Hahn Th. (ed.) International tables for crystallography. Vol. A. Space-group symmetry
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P4
2
mc C
7
4v
4mm Tetragonal
No. 105 P4
2
mc Patterson symmetry P4/mmm
Origin on 2mm on 4
2
mc
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
) 0,0,z (4) 4
−
(0,0,
1
2
) 0,0, z
(5) mx,0,z (6) m 0,y, z (7) cx, ¯x,z (8) cx,x, z
394
International Tables for Crystallography (2006). Vol. A, Space group 105, pp. 394–395.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 105 P4
2
mc
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 +
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
hhl : l = 2n
00l : l = 2n
Special: as above, plus
4 e . m . x,
1
2
,z ¯x,
1
2
,z
1
2
,x,z +
1
2
1
2
, ¯x,z +
1
2
no extra conditions
4 d . m . x , 0,z ¯x,0, z 0,x,z +
1
2
0, ¯x,z +
1
2
no extra conditions
2 c 2 mm. 0,
1
2
,z
1
2
,0, z +
1
2
hkl : h + k + l = 2n
2 b 2 mm.
1
2
,
1
2
,z
1
2
,
1
2
,z +
1
2
hkl : l = 2n
2 a 2 mm. 0,0,z 0,0,z +
1
2
hkl : l = 2n
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
=
1
2
c
Origin at x,x , 0
Maximal non-isomorphic subgroups
I
[2] P4
2
11(P4
2
, 77) 1; 2; 3; 4
[2] P21c (Ccc2, 37) 1; 2; 7; 8
[2] P2m1(Pmm2, 25) 1; 2; 5; 6
IIa none
IIb [2] C 4
2
md (a
= 2a,b
= 2b)(P4
2
nm, 102); [2] C4
2
mc (a
= 2a,b
= 2b)(P4
2
cm, 101)
Maximal isomorphic subgroups of lowest index
IIc
[3] P4
2
mc (c
= 3c) (105); [9] P4
2
mc(a
= 3a,b
= 3b) (105)
Minimal non-isomorphic supergroups
I
[2] P4
2
/mmc (131); [2] P4
2
/nmc(137)
II [2] C 4
2
mc(P4
2
cm, 101); [2] I 4mm (107); [2] P4mm (c
=
1
2
c) (99)
395
P4
2
bc C
8
4v
4mm Tetragonal
No. 106 P4
2
bc Patterson symmetry P4/mmm
Origin on 2 on 4
2
1n
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
) 0,0,z (4) 4
−
(0,0,
1
2
) 0,0,z
(5) ax,
1
4
,z (6) b
1
4
,y,z (7) cx+
1
2
, ¯x, z (8) n(
1
2
,
1
2
,
1
2
) x,x,z
396
International Tables for Crystallography (2006). Vol. A, Space group 106, pp. 396–397.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 106 P4
2
bc
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 c 1(1)x,y,z (2) ¯x, ¯y,z (3) ¯y,x, z +
1
2
(4) y, ¯x, z +
1
2
(5) x +
1
2
, ¯y +
1
2
,z (6) ¯x +
1
2
,y +
1
2
,z (7) ¯y+
1
2
, ¯x +
1
2
,z +
1
2
(8) y +
1
2
,x +
1
2
,z +
1
2
0kl : k = 2n
hhl : l = 2n
00l : l = 2n
h00 : h = 2n
Special: as above, plus
4 b 2 .. 0,
1
2
,z
1
2
,0, z +
1
2
1
2
,0, z 0,
1
2
,z +
1
2
hkl : h + k,l = 2n
4 a 2 .. 0,0,z 0,0,z +
1
2
1
2
,
1
2
,z
1
2
,
1
2
,z +
1
2
hkl : h + k,l = 2n
Symmetry of special projections
Along [001] p4gm
a
= ab
= b
Origin at 0,0,z
Along [100] p1m1
a
=
1
2
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] P4
2
11(P4
2
, 77) 1; 2; 3; 4
[2] P21c (Ccc2, 37) 1; 2; 7; 8
[2] P2b1(Pba2, 32) 1; 2; 5; 6
IIa none
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] P4
2
bc (c
= 3c) (106); [9] P4
2
bc (a
= 3a,b
= 3b) (106)
Minimal non-isomorphic supergroups
I
[2] P4
2
/nbc(133); [2] P4
2
/mbc (135)
II [2] C 4
2
mc(P4
2
cm, 101); [2] I 4cm (108); [2] P4bm(c
=
1
2
c) (100)
397
I 4mm C
9
4v
4mm Tetragonal
No. 107 I 4mm
Patterson symmetry I 4/mmm
Origin on 4mm
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
2
;0≤ z ≤
1
2
; x ≤ y
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) mx,0,z (6) m 0,y,z (7) mx, ¯x, z (8) mx,x,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
(5) n(
1
2
,0,
1
2
) x,
1
4
,z (6) n(0,
1
2
,
1
2
)
1
4
,y,z (7) cx+
1
2
, ¯x, z (8) n(
1
2
,
1
2
,
1
2
) x,x,z
398
International Tables for Crystallography (2006). Vol. A, Space group 107, pp. 398–399.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 107 I 4mm
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 e 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 = 2n
h00 : h
= 2n
Special: as above, plus
8 d . m . x , 0,z ¯x, 0,z 0,x,z 0, ¯x,z no extra conditions
8 c ..mx, x,z ¯x, ¯x,z ¯x,x,zx, ¯x,z no extra conditions
4 b 2 mm. 0,
1
2
,z
1
2
,0, zhkl: l = 2n
2 a 4 mm 0 , 0,z 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] 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 411(I 4, 79) (1; 2; 3; 4)+
[2] I 2m1(Imm2, 44) (1; 2; 5; 6)+
[2] I 21m (Fmm2, 42) (1; 2; 7; 8)+
IIa [2] P4
2
mc (105) 1; 2; 5; 6; (3; 4; 7; 8)+(
1
2
,
1
2
,
1
2
)
[2] P4nc (104) 1; 2; 3; 4; (5; 6; 7; 8)+(
1
2
,
1
2
,
1
2
)
[2] P4
2
nm (102) 1; 2; 7; 8; (3; 4; 5; 6)+(
1
2
,
1
2
,
1
2
)
[2] P4mm (99) 1; 2; 3; 4; 5; 6; 7; 8
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] I 4mm (c
= 3c) (107); [9] I 4mm (a
= 3a,b
= 3b) (107)
Minimal non-isomorphic supergroups
I
[2] I 4/mmm(139)
II [2] C 4mm (c
=
1
2
c)(P4mm, 99)
399
I 4cm C
10
4v
4mm Tetragonal
No. 108 I 4cm
Patterson symmetry I 4/mmm
Origin on 4ce
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
2
;0≤ z ≤
1
2
; y ≤
1
2
− x
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) cx,0,z (6) c 0,y,z (7) cx, ¯x, z (8) cx,x,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
(5) ax,
1
4
,z (6) b
1
4
,y,z (7) mx+
1
2
, ¯x, z (8) g(
1
2
,
1
2
,0) x, x,z
400
International Tables for Crystallography (2006). Vol. A, Space group 108, pp. 400–401.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 108 I 4cm
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 d 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
hkl : h + k + l = 2n
hk0: h + k = 2n
0kl : k,l = 2n
hhl : l = 2n
00l : l = 2n
h00 : h = 2n
Special: as above, plus
8 c ..mx, x +
1
2
,z ¯x, ¯x +
1
2
,z ¯x +
1
2
,x,zx+
1
2
, ¯x,z no extra conditions
4 b 2 . mm
1
2
,0, z 0,
1
2
,zhkl: l = 2n
4 a 4 .. 0,0, z 0,0,z +
1
2
hkl : l = 2n
Symmetry of special projections
Along [001] p4mm
a
=
1
2
(a − b) b
=
1
2
(a + b)
Origin at 0,0,z
Along [100] p1m1
a
=
1
2
bb
=
1
2
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 411(I 4, 79) (1; 2; 3; 4)+
[2] I 2c1(Iba2, 45) (1; 2; 5; 6)+
[2] I 21m (Fmm2, 42) (1; 2; 7; 8)+
IIa [2] P4
2
bc (106) 1; 2; 7; 8; (3; 4; 5; 6)+(
1
2
,
1
2
,
1
2
)
[2] P4cc (103) 1; 2; 3; 4; 5; 6; 7; 8
[2] P4
2
cm (101) 1; 2; 5; 6; (3; 4; 7; 8)+(
1
2
,
1
2
,
1
2
)
[2] P4bm (100) 1; 2; 3; 4; (5; 6; 7; 8)+(
1
2
,
1
2
,
1
2
)
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] I 4cm (c
= 3c) (108); [9] I 4cm (a
= 3a,b
= 3b) (108)
Minimal non-isomorphic supergroups
I
[2] I 4/mcm(140)
II [2] C 4mm (c
=
1
2
c)(P4mm, 99)
401
I 4
1
md C
11
4v
4mm Tetragonal
No. 109 I 4
1
md Patterson symmetry I 4/mmm
Origin on 2mm on 2m1
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) mx,0,z (6) n(0,
1
2
,
1
2
)
1
4
,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
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) n(
1
2
,0,
1
2
) x,
1
4
,z (6) m 0,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
402
International Tables for Crystallography (2006). Vol. A, Space group 109, pp. 402–403.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 109 I 4
1
md
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 c 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 (6) ¯x +
1
2
,y +
1
2
,z +
1
2
(7) ¯y, ¯x +
1
2
,z +
1
4
(8) y +
1
2
,x,z +
3
4
hkl : h + k + l = 2n
hk0: h + k = 2n
0kl : k + l = 2 n
hhl :2h + l = 4n
00l : l = 4n
h00 : h = 2n
h
¯
h0: h = 2n
Special: as above, plus
8 b . m . 0,y,z
1
2
, ¯y +
1
2
,z +
1
2
¯y,
1
2
,z +
1
4
y +
1
2
,0, z +
3
4
no extra conditions
4 a 2 mm. 0, 0,z 0,
1
2
,z +
1
4
hkl : l = 2n + 1
or 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] c1m1
a
= bb
= 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 2m1(Imm2, 44) (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
md (c
= 3c) (109); [9] I 4
1
md (a
= 3a,b
= 3b) (109)
Minimal non-isomorphic supergroups
I
[2] I 4
1
/amd (141)
II [2] C 4
2
md (c
=
1
2
c)(P4
2
nm, 102)
403