Hahn Th. (ed.) International tables for crystallography. Vol. A. Space-group symmetry
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P42
1
2 D
2
4
422 Tetragonal
No. 90 P42
1
2 Patterson symmetry P4/mmm
Origin at 222 at 212
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,
1
2
,z (4) 4
−
1
2
,0, z
(5) 2(0,
1
2
,0)
1
4
,y,0(6)2(
1
2
,0, 0) x,
1
4
,0(7)2x,x , 0(8)2x, ¯x,0
364
International Tables for Crystallography (2006). Vol. A, Space group 90, pp. 364–365.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 90 P42
1
2
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 g 1(1)x,y,z (2) ¯x, ¯y,z (3) ¯y +
1
2
,x +
1
2
,z (4) y +
1
2
, ¯x +
1
2
,z
(5) ¯x+
1
2
,y +
1
2
, ¯z (6) x +
1
2
, ¯y +
1
2
, ¯z (7) y,x, ¯z (8) ¯y, ¯x, ¯z
h00 : h = 2n
Special: as above, plus
4 f ..2 x,x,
1
2
¯x, ¯x,
1
2
¯x +
1
2
,x +
1
2
,
1
2
x +
1
2
, ¯x +
1
2
,
1
2
0kl : k = 2n
4 e ..2 x,x, 0¯x, ¯x,0¯x +
1
2
,x +
1
2
,0 x +
1
2
, ¯x +
1
2
,00kl : k = 2n
4 d 2 .. 0,0,z
1
2
,
1
2
,z
1
2
,
1
2
, ¯z 0, 0, ¯zhkl: h + k = 2n
2 c 4 .. 0,
1
2
,z
1
2
,0, ¯zhk0: h + k = 2n
2 b 2 . 22 0,0,
1
2
1
2
,
1
2
,
1
2
hkl : h + k = 2n
2 a 2 . 22 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,
1
2
,z
Along [100] p2mg
a
= bb
= c
Origin at x,
1
4
,0
Along [110] p2mm
a
=
1
2
(−a + b) b
= c
Origin at x,x , 0
Maximal non-isomorphic subgroups
I
[2] P411 (P4, 75) 1; 2; 3; 4
[2] P212 (C222, 21) 1; 2; 7; 8
[2] P22
1
1(P2
1
2
1
2, 18) 1; 2; 5; 6
IIa none
IIb [2] P4
2
2
1
2(c
= 2c) (94)
Maximal isomorphic subgroups of lowest index
IIc
[2] P42
1
2(c
= 2c) (90); [9] P42
1
2(a
= 3a,b
= 3b) (90)
Minimal non-isomorphic supergroups
I
[2] P4/mbm(127); [2] P4/mnc (128); [2] P4/nmm(129); [2] P4/ncc(130)
II [2] C 422(P422, 89); [2] I 422(97)
365
P4
1
22 D
3
4
422 Tetragonal
No. 91 P4
1
22 Patterson symmetry P4/mmm
Origin on 2[010] at 4
1
(1,2) 1
Asymmetric unit 0 ≤ x ≤ 1; 0 ≤ y ≤ 1; 0 ≤ z ≤
1
8
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
(5) 2 0,y, 0(6)2x,0,
1
4
(7) 2 x,x,
3
8
(8) 2 x, ¯x,
1
8
366
International Tables for Crystallography (2006). Vol. A, Space group 91, pp. 366–367.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 91 P4
1
22
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 d 1(1)x,y,z (2) ¯x, ¯y,z +
1
2
(3) ¯y,x,z +
1
4
(4) y, ¯x,z +
3
4
(5) ¯x, y, ¯z (6) x, ¯y, ¯z +
1
2
(7) y,x, ¯z +
3
4
(8) ¯y, ¯x, ¯z +
1
4
00l : l = 4n
Special: as above, plus
4 c ..2 x,x,
3
8
¯x, ¯x,
7
8
¯x,x,
5
8
x, ¯x,
1
8
0kl : l = 2n + 1
or l = 4n
4 b . 2 .
1
2
,y,0
1
2
, ¯y,
1
2
¯y,
1
2
,
1
4
y,
1
2
,
3
4
hhl : l = 2n + 1
or l = 4n
4 a . 2 . 0,y,00, ¯y,
1
2
¯y,0,
1
4
y, 0,
3
4
hhl : l = 2n + 1
or l = 4n
Symmetry of special projections
Along [001] p4mm
a
= ab
= b
Origin at 0,0,z
Along [100] p2gm
a
= bb
= c
Origin at x,0,
1
4
Along [110] p2gm
a
=
1
2
(−a + b) b
= c
Origin at x,x ,
3
8
Maximal non-isomorphic subgroups
I
[2] P4
1
11(P4
1
, 76) 1; 2; 3; 4
[2] P2
1
21(P222
1
, 17) 1; 2; 5; 6
[2] P2
1
12(C222
1
, 20) 1; 2; 7; 8
IIa none
IIb [2] C 4
1
22
1
(a
= 2a,b
= 2b)(P4
1
2
1
2, 92)
Maximal isomorphic subgroups of lowest index
IIc
[2] C 4
1
22(a
= 2a,b
= 2b)(P4
1
22, 91); [3] P4
3
22(c
= 3c) (95); [5] P4
1
22(c
= 5c) (91)
Minimal non-isomorphic supergroups
I
none
II [2] I 4
1
22 (98); [2] P4
2
22(c
=
1
2
c) (93)
367
P4
1
2
1
2 D
4
4
422 Tetragonal
No. 92 P4
1
2
1
2 Patterson symmetry P4/mmm
Origin on 2[110] at 2
1
1(1, 2)
Asymmetric unit 0 ≤ x ≤ 1; 0 ≤ y ≤ 1; 0 ≤ z ≤
1
8
Symmetry operations
(1) 1 (2) 2(0,0,
1
2
) 0,0,z (3) 4
+
(0,0,
1
4
) 0,
1
2
,z (4) 4
−
(0,0,
3
4
)
1
2
,0, z
(5) 2(0,
1
2
,0)
1
4
,y,
1
8
(6) 2(
1
2
,0, 0) x,
1
4
,
3
8
(7) 2 x,x,0(8)2x, ¯x,
1
4
368
International Tables for Crystallography (2006). Vol. A, Space group 92, pp. 368–369.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 92 P4
1
2
1
2
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 b 1(1)x,y,z (2) ¯x, ¯y,z +
1
2
(3) ¯y +
1
2
,x +
1
2
,z +
1
4
(4) y +
1
2
, ¯x +
1
2
,z +
3
4
(5) ¯x +
1
2
,y +
1
2
, ¯z +
1
4
(6) x +
1
2
, ¯y +
1
2
, ¯z +
3
4
(7) y,x, ¯z (8) ¯y, ¯x, ¯z +
1
2
00l : l = 4n
h00 : h = 2n
Special: as above, plus
4 a ..2 x,x,0¯x, ¯x,
1
2
¯x +
1
2
,x +
1
2
,
1
4
x +
1
2
, ¯x +
1
2
,
3
4
0kl : l = 2n + 1
or 2k + l = 4n
Symmetry of special projections
Along [001] p4gm
a
= ab
= b
Origin at 0,
1
2
,z
Along [100] p2gg
a
= bb
= c
Origin at x,
1
4
,
3
8
Along [110] p2gm
a
=
1
2
(−a + b) b
= c
Origin at x, x,0
Maximal non-isomorphic subgroups
I
[2] P4
1
11(P4
1
, 76) 1; 2; 3; 4
[2] P2
1
12(C222
1
, 20) 1; 2; 7; 8
[2] P2
1
2
1
1(P2
1
2
1
2
1
, 19) 1; 2; 5; 6
IIa none
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] P4
3
2
1
2(c
= 3c) (96); [5] P4
1
2
1
2(c
= 5c) (92); [9] P4
1
2
1
2(a
= 3a,b
= 3b) (92)
Minimal non-isomorphic supergroups
I
[3] P4
1
32 (213)
II [2] C 4
1
22(P4
1
22, 91); [2] I 4
1
22 (98); [2] P4
2
2
1
2(c
=
1
2
c) (94)
369
P4
2
22 D
5
4
422 Tetragonal
No. 93 P4
2
22 Patterson symmetry P4/mmm
Origin at 222 at 4
2
21
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤ 1; 0 ≤ z ≤
1
4
Symmetry operations
(1) 1 (2) 2 0,0, z (3) 4
+
(0,0,
1
2
) 0,0,z (4) 4
−
(0,0,
1
2
) 0,0,z
(5) 2 0,y, 0(6)2x,0,0(7)2x, x,
1
4
(8) 2 x, ¯x,
1
4
370
International Tables for Crystallography (2006). Vol. A, Space group 93, pp. 370–371.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 93 P4
2
22
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 p 1(1)x,y,z (2) ¯x, ¯y,z (3) ¯y,x, z +
1
2
(4) y, ¯x, z +
1
2
(5) ¯x, y, ¯z (6) x, ¯y, ¯z (7) y,x, ¯z +
1
2
(8) ¯y, ¯x, ¯z +
1
2
00l : l = 2n
Special: as above, plus
4 o ..2 x,x,
3
4
¯x, ¯x,
3
4
¯x,x,
1
4
x, ¯x,
1
4
0kl : l = 2n
4 n ..2 x,x,
1
4
¯x, ¯x,
1
4
¯x,x,
3
4
x, ¯x,
3
4
0kl : l = 2n
4 m . 2 . x,
1
2
,0¯x,
1
2
,0
1
2
,x,
1
2
1
2
, ¯x,
1
2
hhl : l = 2n
4 l . 2 . x,0,
1
2
¯x,0,
1
2
0,x, 00, ¯x,0 hhl : l = 2n
4 k . 2 . x,
1
2
,
1
2
¯x,
1
2
,
1
2
1
2
,x,0
1
2
, ¯x,0 hhl : l = 2n
4 j . 2 . x, 0,0¯x,0,00,x,
1
2
0, ¯x,
1
2
hhl : l = 2n
4 i 2 .. 0,
1
2
,z
1
2
,0, z +
1
2
0,
1
2
, ¯z
1
2
,0, ¯z +
1
2
hkl : h + k + l = 2n
4 h 2 ..
1
2
,
1
2
,z
1
2
,
1
2
,z +
1
2
1
2
,
1
2
, ¯z
1
2
,
1
2
, ¯z +
1
2
hkl : l = 2n
4 g 2 .. 0,0,z 0,0,z +
1
2
0,0, ¯z 0,0 , ¯z +
1
2
hkl : l = 2n
2 f 2 . 22
1
2
,
1
2
,
1
4
1
2
,
1
2
,
3
4
hkl : l = 2n
2 e 2 . 22 0,0,
1
4
0,0,
3
4
hkl : l = 2n
2 d 222. 0,
1
2
,
1
2
1
2
,0, 0 hkl : h + k + l = 2n
2 c 222. 0,
1
2
,0
1
2
,0,
1
2
hkl : h + k + l = 2n
2 b 222.
1
2
,
1
2
,0
1
2
,
1
2
,
1
2
hkl : l = 2n
2 a 222. 0,0, 00,0,
1
2
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,0,0
Along [110] p2mm
a
=
1
2
(−a + b) b
= c
Origin at x,x ,
1
4
Maximal non-isomorphic subgroups
I
[2] P4
2
11(P4
2
, 77) 1; 2; 3; 4
[2] P212 (C222, 21) 1; 2; 7; 8
[2] P221 (P222, 16) 1; 2; 5; 6
IIa none
IIb [2] P4
3
22(c
= 2c) (95); [2] P4
1
22(c
= 2c) (91); [2] C4
2
22
1
(a
= 2a,b
= 2b)(P4
2
2
1
2, 94);
[2] F 4
1
22(a
= 2a,b
= 2b,c
= 2c)(I 4
1
22, 98)
Maximal isomorphic subgroups of lowest index
IIc
[2] C 4
2
22(a
= 2a,b
= 2b)(P4
2
22, 93); [3] P4
2
22(c
= 3c) (93)
Minimal non-isomorphic supergroups
I
[2] P4
2
/mmc (131); [2] P4
2
/mcm (132); [2] P4
2
/nbc(133); [2] P4
2
/nnm(134); [3] P4
2
32 (208)
II [2] I 422 (97); [2] P422 (c
=
1
2
c) (89)
371
P4
2
2
1
2 D
6
4
422 Tetragonal
No. 94 P4
2
2
1
2 Patterson symmetry P4/mmm
Origin at 222 at 212
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,
1
2
) 0,
1
2
,z (4) 4
−
(0,0,
1
2
)
1
2
,0, z
(5) 2(0,
1
2
,0)
1
4
,y,
1
4
(6) 2(
1
2
,0, 0) x,
1
4
,
1
4
(7) 2 x,x,0(8)2x, ¯x,0
372
International Tables for Crystallography (2006). Vol. A, Space group 94, pp. 372–373.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 94 P4
2
2
1
2
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 g 1(1)x,y,z (2) ¯x, ¯y,z (3) ¯y +
1
2
,x +
1
2
,z +
1
2
(4) y +
1
2
, ¯x +
1
2
,z +
1
2
(5) ¯x +
1
2
,y +
1
2
, ¯z +
1
2
(6) x +
1
2
, ¯y +
1
2
, ¯z +
1
2
(7) y,x, ¯z (8) ¯y, ¯x, ¯z
00l : l = 2n
h00 : h = 2n
Special: as above, plus
4 f ..2 x,x,
1
2
¯x, ¯x,
1
2
¯x +
1
2
,x +
1
2
,0 x +
1
2
, ¯x +
1
2
,00kl : k + l = 2n
4 e ..2 x,x, 0¯x, ¯x,0¯x +
1
2
,x +
1
2
,
1
2
x +
1
2
, ¯x +
1
2
,
1
2
0kl : k + l = 2 n
4 d 2 .. 0,
1
2
,z 0,
1
2
,z +
1
2
1
2
,0, ¯z +
1
2
1
2
,0, ¯zhkl: l = 2n
hk0: h + k = 2n
4 c 2 .. 0,0,z
1
2
,
1
2
,z +
1
2
1
2
,
1
2
, ¯z +
1
2
0,0, ¯zhkl: h + k + l = 2n
2 b 2 . 22 0,0,
1
2
1
2
,
1
2
,0 hkl : h + k + l = 2n
2 a 2 . 22 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,
1
2
,z
Along [100] p2mg
a
= bb
= c
Origin at x,
1
4
,
1
4
Along [110] p2mm
a
=
1
2
(−a + b) b
= c
Origin at x,x , 0
Maximal non-isomorphic subgroups
I
[2] P4
2
11(P4
2
, 77) 1; 2; 3; 4
[2] P212 (C222, 21) 1; 2; 7; 8
[2] P22
1
1(P2
1
2
1
2, 18) 1; 2; 5; 6
IIa none
IIb [2] P4
3
2
1
2(c
= 2c) (96); [2] P4
1
2
1
2(c
= 2c) (92)
Maximal isomorphic subgroups of lowest index
IIc
[3] P4
2
2
1
2(c
= 3c) (94); [9] P4
2
2
1
2(a
= 3a,b
= 3b) (94)
Minimal non-isomorphic supergroups
I
[2] P4
2
/mbc (135); [2] P4
2
/mnm (136); [2] P4
2
/nmc(137); [2] P4
2
/ncm(138)
II [2] C 4
2
22(P4
2
22, 93); [2] I 422 (97); [2] P42
1
2(c
=
1
2
c) (90)
373