Hahn Th. (ed.) International tables for crystallography. Vol. A. Space-group symmetry
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P4
3
22 D
7
4
422 Tetragonal
No. 95 P4
3
22 Patterson symmetry P4/mmm
Origin on 2[010] at 4
3
(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,
3
4
) 0,0,z (4) 4
−
(0,0,
1
4
) 0,0,z
(5) 2 0,y, 0(6)2x,0,
1
4
(7) 2 x,x,
1
8
(8) 2 x, ¯x,
3
8
374
International Tables for Crystallography (2006). Vol. A, Space group 95, pp. 374–375.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 95 P4
3
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 +
3
4
(4) y, ¯x,z +
1
4
(5) ¯x, y, ¯z (6) x, ¯y, ¯z +
1
2
(7) y,x, ¯z +
1
4
(8) ¯y, ¯x, ¯z +
3
4
00l : l = 4n
Special: as above, plus
4 c ..2 x,x,
5
8
¯x, ¯x,
1
8
¯x,x,
3
8
x, ¯x,
7
8
0kl : l = 2n + 1
or l = 4n
4 b . 2 .
1
2
,y,0
1
2
, ¯y,
1
2
¯y,
1
2
,
3
4
y,
1
2
,
1
4
hhl : l = 2n + 1
or l = 4n
4 a . 2 . 0,y,00, ¯y,
1
2
¯y,0,
3
4
y, 0,
1
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 ,
1
8
Maximal non-isomorphic subgroups
I
[2] P4
3
11(P4
3
, 78) 1; 2; 3; 4
[2] P2
1
12(C222
1
, 20) 1; 2; 7; 8
[2] P2
1
21(P222
1
, 17) 1; 2; 5; 6
IIa none
IIb [2] C 4
3
22
1
(a
= 2a,b
= 2b)(P4
3
2
1
2, 96)
Maximal isomorphic subgroups of lowest index
IIc
[2] C 4
3
22(a
= 2a,b
= 2b)(P4
3
22, 95); [3] P4
1
22(c
= 3c) (91); [5] P4
3
22(c
= 5c) (95)
Minimal non-isomorphic supergroups
I
none
II [2] I 4
1
22 (98); [2] P4
2
22(c
=
1
2
c) (93)
375
P4
3
2
1
2 D
8
4
422 Tetragonal
No. 96 P4
3
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,
3
4
) 0,
1
2
,z (4) 4
−
(0,0,
1
4
)
1
2
,0, z
(5) 2(0,
1
2
,0)
1
4
,y,
3
8
(6) 2(
1
2
,0, 0) x,
1
4
,
1
8
(7) 2 x,x,0(8)2x, ¯x,
1
4
376
International Tables for Crystallography (2006). Vol. A, Space group 96, pp. 376–377.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 96 P4
3
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 +
3
4
(4) y +
1
2
, ¯x +
1
2
,z +
1
4
(5) ¯x +
1
2
,y +
1
2
, ¯z +
3
4
(6) x +
1
2
, ¯y +
1
2
, ¯z +
1
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
,
3
4
x +
1
2
, ¯x +
1
2
,
1
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
,
1
8
Along [110] p2gm
a
=
1
2
(−a + b) b
= c
Origin at x, x,0
Maximal non-isomorphic subgroups
I
[2] P4
3
11(P4
3
, 78) 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
1
2
1
2(c
= 3c) (92); [5] P4
3
2
1
2(c
= 5c) (96); [9] P4
3
2
1
2(a
= 3a,b
= 3b) (96)
Minimal non-isomorphic supergroups
I
[3] P4
3
32 (212)
II [2] C 4
3
22(P4
3
22, 95); [2] I 4
1
22 (98); [2] P4
2
2
1
2(c
=
1
2
c) (94)
377
I 422 D
9
4
422 Tetragonal
No. 97 I 422
Patterson symmetry I 4/mmm
Origin at 422
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,z (3) 4
+
0,0,z (4) 4
−
0,0,z
(5) 2 0,y, 0(6)2x,0,0(7)2x, x,0(8)2x, ¯x,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) 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(
1
2
,
1
2
,0) x,x,
1
4
(8) 2 x, ¯x +
1
2
,
1
4
378
International Tables for Crystallography (2006). Vol. A, Space group 97, pp. 378–379.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 97 I 422
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) ¯y,x, z (4) y, ¯x,z
(5) ¯x, y, ¯z (6) x, ¯y, ¯z (7) y,x, ¯z (8) ¯y, ¯x, ¯z
hkl : h + k + l = 2n
hk0: h + k = 2n
0kl : k + l = 2n
hhl : l = 2n
00l : l = 2
n
h00 : h = 2n
Special: as above, plus
8 j ..2 x,x +
1
2
,
1
4
¯x, ¯x +
1
2
,
1
4
¯x +
1
2
,x,
1
4
x +
1
2
, ¯x,
1
4
0kl : k = 2n
8 i . 2 . x,0,
1
2
¯x,0,
1
2
0,x,
1
2
0, ¯x,
1
2
no extra conditions
8 h . 2 . x,0, 0¯x,0,00, x,00, ¯x,0 no extra conditions
8 g ..2 x,x, 0¯x, ¯x,0¯x,x,0 x, ¯x,0 no extra conditions
8 f 2 .. 0,
1
2
,z
1
2
,0, z 0,
1
2
, ¯z
1
2
,0, ¯zhkl: l = 2n
4 e 4 .. 0,0,z 0,0, ¯z no extra conditions
4 d 2 . 22 0,
1
2
,
1
4
1
2
,0,
1
4
hkl : l = 2n
4 c 222. 0,
1
2
,0
1
2
,0, 0 hkl : l = 2n
2 b 422 0,0,
1
2
no extra conditions
2 a 422 0,0,0 no extra conditions
Symmetry of special projections
Along [001] p4mm
a
=
1
2
(a − b) b
=
1
2
(a + b)
Origin at 0,0,z
Along [100] c2mm
a
= bb
= c
Origin at x,0,0
Along [110] p2mm
a
=
1
2
(−a + b) b
=
1
2
c
Origin at x,x , 0
Maximal non-isomorphic subgroups
I
[2] I 411(I 4, 79) (1; 2; 3; 4)+
[2] I 221(I 222, 23) (1; 2; 5; 6)+
[2] I 212(F 222, 22) (1; 2; 7; 8)+
IIa [2] P4
2
2
1
2 (94) 1; 2; 7; 8; (3; 4; 5; 6)+(
1
2
,
1
2
,
1
2
)
[2] P4
2
22 (93) 1; 2; 5; 6; (3; 4; 7; 8)+(
1
2
,
1
2
,
1
2
)
[2] P42
1
2 (90) 1; 2; 3; 4; (5; 6; 7; 8)+(
1
2
,
1
2
,
1
2
)
[2] P422 (89) 1; 2; 3; 4; 5; 6; 7; 8
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] I 422(c
= 3c) (97); [9] I 422 (a
= 3a,b
= 3b) (97)
Minimal non-isomorphic supergroups
I
[2] I 4/mmm(139); [2] I 4/mcm(140); [3] F 432 (209); [3] I 432 (211)
II [2] C 422(c
=
1
2
c)(P422, 89)
379
I 4
1
22 D
10
4
422 Tetragonal
No. 98 I 4
1
22 Patterson symmetry I 4/mmm
Origin at 222 at 212
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤ 1; 0 ≤ z ≤
1
8
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) 2
1
4
,y,
3
8
(6) 2 x,
1
4
,
1
8
(7) 2(
1
2
,
1
2
,0) x,x,
1
4
(8) 2 x, ¯x,0
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) 2(0,
1
2
,0) 0,y,
1
8
(6) 2(
1
2
,0, 0) x,0,
3
8
(7) 2 x , x,0(8)2x, ¯x +
1
2
,
1
4
380
International Tables for Crystallography (2006). Vol. A, Space group 98, pp. 380–381.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 98 I 4
1
22
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 g 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 +
1
2
,y, ¯z +
3
4
(6) x, ¯y +
1
2
, ¯z +
1
4
(7) y +
1
2
,x +
1
2
, ¯z +
1
2
(8) ¯y, ¯x, ¯z
hkl : h + k + l = 2n
hk0: h + k = 2n
0kl : k + l = 2 n
hhl : l = 2n
00l : l = 4n
h00 : h = 2n
Special: as above, plus
8 f . 2 . x,
1
4
,
1
8
¯x +
1
2
,
1
4
,
5
8
3
4
,x +
1
2
,
3
8
3
4
, ¯x,
7
8
hhl : l = 4n
8 e ..2¯x,x, 0 x +
1
2
, ¯x +
1
2
,
1
2
¯x, ¯x +
1
2
,
1
4
x +
1
2
,x,
3
4
0kl : k = 2n + 1
or l = 4n
8 d ..2 x,x,0¯x +
1
2
, ¯x +
1
2
,
1
2
¯x,x +
1
2
,
1
4
x +
1
2
, ¯x,
3
4
0kl : k = 2n + 1
or l = 4n
8 c 2 .. 0,0,z 0,
1
2
,z +
1
4
1
2
,0, ¯z +
3
4
1
2
,
1
2
, ¯z +
1
2
hkl : l = 2n + 1
or 2h + l = 4n
4 b 2 . 22 0,0,
1
2
0,
1
2
,
3
4
hkl : l = 2n + 1
or 2h + l = 4n
4 a 2 . 22 0,0,00,
1
2
,
1
4
hkl : l = 2n + 1
or 2h + l = 4n
Symmetry of special projections
Along [001] p4mm
a
=
1
2
(a − b) b
=
1
2
(a + b)
Origin at
1
4
,
1
4
,z
Along [100] c2mm
a
= bb
= c
Origin at x,0,
3
8
Along [110] p2mm
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 2
1
21(I 2
1
2
1
2
1
, 24) (1; 2; 5; 6)+
[2] I 2
1
12(F 222, 22) (1; 2; 7; 8)+
IIa [2] P4
3
2
1
2 (96) 1; 2; 7; 8; (3; 4; 5; 6)+(
1
2
,
1
2
,
1
2
)
[2] P4
3
22 (95) 1; 2; 5; 6; (3; 4; 7; 8)+(
1
2
,
1
2
,
1
2
)
[2] P4
1
2
1
2 (92) 1; 2; 3; 4; (5; 6; 7; 8)+(
1
2
,
1
2
,
1
2
)
[2] P4
1
22 (91) 1; 2; 3; 4; 5; 6; 7; 8
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] I 4
1
22(c
= 3c) (98); [9] I 4
1
22(a
= 3a,b
= 3b) (98)
Minimal non-isomorphic supergroups
I
[2] I 4
1
/amd (141); [2] I 4
1
/acd (142); [3] F 4
1
32 (210); [3] I 4
1
32 (214)
II [2] C 4
2
22(c
=
1
2
c)(P4
2
22, 93)
381
P4mm C
1
4v
4mm Tetragonal
No. 99 P4mm
Patterson symmetry P4/mmm
Origin on 4mm
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 (4) 4
−
0,0,z
(5) mx,0,z (6) m 0,y, z (7) mx, ¯x,z (8) mx,x, z
382
International Tables for Crystallography (2006). Vol. A, Space group 99, pp. 382–383.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 99 P4mm
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,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 f . m . x,
1
2
,z ¯x,
1
2
,z
1
2
,x,z
1
2
, ¯x, z no extra conditions
4 e . m . x, 0,z ¯x,0,z 0,x,z 0, ¯x,z no extra conditions
4 d ..mx,x,z ¯x, ¯x,z ¯x,x,zx, ¯x,z no extra conditions
2 c 2 mm.
1
2
,0, z 0,
1
2
,zhkl: h + k = 2n
1 b 4 mm
1
2
,
1
2
,z no extra conditions
1 a 4 mm 0,0, z no extra conditions
Symmetry of special projections
Along [001] p4mm
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] P411 (P4, 75) 1; 2; 3; 4
[2] P21m (Cmm2, 35) 1; 2; 7; 8
[2] P2m1(Pmm2, 25) 1; 2; 5; 6
IIa none
IIb [2] P4
2
mc (c
= 2c) (105); [2] P4cc (c
= 2c) (103); [2] P4
2
cm (c
= 2c) (101); [2] C4md (a
= 2a,b
= 2b)(P4bm, 100);
[2] F 4mc (a
= 2a,b
= 2b,c
= 2c)(I 4cm, 108); [2] F 4mm (a
= 2a,b
= 2b,c
= 2c)(I 4mm, 107)
Maximal isomorphic subgroups of lowest index
IIc
[2] P4mm (c
= 2c) (99); [2] C4mm (a
= 2a,b
= 2b)(P4mm, 99)
Minimal non-isomorphic supergroups
I
[2] P4/mmm(123); [2] P4/nmm(129)
II [2] I 4mm (107)
383