Hahn Th. (ed.) International tables for crystallography. Vol. A. Space-group symmetry
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P4
2
C
3
4
4 Tetragonal
No. 77 P4
2
Patterson symmetry P4/m
Origin on 2 on 4
2
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,
1
2
) 0,0, z (4) 4
−
(0,0,
1
2
) 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 +
1
2
(4) y, ¯x, z +
1
2
00l : l = 2n
Special: as above, plus
2 c 2 .. 0,
1
2
,z
1
2
,0, z +
1
2
hkl : h + k + l = 2n
2 b 2 ..
1
2
,
1
2
,z
1
2
,
1
2
,z +
1
2
hkl : l = 2n
2 a 2 .. 0,0,z 0,0,z +
1
2
hkl : l = 2n
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
3
(c
= 2c) (78); [2] P4
1
(c
= 2c) (76); [2] F 4
1
(a
= 2a,b
= 2b,c
= 2c)(I 4
1
, 80)
Maximal isomorphic subgroups of lowest index
IIc
[2] C 4
2
(a
= 2a,b
= 2b)(P4
2
, 77); [3] P4
2
(c
= 3c) (77)
Minimal non-isomorphic supergroups
I
[2] P4
2
/m (84); [2] P4
2
/n (86); [2] P4
2
22 (93); [2] P4
2
2
1
2 (94); [2] P4
2
cm(101); [2] P4
2
nm(102); [2] P4
2
mc (105);
[2] P4
2
bc (106)
II [2] I 4 (79); [2] P4(c
=
1
2
c) (75)
334
International Tables for Crystallography (2006). Vol. A, Space group 77, p. 334.
Copyright © 2006 International Union of Crystallography
Tetragonal 4 C
4
4
P4
3
Patterson symmetry P4/m P4
3
No. 78
Origin on 4
3
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,
3
4
) 0,0, z (4) 4
−
(0,0,
1
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 +
3
4
(4) y, ¯x,z +
1
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
3
(a
= 2a,b
= 2b)(P4
3
, 78); [3] P4
1
(c
= 3c) (76); [5] P4
3
(c
= 5c) (78)
Minimal non-isomorphic supergroups
I
[2] P4
3
22 (95); [2] P4
3
2
1
2 (96)
II [2] I 4
1
(80); [2] P4
2
(c
=
1
2
c) (77)
335
International Tables for Crystallography (2006). Vol. A, Space group 78, p. 335.
Copyright © 2006 International Union of Crystallography
I 4 C
5
4
4 Tetragonal
No. 79 I 4
Patterson symmetry I 4/m
Origin on 4
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
2
;0≤ z ≤
1
2
Symmetry operations
For (0,0,0)+ set
(1) 1 (2) 2 0,0, z (3) 4
+
0,0,z (4) 4
−
0,0,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) 4
+
(0,0,
1
2
) 0,
1
2
,z (4) 4
−
(0,0,
1
2
)
1
2
,0, z
336
International Tables for Crystallography (2006). Vol. A, Space group 79, pp. 336–337.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 79 I 4
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); t(
1
2
,
1
2
,
1
2
); (2); (3)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates
(0,0, 0)+ (
1
2
,
1
2
,
1
2
)+
Reflection conditions
General:
8 c 1(1)x,y,z (2) ¯x, ¯y,z (3) ¯y,x,z (4) y, ¯x,zhkl: h + k + l = 2n
hk0: h + k = 2n
0kl : k + l = 2 n
hhl : l = 2n
00l : l = 2n
h00 : h = 2n
Special: as above, plus
4 b 2 .. 0,
1
2
,z
1
2
,0, zhkl: l = 2n
2 a 4 .. 0,0,z 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] c1m1
a
= bb
= c
Origin at x,0,0
Along [110] p1m1
a
=
1
2
(−a + b) b
=
1
2
c
Origin at x,x , 0
Maximal non-isomorphic subgroups
I
[2] I 2(C2, 5) (1; 2)+
IIa [2] P4
2
(77) 1; 2; (3; 4)+(
1
2
,
1
2
,
1
2
)
[2] P4 (75) 1; 2; 3; 4
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] I 4(c
= 3c) (79); [5] I 4(a
= a + 2b,b
= −2a + b or a
= a − 2b,b
= 2a + b) (79)
Minimal non-isomorphic supergroups
I
[2] I 4/m (87); [2] I 422 (97); [2] I 4mm (107); [2] I 4cm (108)
II [2] C 4(c
=
1
2
c)(P4, 75)
337
I 4
1
C
6
4
4 Tetragonal
No. 80 I 4
1
Patterson symmetry I 4/m
Origin on 2
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤ 1; 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
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
338
International Tables for Crystallography (2006). Vol. A, Space group 80, pp. 338–339.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 80 I 4
1
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); t(
1
2
,
1
2
,
1
2
); (2); (3)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates
(0,0, 0)+ (
1
2
,
1
2
,
1
2
)+
Reflection conditions
General:
8 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
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
4 a 2 .. 0,0,z 0,
1
2
,z +
1
4
hkl : l = 2n + 1
or 2h + l = 4n
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] c1m1
a
= bb
= c
Origin at x,0,0
Along [110] p1m1
a
=
1
2
(−a + b) b
=
1
2
c
Origin at x,x , 0
Maximal non-isomorphic subgroups
I
[2] I 2(C2, 5) (1; 2)+
IIa [2] P4
3
(78) 1; 2; (3; 4)+(
1
2
,
1
2
,
1
2
)
[2] P4
1
(76) 1; 2; 3; 4
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] I 4
1
(c
= 3c) (80); [5] I 4
1
(a
= a + 2b,b
= −2a + b or a
= a − 2b,b
= 2a + b) (80)
Minimal non-isomorphic supergroups
I
[2] I 4
1
/a (88); [2] I 4
1
22 (98); [2] I 4
1
md (109); [2] I 4
1
cd(110)
II [2] C 4
2
(c
=
1
2
c)(P4
2
, 77)
339
P
¯
4 S
1
4
¯
4 Tetragonal
No. 81 P
¯
4
Patterson symmetry P4/m
Origin at
¯
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;0,0, 0(4)
¯
4
−
0,0,z;0,0, 0
340
International Tables for Crystallography (2006). Vol. A, Space group 81, pp. 340–341.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 81 P
¯
4
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 h 1(1)x, y,z (2) ¯x, ¯y,z (3) y, ¯x, ¯z (4) ¯y,x, ¯z no conditions
Special:
2 g 2 .. 0,
1
2
,z
1
2
,0, ¯zhk0: h + k = 2n
2 f 2 ..
1
2
,
1
2
,z
1
2
,
1
2
, ¯z no extra conditions
2 e 2 .. 0, 0,z 0,0, ¯z no extra conditions
1 d
¯
4 ..
1
2
,
1
2
,
1
2
no extra conditions
1 c
¯
4 ..
1
2
,
1
2
,0 no extra conditions
1 b
¯
4 .. 0,0,
1
2
no extra conditions
1 a
¯
4 .. 0,0,0 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] F
¯
4(a
= 2a,b
= 2b,c
= 2c)(I
¯
4, 82)
Maximal isomorphic subgroups of lowest index
IIc
[2] P
¯
4(c
= 2c) (81); [2] C
¯
4(a
= 2a,b
= 2b)(P
¯
4, 81)
Minimal non-isomorphic supergroups
I
[2] P4/m (83); [2] P4
2
/m (84); [2] P4/n (85); [2] P4
2
/n (86); [2] P
¯
42m (111); [2] P
¯
42c (112); [2] P
¯
42
1
m (113); [2] P
¯
42
1
c (114);
[2] P
¯
4m2 (115); [2] P
¯
4c2 (116); [2] P
¯
4b2 (117); [2] P
¯
4n2 (118)
II [2] I
¯
4 (82)
341
I
¯
4 S
2
4
¯
4 Tetragonal
No. 82 I
¯
4
Patterson symmetry I 4/m
Origin at
¯
4
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
2
;0≤ z ≤
1
2
Symmetry operations
For (0,0,0)+ set
(1) 1 (2) 2 0,0, z (3)
¯
4
+
0,0,z;0,0, 0(4)
¯
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
+
1
2
,0, z;
1
2
,0,
1
4
(4)
¯
4
−
0,
1
2
,z;0,
1
2
,
1
4
342
International Tables for Crystallography (2006). Vol. A, Space group 82, pp. 342–343.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 82 I
¯
4
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); t(
1
2
,
1
2
,
1
2
); (2); (3)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates
(0,0, 0)+ (
1
2
,
1
2
,
1
2
)+
Reflection conditions
General:
8 g 1(1)x, y,z (2) ¯x, ¯y,z (3) y, ¯x, ¯z (4) ¯y,x, ¯zhkl: h + k + l = 2n
hk0: h + k = 2n
0kl : k + l = 2 n
hhl : l = 2n
00l : l = 2n
h00 : h = 2n
Special: no extra conditions
4 f 2 .. 0,
1
2
,z
1
2
,0, ¯z
4 e 2 .. 0, 0,z 0,0, ¯z
2 d
¯
4 .. 0,
1
2
,
3
4
2 c
¯
4 .. 0,
1
2
,
1
4
2 b
¯
4 .. 0,0,
1
2
2 a
¯
4 .. 0,0,0
Symmetry of special projections
Along [001] p4
a
=
1
2
(a − b) b
=
1
2
(a + b)
Origin at 0,0,z
Along [100] c1m1
a
= bb
= c
Origin at x,0,0
Along [110] p1m1
a
=
1
2
(−a + b) b
=
1
2
c
Origin at x,x , 0
Maximal non-isomorphic subgroups
I
[2] I 2(C2, 5) (1; 2)+
IIa [2] P
¯
4 (81) 1; 2; 3; 4
[2] P
¯
4 (81) 1; 2; (3; 4)+(
1
2
,
1
2
,
1
2
)
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] I
¯
4(c
= 3c) (82); [5] I
¯
4(a
= a + 2b,b
= −2a + b or a
= a − 2b,b
= 2a + b) (82)
Minimal non-isomorphic supergroups
I
[2] I 4/m (87); [2] I 4
1
/a (88); [2] I
¯
4m2 (119); [2] I
¯
4c2 (120); [2] I
¯
42m (121); [2] I
¯
42d (122)
II [2] C
¯
4(c
=
1
2
c)(P
¯
4, 81)
343