Hahn Th. (ed.) International tables for crystallography. Vol. A. Space-group symmetry
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P4
2
/nC
4
4h
4/m Tetragonal
No. 86 P4
2
/n Patterson symmetry P4/m
ORIGIN CHOICE 2
Origin at
¯
1onn,at
1
4
,
1
4
,
1
4
from
¯
4
Asymmetric unit −
1
4
≤ x ≤
1
4
; −
1
4
≤ y ≤
1
4
;0≤ z ≤
1
2
Symmetry operations
(1) 1 (2) 2
1
4
,
1
4
,z (3) 4
+
(0,0,
1
2
) −
1
4
,
1
4
,z (4) 4
−
(0,0,
1
2
)
1
4
,−
1
4
,z
(5)
¯
10, 0,0(6)n(
1
2
,
1
2
,0) x,y,0(7)
¯
4
+
1
4
,
1
4
,z;
1
4
,
1
4
,
1
4
(8)
¯
4
−
1
4
,
1
4
,z;
1
4
,
1
4
,
1
4
354
CONTINUED No. 86 P4
2
/n
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+
1
2
, ¯y +
1
2
,z (3) ¯y,x +
1
2
,z +
1
2
(4) y +
1
2
, ¯x, z +
1
2
(5) ¯x, ¯y, ¯z (6) x +
1
2
,y +
1
2
, ¯z (7) y, ¯x +
1
2
, ¯z +
1
2
(8) ¯y +
1
2
,x, ¯z +
1
2
hk0: h + k = 2n
00l : l = 2n
h00 : h = 2n
Special: as above, plus
4 f 2 ..
1
4
,
1
4
,z
3
4
,
3
4
,z +
1
2
3
4
,
3
4
, ¯z
1
4
,
1
4
, ¯z +
1
2
hkl : h + k + l = 2n
4 e 2 ..
3
4
,
1
4
,z
3
4
,
1
4
,z +
1
2
1
4
,
3
4
, ¯z
1
4
,
3
4
, ¯z +
1
2
hkl : l = 2n
4 d
¯
10, 0,
1
2
1
2
,
1
2
,
1
2
0,
1
2
,0
1
2
,0, 0 hkl : h + k,h + l, k + l = 2n
4 c
¯
10, 0,0
1
2
,
1
2
,00,
1
2
,
1
2
1
2
,0,
1
2
hkl : h + k,h + l,k + l = 2n
2 b
¯
4 ..
1
4
,
1
4
,
3
4
3
4
,
3
4
,
1
4
hkl : h + k + l = 2n
2 a
¯
4 ..
1
4
,
1
4
,
1
4
3
4
,
3
4
,
3
4
hkl : h + k + l = 2n
Symmetry of special projections
Along [001] p4
a
=
1
2
(a − b) b
=
1
2
(a + b)
Origin at
1
4
,
1
4
,z
Along [100] p2mg
a
= bb
= c
Origin at x,0,0
Along [110] p2mm
a
=
1
2
(−a + b) b
= c
Origin at x, x,0
Maximal non-isomorphic subgroups
I
[2] P
¯
4 (81) 1; 2; 7; 8
[2] P4
2
(77) 1; 2; 3; 4
[2] P2/n (P2/c , 13) 1; 2; 5; 6
IIa none
IIb [2] F 4
1
/d (a
= 2a,b
= 2b,c
= 2c)(I 4
1
/a, 88)
Maximal isomorphic subgroups of lowest index
IIc
[3] P4
2
/n (c
= 3c) (86); [5] P4
2
/n (a
= a + 2b,b
= −2a + b or a
= a − 2b,b
= 2a + b) (86)
Minimal non-isomorphic supergroups
I
[2] P4
2
/nbc(133); [2] P4
2
/nnm(134); [2] P4
2
/nmc(137); [2] P4
2
/ncm(138)
II [2] C 4
2
/m (P4
2
/m, 84); [2] I 4/m (87); [2] P4/n (c
=
1
2
c) (85)
355
I 4/mC
5
4h
4/m Tetragonal
No. 87 I 4/m
Patterson symmetry I 4/m
Origin at centre (4/m)
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)
¯
10, 0,0(6)mx,y,0(7)
¯
4
+
0,0,z;0,0, 0(8)
¯
4
−
0,0,z;0,0, 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) 4
+
(0,0,
1
2
) 0,
1
2
,z (4) 4
−
(0,0,
1
2
)
1
2
,0, z
(5)
¯
1
1
4
,
1
4
,
1
4
(6) n(
1
2
,
1
2
,0) x,y,
1
4
(7)
¯
4
+
1
2
,0, z;
1
2
,0,
1
4
(8)
¯
4
−
0,
1
2
,z;0,
1
2
,
1
4
356
International Tables for Crystallography (2006). Vol. A, Space group 87, pp. 356–357.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 87 I 4/m
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 i 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 hm.. x,y,0¯x, ¯y,0¯y,x,0 y, ¯x,0 no extra conditions
8 g 2 .. 0,
1
2
,z
1
2
,0, z 0,
1
2
, ¯z
1
2
,0, ¯zhkl: l = 2n
8 f
¯
1
1
4
,
1
4
,
1
4
3
4
,
3
4
,
1
4
3
4
,
1
4
,
1
4
1
4
,
3
4
,
1
4
hkl : k,l = 2n
4 e 4 .. 0, 0,z 0,0, ¯z no extra conditions
4 d
¯
4 .. 0,
1
2
,
1
4
1
2
,0,
1
4
hkl : l = 2n
4 c 2/m .. 0,
1
2
,0
1
2
,0, 0 hk l : l = 2n
2 b 4/ m .. 0, 0,
1
2
no extra conditions
2 a 4/ m .. 0, 0,0 no extra conditions
Symmetry of special projections
Along [001] p4
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
¯
4 (82) (1; 2; 7; 8)+
[2] I 4 (79) (1; 2; 3; 4)+
[2] I 2/m (C2/m, 12) (1; 2; 5; 6)+
IIa [2] P4
2
/n (86) 1; 2; 7; 8; (3; 4; 5; 6)+(
1
2
,
1
2
,
1
2
)
[2] P4/n (85) 1; 2; 3; 4; (5; 6; 7; 8)+(
1
2
,
1
2
,
1
2
)
[2] P4
2
/m (84) 1; 2; 5; 6; (3; 4; 7; 8)+(
1
2
,
1
2
,
1
2
)
[2] P4/m (83) 1; 2; 3; 4; 5; 6; 7; 8
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] I 4/m (c
= 3c) (87); [5] I 4/m (a
= a + 2b,b
= −2a + b or a
= a − 2b,b
= 2a + b) (87)
Minimal non-isomorphic supergroups
I
[2] I 4/mmm(139); [2] I 4/mcm(140)
II [2] C 4/m (c
=
1
2
c)(P4/m, 83)
357
I 4
1
/aC
6
4h
4/m Tetragonal
No. 88 I 4
1
/a Patterson symmetry I 4/m
ORIGIN CHOICE 1
Origin at
¯
4, at 0,−
1
4
,−
1
8
from
¯
1
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,
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)
¯
10,
1
4
,
1
8
(6) ax,y,
3
8
(7)
¯
4
+
0,0,z;0,0, 0(8)
¯
4
−
0,
1
2
,z;0,
1
2
,
1
4
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)
¯
1
1
4
,0,
3
8
(6) bx,y,
1
8
(7)
¯
4
+
1
2
,0, z;
1
2
,0,
1
4
(8)
¯
4
−
0,0,z;0,0, 0
358
International Tables for Crystallography (2006). Vol. A, Space group 88, pp. 358–361.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 88 I 4
1
/a
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 +
1
2
,z +
1
2
(3) ¯y,x +
1
2
,z +
1
4
(4) y +
1
2
, ¯x, z +
3
4
(5) ¯x, ¯y +
1
2
, ¯z +
1
4
(6) x +
1
2
,y, ¯z +
3
4
(7) y, ¯x, ¯z (8) ¯y +
1
2
,x +
1
2
, ¯z +
1
2
hkl : h + k + l = 2n
hk0: h, k = 2n
0kl : k + l = 2 n
hhl : l = 2n
00l : l = 4n
h00 : h = 2n
h
¯
h0: h = 2n
Special: as above, plus
8 e 2 .. 0,0,z 0,
1
2
,z +
1
4
0,
1
2
, ¯z +
1
4
0,0, ¯zhkl: l = 2n + 1
or 2h + l = 4n
8 d
¯
10,
1
4
,
5
8
1
2
,
1
4
,
1
8
3
4
,
1
2
,
7
8
3
4
,0,
3
8
8 c
¯
10,
1
4
,
1
8
1
2
,
1
4
,
5
8
3
4
,
1
2
,
3
8
3
4
,0,
7
8
hkl : l = 2n + 1
or h, k = 2n, h + k + l = 4n
4 b
¯
4 .. 0,0,
1
2
0,
1
2
,
3
4
4 a
¯
4 .. 0,0,00,
1
2
,
1
4
hkl : l = 2n + 1
or 2h + l = 4n
Symmetry of special projections
Along [001] p4
a
=
1
2
ab
=
1
2
b
Origin at 0,0,z
Along [100] c2mm
a
= bb
= c
Origin at x,0,
3
8
Along [110] p2mg
a
=
1
2
(−a + b) b
=
1
2
c
Origin at x,x +
1
4
,
1
8
Maximal non-isomorphic subgroups
I
[2] I
¯
4 (82) (1; 2; 7; 8)+
[2] I 4
1
(80) (1; 2; 3; 4)+
[2] I 2/a (C2/c, 15) (1; 2; 5; 6)+
IIa none
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] I 4
1
/a (c
= 3c) (88); [5] I 4
1
/a (a
= a + 2b,b
= −2a + b or a
= a − 2b,b
= 2a+ b) (88)
Minimal non-isomorphic supergroups
I
[2] I 4
1
/amd (141); [2] I 4
1
/acd (142)
II [2] C 4
2
/a (c
=
1
2
c)(P4
2
/n, 86)
359
I 4
1
/aC
6
4h
4/m Tetragonal
No. 88 I 4
1
/a Patterson symmetry I 4/m
ORIGIN CHOICE 2
Origin at
¯
1 on glide plane b,at0,
1
4
,
1
8
from
¯
4
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,
1
2
)
1
4
,0, z (3) 4
+
(0,0,
1
4
)
1
4
,
1
2
,z (4) 4
−
(0,0,
3
4
)
3
4
,0, z
(5)
¯
10, 0,0(6)ax,y,
1
4
(7)
¯
4
+
1
2
,
1
4
,z;
1
2
,
1
4
,
3
8
(8)
¯
4
−
0,
1
4
,z;0,
1
4
,
1
8
For (
1
2
,
1
2
,
1
2
)+ set
(1) t(
1
2
,
1
2
,
1
2
) (2) 2 0,
1
4
,z (3) 4
+
(0,0,
3
4
) −
1
4
,
1
2
,z (4) 4
−
(0,0,
1
4
)
1
4
,0, z
(5)
¯
1
1
4
,
1
4
,
1
4
(6) bx,y, 0(7)
¯
4
+
1
2
,−
1
4
,z;
1
2
,−
1
4
,
1
8
(8)
¯
4
−
0,
3
4
,z;0,
3
4
,
3
8
360
CONTINUED No. 88 I 4
1
/a
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) ¯y+
3
4
,x +
1
4
,z +
1
4
(4) y +
3
4
, ¯x +
3
4
,z +
3
4
(5) ¯x, ¯y, ¯z (6) x +
1
2
,y, ¯z +
1
2
(7) y +
1
4
, ¯x +
3
4
, ¯z +
3
4
(8) ¯y +
1
4
,x +
1
4
, ¯z +
1
4
hkl : h + k + l = 2n
hk0: h, k = 2n
0kl : k + l = 2 n
hhl : l = 2n
00l : l = 4n
h00 : h = 2n
h
¯
h0: h = 2n
Special: as above, plus
8 e 2 .. 0,
1
4
,z
1
2
,
1
4
,z +
1
4
0,
3
4
, ¯z
1
2
,
3
4
, ¯z +
3
4
hkl : l = 2n + 1
or 2h + l = 4n
8 d
¯
10, 0,
1
2
1
2
,0, 0
3
4
,
1
4
,
3
4
3
4
,
3
4
,
1
4
8 c
¯
10, 0,0
1
2
,0,
1
2
3
4
,
1
4
,
1
4
3
4
,
3
4
,
3
4
hkl : l = 2n + 1
or h, k = 2n, h + k + l = 4n
4 b
¯
4 .. 0,
1
4
,
5
8
1
2
,
1
4
,
7
8
4 a
¯
4 .. 0,
1
4
,
1
8
1
2
,
1
4
,
3
8
hkl : l = 2n + 1
or 2h + l = 4n
Symmetry of special projections
Along [001] p4
a
=
1
2
ab
=
1
2
b
Origin at
1
4
,0, z
Along [100] c2mm
a
= bb
= c
Origin at x,
1
4
,
1
4
Along [110] p2mg
a
=
1
2
(−a + b) b
=
1
2
c
Origin at x,x, 0
Maximal non-isomorphic subgroups
I
[2] I
¯
4 (82) (1; 2; 7; 8)+
[2] I 4
1
(80) (1; 2; 3; 4)+
[2] I 2/a (C2/c, 15) (1; 2; 5; 6)+
IIa none
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] I 4
1
/a (c
= 3c) (88); [5] I 4
1
/a (a
= a + 2b,b
= −2a + b or a
= a − 2b,b
= 2a+ b) (88)
Minimal non-isomorphic supergroups
I
[2] I 4
1
/amd (141); [2] I 4
1
/acd (142)
II [2] C 4
2
/a (c
=
1
2
c)(P4
2
/n, 86)
361
P422 D
1
4
422 Tetragonal
No. 89 P422
Patterson symmetry P4/mmm
Origin at 422
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 (4) 4
−
0,0,z
(5) 2 0,y, 0(6)2x,0,0(7)2x, x,0(8)2x, ¯x,0
362
International Tables for Crystallography (2006). Vol. A, Space group 89, pp. 362–363.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 89 P422
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 (4) y, ¯x,z
(5) ¯x, y, ¯z (6) x, ¯y, ¯z (7) y,x, ¯z (8) ¯y, ¯x, ¯z
no conditions
Special:
4 o . 2 . x,
1
2
,0¯x,
1
2
,0
1
2
,x,0
1
2
, ¯x, 0 no extra conditions
4 n . 2 . x,0,
1
2
¯x,0,
1
2
0,x,
1
2
0, ¯x,
1
2
no extra conditions
4 m . 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 l . 2 . x,0,0¯x, 0,00,x, 00, ¯x,0 no extra conditions
4 k ..2 x,x,
1
2
¯x, ¯x,
1
2
¯x,x,
1
2
x, ¯x,
1
2
no extra conditions
4 j ..2 x, x,0¯x, ¯x,0¯x,x,0 x, ¯x,0 no extra conditions
4 i 2 .. 0,
1
2
,z
1
2
,0, z 0,
1
2
, ¯z
1
2
,0, ¯zhkl: h + k = 2n
2 h 4 ..
1
2
,
1
2
,z
1
2
,
1
2
, ¯z no extra conditions
2 g 4 .. 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 422
1
2
,
1
2
,
1
2
no extra conditions
1 c 422
1
2
,
1
2
,0 no extra conditions
1 b 422 0,0,
1
2
no extra conditions
1 a 422 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] 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] P221 (P222, 16) 1; 2; 5; 6
IIa none
IIb [2] P4
2
22(c
= 2c) (93); [2] C422
1
(a
= 2a,b
= 2b)(P42
1
2, 90); [2] F 422(a
= 2a,b
= 2b,c
= 2c)(I 422, 97)
Maximal isomorphic subgroups of lowest index
IIc
[2] P422 (c
= 2c) (89); [2] C422 (a
= 2a,b
= 2b)(P422, 89)
Minimal non-isomorphic supergroups
I
[2] P4/mmm(123); [2] P4/mcc (124); [2] P4 /nbm(125); [2] P4/nnc (126); [3] P432 (207)
II [2] I 422(97)
363