Hahn Th. (ed.) International tables for crystallography. Vol. A. Space-group symmetry
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I 4
1
cd C
12
4v
4mm Tetragonal
No. 110 I 4
1
cd Patterson symmetry I 4/mmm
Origin on 2c1
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
2
;0≤ z ≤
1
4
Symmetry operations
For (0,0,0)+ set
(1) 1 (2) 2(0,0,
1
2
)
1
4
,
1
4
,z (3) 4
+
(0,0,
1
4
) −
1
4
,
1
4
,z (4) 4
−
(0,0,
3
4
)
1
4
,−
1
4
,z
(5) cx,0,z (6) b
1
4
,y,z (7) d(−
1
4
,
1
4
,
3
4
) x +
1
4
, ¯x, z (8) d(
1
4
,
1
4
,
1
4
) x +
1
4
,x,z
For (
1
2
,
1
2
,
1
2
)+ set
(1) t(
1
2
,
1
2
,
1
2
) (2) 2 0, 0,z (3) 4
+
(0,0,
3
4
)
1
4
,
1
4
,z (4) 4
−
(0,0,
1
4
)
1
4
,
1
4
,z
(5) ax,
1
4
,z (6) c 0,y,z (7) d(
1
4
,−
1
4
,
1
4
) x +
1
4
, ¯x, z (8) d(
1
4
,
1
4
,
3
4
) x −
1
4
,x,z
404
International Tables for Crystallography (2006). Vol. A, Space group 110, pp. 404–405.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 110 I 4
1
cd
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 b 1(1)x,y, z (2) ¯x +
1
2
, ¯y +
1
2
,z +
1
2
(3) ¯y,x +
1
2
,z +
1
4
(4) y +
1
2
, ¯x, z +
3
4
(5) x, ¯y, z +
1
2
(6) ¯x+
1
2
,y +
1
2
,z (7) ¯y, ¯x +
1
2
,z +
3
4
(8) y +
1
2
,x,z +
1
4
hkl : h + k + l = 2n
hk0: h + k = 2n
0kl : k,l = 2n
hhl :2h + l = 4n
00l : l = 4n
h00 : h = 2n
h
¯
h0: h = 2n
Special: as above, plus
8 a 2 .. 0, 0,z 0,
1
2
,z +
1
4
0,0,z +
1
2
0,
1
2
,z +
3
4
hkl :2h + l = 4n
Symmetry of special projections
Along [001] p4gm
a
=
1
2
(a − b) b
=
1
2
(a + b)
Origin at
1
4
,
1
4
,z
Along [100] p1m1
a
=
1
2
bb
=
1
2
c
Origin at x,0,0
Along [110] c1m1
a
=
1
2
(−a + b) b
=
1
2
c
Origin at x,x , 0
Maximal non-isomorphic subgroups
I
[2] I 4
1
11(I 4
1
, 80) (1; 2; 3; 4)+
[2] I 2c1(Iba2, 45) (1; 2; 5; 6)+
[2] I 21d (Fdd2, 43) (1; 2; 7; 8)+
IIa none
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] I 4
1
cd(c
= 3c) (110); [9] I 4
1
cd(a
= 3a,b
= 3b) (110)
Minimal non-isomorphic supergroups
I
[2] I 4
1
/acd (142)
II [2] C 4
2
md (c
=
1
2
c)(P4
2
nm, 102)
405
P
¯
42mD
1
2d
¯
42m Tetragonal
No. 111 P
¯
42m
Patterson symmetry P4/mmm
Origin at
¯
42m
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
2
;0≤ z ≤ 1; x ≤ y
Symmetry operations
(1) 1 (2) 2 0,0, z (3)
¯
4
+
0,0,z;0,0,0(4)
¯
4
−
0,0,z;0,0, 0
(5) 2 0,y, 0(6)2x,0,0(7)mx, ¯x, z (8) mx, x,z
406
International Tables for Crystallography (2006). Vol. A, Space group 111, pp. 406–407.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 111 P
¯
42m
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
General:
8 o 1(1)x, y,z (2) ¯x, ¯y,z (3) y, ¯x, ¯z (4) ¯y,x, ¯z
(5) ¯x, y, ¯z (6) x, ¯y, ¯z (7) ¯y, ¯x,z (8) y,x,z
no conditions
Special:
4 n ..mx,x, z ¯x, ¯x,zx, ¯x, ¯z ¯x,x, ¯z no extra conditions
4 m 2 .. 0,
1
2
,z
1
2
,0, ¯z 0,
1
2
, ¯z
1
2
,0, zhkl: h + k = 2n
4 l . 2 . x,
1
2
,0¯x,
1
2
,0
1
2
, ¯x, 0
1
2
,x,0 no extra conditions
4 k . 2 . x, 0,
1
2
¯x,0,
1
2
0, ¯x,
1
2
0,x,
1
2
no extra conditions
4 j . 2 . x,
1
2
,
1
2
¯x,
1
2
,
1
2
1
2
, ¯x,
1
2
1
2
,x,
1
2
no extra conditions
4 i . 2 . x,0,0¯x,0, 00, ¯x,00,x,0 no extra conditions
2 h 2 . mm
1
2
,
1
2
,z
1
2
,
1
2
, ¯z no extra conditions
2 g 2 . mm 0,0,z 0,0, ¯z no extra conditions
2 f 222.
1
2
,0,
1
2
0,
1
2
,
1
2
hkl : h + k = 2n
2 e 222.
1
2
,0, 00,
1
2
,0 hkl : h + k = 2n
1 d
¯
42m
1
2
,
1
2
,0 no extra conditions
1 c
¯
42m 0,0,
1
2
no extra conditions
1 b
¯
42m
1
2
,
1
2
,
1
2
no extra conditions
1 a
¯
42m 0,0,0 no extra conditions
Symmetry of special projections
Along [001] p4mm
a
= ab
= b
Origin at 0,0,z
Along [100] p2mm
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] P
¯
411 (P
¯
4, 81) 1; 2; 3; 4
[2] P21m (Cmm2, 35) 1; 2; 7; 8
[2] P221 (P222, 16) 1; 2; 5; 6
IIa none
IIb [2] P
¯
42c (c
= 2c) (112); [2] C
¯
42d (a
= 2a,b
= 2b)(P
¯
4b2, 117); [2] C
¯
42m (a
= 2a,b
= 2b)(P
¯
4m2, 115);
[2] F
¯
42c (a
= 2a,b
= 2b,c
= 2c)(I
¯
4c2, 120); [2] F
¯
42m (a
= 2a,b
= 2b,c
= 2c)(I
¯
4m2, 119)
Maximal isomorphic subgroups of lowest index
IIc
[2] P
¯
42m (c
= 2c) (111); [9] P
¯
42m (a
= 3a,b
= 3b) (111)
Minimal non-isomorphic supergroups
I
[2] P4/mmm(123); [2] P4/nbm(125); [2] P4
2
/mcm (132); [2] P4
2
/nnm(134); [3] P
¯
43m (215)
II [2] C
¯
42m (P
¯
4m2, 115); [2] I
¯
42m (121)
407
P
¯
42cD
2
2d
¯
42m Tetragonal
No. 112 P
¯
42c
Patterson symmetry P4/mmm
Origin at
¯
41c
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
2
;0≤ z ≤
1
2
Symmetry operations
(1) 1 (2) 2 0,0,z (3)
¯
4
+
0,0,z;0,0, 0(4)
¯
4
−
0,0,z;0,0, 0
(5) 2 0,y,
1
4
(6) 2 x,0,
1
4
(7) cx, ¯x, z (8) cx,x, z
408
International Tables for Crystallography (2006). Vol. A, Space group 112, pp. 408–409.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 112 P
¯
42c
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
General:
8 n 1(1)x, y,z (2) ¯x, ¯y,z (3) y, ¯x, ¯z (4) ¯y,x, ¯z
(5) ¯x, y, ¯z +
1
2
(6) x, ¯y, ¯z +
1
2
(7) ¯y, ¯x,z +
1
2
(8) y,x,z +
1
2
hhl : l = 2n
00l : l = 2n
Special: as above, plus
4 m 2 .. 0,
1
2
,z
1
2
,0, ¯z 0,
1
2
, ¯z +
1
2
1
2
,0, z +
1
2
hkl : h + k + l = 2n
4 l 2 ..
1
2
,
1
2
,z
1
2
,
1
2
, ¯z
1
2
,
1
2
, ¯z +
1
2
1
2
,
1
2
,z +
1
2
hkl : l = 2n
4 k 2 .. 0,0,z 0,0, ¯z 0,0, ¯z +
1
2
0,0,z +
1
2
hkl : l = 2n
4 j . 2 . 0, y,
1
4
0, ¯y,
1
4
y, 0,
3
4
¯y,0,
3
4
no extra conditions
4 i . 2 . x,
1
2
,
1
4
¯x,
1
2
,
1
4
1
2
, ¯x,
3
4
1
2
,x,
3
4
no extra conditions
4 h . 2 .
1
2
,y,
1
4
1
2
, ¯y,
1
4
y,
1
2
,
3
4
¯y,
1
2
,
3
4
no extra conditions
4 g . 2 . x,0,
1
4
¯x,0,
1
4
0, ¯x,
3
4
0,x,
3
4
no extra conditions
2 f
¯
4 ..
1
2
,
1
2
,0
1
2
,
1
2
,
1
2
hkl : l = 2n
2 e
¯
4 .. 0,0, 00,0,
1
2
hkl : l = 2n
2 d 222. 0,
1
2
,
1
4
1
2
,0,
3
4
hkl : h + k + l = 2n
2 c 222.
1
2
,
1
2
,
1
4
1
2
,
1
2
,
3
4
hkl : l = 2n
2 b 222.
1
2
,0,
1
4
0,
1
2
,
3
4
hkl : h + k + l = 2n
2 a 222. 0,0,
1
4
0,0,
3
4
hkl : l = 2n
Symmetry of special projections
Along [001] p4mm
a
= ab
= b
Origin at 0,0,z
Along [100] p2mm
a
= bb
= c
Origin at x,0,
1
4
Along [110] p1m1
a
=
1
2
(−a + b) b
=
1
2
c
Origin at x,x , 0
Maximal non-isomorphic subgroups
I
[2] P
¯
411 (P
¯
4, 81) 1; 2; 3; 4
[2] P21c (Ccc2, 37) 1; 2; 7; 8
[2] P221 (P222, 16) 1; 2; 5; 6
IIa none
IIb [2] C
¯
42d (a
= 2a,b
= 2b)(P
¯
4n2, 118); [2] C
¯
42c (a
= 2a,b
= 2b)(P
¯
4c2, 116)
Maximal isomorphic subgroups of lowest index
IIc
[3] P
¯
42c (c
= 3c) (112); [9] P
¯
42c (a
= 3a,b
= 3b) (112)
Minimal non-isomorphic supergroups
I
[2] P4/mcc(124); [2] P4/nnc(126); [2] P4
2
/mmc (131); [2] P4
2
/nbc(133); [3] P
¯
43n (218)
II [2] C
¯
42c (P
¯
4c2, 116); [2] I
¯
42m (121); [2] P
¯
42m (c
=
1
2
c) (111)
409
P
¯
42
1
mD
3
2d
¯
42m Tetragonal
No. 113 P
¯
42
1
m Patterson symmetry P4/mmm
Origin at
¯
41g
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
2
;0≤ z ≤ 1; y ≤
1
2
− x
Symmetry operations
(1) 1 (2) 2 0,0, z (3)
¯
4
+
0,0,z;0,0,0(4)
¯
4
−
0,0,z;0,0, 0
(5) 2(0,
1
2
,0)
1
4
,y,0(6)2(
1
2
,0, 0) x,
1
4
,0(7)mx+
1
2
, ¯x, z (8) g(
1
2
,
1
2
,0) x,x,z
410
International Tables for Crystallography (2006). Vol. A, Space group 113, pp. 410–411.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 113 P
¯
42
1
m
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
General:
8 f 1(1)x,y, z (2) ¯x, ¯y,z (3) y, ¯x, ¯z (4) ¯y,x, ¯z
(5) ¯x+
1
2
,y +
1
2
, ¯z (6) x +
1
2
, ¯y +
1
2
, ¯z (7) ¯y +
1
2
, ¯x +
1
2
,z (8) y +
1
2
,x +
1
2
,z
h00 : h = 2n
Special: as above, plus
4 e ..mx,x +
1
2
,z ¯x, ¯x +
1
2
,zx+
1
2
, ¯x, ¯z ¯x+
1
2
,x, ¯z no extra conditions
4 d 2 .. 0,0,z 0,0, ¯z
1
2
,
1
2
, ¯z
1
2
,
1
2
,zhkl: h + k = 2n
2 c 2 . mm 0,
1
2
,z
1
2
,0, ¯zhk0: h + k = 2n
2 b
¯
4 .. 0,0,
1
2
1
2
,
1
2
,
1
2
hkl : h + k = 2n
2 a
¯
4 .. 0,0, 0
1
2
,
1
2
,0 hkl : h + k = 2n
Symmetry of special projections
Along [001] p4gm
a
= ab
= b
Origin at 0,0,z
Along [100] p2mg
a
= bb
= c
Origin at x,
1
4
,0
Along [110] p1m1
a
=
1
2
(−a + b) b
= c
Origin at x,x , 0
Maximal non-isomorphic subgroups
I
[2] P
¯
411 (P
¯
4, 81) 1; 2; 3; 4
[2] P21m (Cmm2, 35) 1; 2; 7; 8
[2] P22
1
1(P2
1
2
1
2, 18) 1; 2; 5; 6
IIa none
IIb [2] P
¯
42
1
c (c
= 2c) (114)
Maximal isomorphic subgroups of lowest index
IIc
[2] P
¯
42
1
m (c
= 2c) (113); [9] P
¯
42
1
m (a
= 3a,b
= 3b) (113)
Minimal non-isomorphic supergroups
I
[2] P4/mbm(127); [2] P4/nmm(129); [2] P4
2
/mnm (136); [2] P4
2
/ncm(138)
II [2] C
¯
42m (P
¯
4m2, 115); [2] I
¯
42m (121)
411
P
¯
42
1
cD
4
2d
¯
42m Tetragonal
No. 114 P
¯
42
1
c Patterson symmetry P4/mmm
Origin at
¯
41n
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
2
;0≤ z ≤
1
2
Symmetry operations
(1) 1 (2) 2 0,0,z (3)
¯
4
+
0,0,z;0,0, 0(4)
¯
4
−
0,0,z;0,0, 0
(5) 2(0,
1
2
,0)
1
4
,y,
1
4
(6) 2(
1
2
,0, 0) x,
1
4
,
1
4
(7) cx+
1
2
, ¯x,z (8) n(
1
2
,
1
2
,
1
2
) x,x,z
412
International Tables for Crystallography (2006). Vol. A, Space group 114, pp. 412–413.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 114 P
¯
42
1
c
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
General:
8 e 1(1)x,y,z (2) ¯x, ¯y,z (3) y, ¯x, ¯z (4) ¯y,x, ¯z
(5) ¯x +
1
2
,y +
1
2
, ¯z +
1
2
(6) x +
1
2
, ¯y +
1
2
, ¯z +
1
2
(7) ¯y +
1
2
, ¯x +
1
2
,z +
1
2
(8) y +
1
2
,x +
1
2
,z +
1
2
hhl : l = 2n
00l : l = 2n
h00 : h = 2n
Special: as above, plus
4 d 2 .. 0,
1
2
,z
1
2
,0, ¯z
1
2
,0, ¯z +
1
2
0,
1
2
,z +
1
2
hkl : l = 2n
hk0: h + k = 2n
4 c 2 .. 0, 0,z 0,0, ¯z
1
2
,
1
2
, ¯z +
1
2
1
2
,
1
2
,z +
1
2
hkl : h + k + l = 2n
2 b
¯
4 .. 0,0,
1
2
1
2
,
1
2
,0 hkl : h + k + l = 2n
2 a
¯
4 .. 0,0,0
1
2
,
1
2
,
1
2
hkl : h + k + l = 2n
Symmetry of special projections
Along [001] p4gm
a
= ab
= b
Origin at 0,0,z
Along [100] p2mg
a
= bb
= c
Origin at x,
1
4
,
1
4
Along [110] p1m1
a
=
1
2
(−a + b) b
=
1
2
c
Origin at x, x,0
Maximal non-isomorphic subgroups
I
[2] P
¯
411 (P
¯
4, 81) 1; 2; 3; 4
[2] P21c (Ccc2, 37) 1; 2; 7; 8
[2] P22
1
1(P2
1
2
1
2, 18) 1; 2; 5; 6
IIa none
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] P
¯
42
1
c (c
= 3c) (114); [9] P
¯
42
1
c (a
= 3a,b
= 3b) (114)
Minimal non-isomorphic supergroups
I
[2] P4/mnc(128); [2] P4/ncc(130); [2] P4
2
/mbc (135); [2] P4
2
/nmc(137)
II [2] C
¯
42c (P
¯
4c2, 116); [2] I
¯
42m (121); [2] P
¯
42
1
m (c
=
1
2
c) (113)
413