6; (3; 4; 7; 8)+(
1
2
,
1
2
,
1
2
)
[2] Pmma (51) 1; 2; 7; 8; (3; 4; 5; 6)+(
1
2
,
1
2
,
1
2
)
[2] Pmmb (Pmma, 51) 1; 2; 3; 4; 5; 6; 7; 8
IIb none
(Continued on preceding page)
331
P4 C
1
4
4 Tetragonal
No. 75 P4
Patterson symmetry P4/m
Origin on 4
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
2
;0≤ z ≤ 1
Symmetry operations
(1) 1 (2) 2 0,0,z (3) 4
+
0,0,z (4) 4
−
0,0,z
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 d 1(1)x,y,z (2) ¯x, ¯y,z (3) ¯y,x, z (4) y, ¯x,z no conditions
Special:
2 c 2 .. 0,
1
2
,z
1
2
,0, zhkl: h + k = 2n
1 b 4 ..
1
2
,
1
2
,z no extra conditions
1 a 4 .. 0,0,z no extra conditions
Symmetry of special projections
Along [001] p4
a
= ab
= b
Origin at 0,0,z
Along [100] p1m1
a
= bb
= c
Origin at x,0,0
Along [110] p1m1
a
=
1
2
(−a + b) b
= c
Origin at x,x , 0
Maximal non-isomorphic subgroups
I
[2] P2 (3) 1; 2
IIa none
IIb [2] P4
2
(c
= 2c) (77); [2] F 4(a
= 2a,b
= 2b,c
= 2c)(I 4, 79)
Maximal isomorphic subgroups of lowest index
IIc
[2] P4(c
= 2c) (75); [2] C4(a
= 2a,b
= 2b)(P4, 75)
Minimal non-isomorphic supergroups
I
[2] P4/m (83); [2] P4/n (85); [2] P422 (89); [2] P42
1
2 (90); [2] P4mm (99); [2] P4bm (100); [2] P4cc (103); [2] P4nc(104)
II [2] I 4 (79)
332
International Tables for Crystallography (2006). Vol. A, Space group 75, p. 332.
Copyright © 2006 International Union of Crystallography
Tetragonal 4 C
2
4
P4
1
Patterson symmetry P4/m P4
1
No. 76
Origin on 4
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) 4
+
(0,0,
1
4
) 0,0, z (4) 4
−
(0,0,
3
4
) 0,0,z
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 a 1(1)x, y,z (2) ¯x, ¯y,z +
1
2
(3) ¯y,x,z +
1
4
(4) y, ¯x,z +
3
4
00l : l = 4n
Symmetry of special projections
Along [001] p4
a
= ab
= b
Origin at 0,0,z
Along [100] p1g1
a
= bb
= c
Origin at x,0,0
Along [110] p1g1
a
=
1
2
(−a + b) b
= c
Origin at x,x , 0
Maximal non-isomorphic subgroups
I
[2] P2
1
(4) 1; 2
IIa none
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[2] C 4
1
(a
= 2a,b
= 2b)(P4
1
, 76); [3] P4
3
(c
= 3c) (78); [5] P4
1
(c
= 5c) (76)
Minimal non-isomorphic supergroups
I
[2] P4
1
22 (91); [2] P4
1
2
1
2 (92)
II [2] I 4
1
(80); [2] P4
2
(c
=
1
2
c) (77)
333
International Tables for Crystallography (2006). Vol. A, Space group 76, p. 333.
Copyright © 2006 International Union of Crystallography