Hahn Th. (ed.) International tables for crystallography. Vol. A. Space-group symmetry
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P4
2
/mmc D
9
4h
4/mmm Tetragonal
No. 131 P 4
2
/m 2/m 2/c Patterson symmetry P4/mmm
Origin at centre (mmm) at 4
2
/m2/mc
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
2
;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
(9)
¯
10, 0,0 (10) mx, y,0 (11)
¯
4
+
0,0,z;0,0,
1
4
(12)
¯
4
−
0,0,z;0,0,
1
4
(13) mx,0,z (14) m 0,y,z (15) cx, ¯x,z (16) cx,x,z
Maximal non-isomorphic subgroups
I
[2] P
¯
4m2 (115) 1; 2; 7; 8; 11; 12; 13; 14
[2] P
¯
42c (112) 1; 2; 5; 6; 11; 12; 15; 16
[2] P4
2
mc (105) 1; 2; 3; 4; 13; 14; 15; 16
[2] P4
2
22 (93) 1; 2; 3; 4; 5; 6; 7; 8
[2] P4
2
/m11 (P4
2
/m, 84) 1; 2; 3; 4; 9; 10; 11; 12
[2] P2/m12/c (Cccm, 66) 1; 2; 7; 8; 9; 10; 15; 16
[2] P2/m2/m1(Pmmm, 47) 1; 2; 5; 6; 9; 10; 13; 14
IIa none
IIb [2] C 4
2
/emc (a
= 2a,b
= 2b)(P4
2
/ncm, 138); [2]C4
2
/mmd (a
= 2a,b
= 2b)(P4
2
/mnm, 136);
[2] C 4
2
/emd (a
= 2a,b
= 2b)(P4
2
/nnm, 134); [2] C4
2
/mmc (a
= 2a,b
= 2b)(P4
2
/mcm, 132)
Maximal isomorphic subgroups of lowest index
IIc
[3] P4
2
/mmc (c
= 3c) (131); [9] P4
2
/mmc (a
= 3a,b
= 3b) (131)
Minimal non-isomorphic supergroups
I
[3] Pm
¯
3n (223)
II [2] C 4
2
/mmc (P4
2
/mcm, 132); [2] I 4/mmm(139); [2] P4/mmm(c
=
1
2
c) (123)
454
International Tables for Crystallography (2006). Vol. A, Space group 131, pp. 454–455.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 131 P4
2
/mmc
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3); (5); (9)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
General:
16 r 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
(9) ¯x, ¯y, ¯z (10) x, y, ¯z (11) y, ¯x, ¯z +
1
2
(12) ¯y,x, ¯z +
1
2
(13) x, ¯y,z (14) ¯x,y,z (15) ¯y, ¯x,z +
1
2
(16) y,x , z +
1
2
hhl : l = 2n
00l : l = 2n
Special: as above, plus
8 qm.. x,y,0¯x, ¯y,0¯y,x,
1
2
y, ¯x,
1
2
¯x,y,0 x , ¯y, 0 y,x,
1
2
¯y, ¯x,
1
2
no extra conditions
8 p . m .
1
2
,y,z
1
2
, ¯y, z ¯y,
1
2
,z +
1
2
y,
1
2
,z +
1
2
1
2
,y, ¯z
1
2
, ¯y, ¯zy,
1
2
, ¯z +
1
2
¯y,
1
2
, ¯z +
1
2
no extra conditions
8 o . m . 0,y,z 0, ¯y,z ¯y, 0,z +
1
2
y, 0,z +
1
2
0,y, ¯z 0, ¯y, ¯zy,0, ¯z +
1
2
¯y,0, ¯z +
1
2
no extra conditions
8 n ..2 x,x,
1
4
¯x, ¯x,
1
4
¯x,x,
3
4
x, ¯x,
3
4
¯x, ¯x,
3
4
x,x,
3
4
x, ¯x,
1
4
¯x,x,
1
4
hkl : l = 2n
4 mm2 m. x,
1
2
,0¯x,
1
2
,0
1
2
,x,
1
2
1
2
, ¯x,
1
2
no extra conditions
4 lm2 m. x,0,
1
2
¯x,0,
1
2
0,x, 00, ¯x,0 no extra conditions
4 km2 m. x,
1
2
,
1
2
¯x,
1
2
,
1
2
1
2
,x,0
1
2
, ¯x, 0 no extra conditions
4 jm2 m. x, 0,0¯x,0,00,x,
1
2
0, ¯x,
1
2
no extra conditions
4 i 2 mm. 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 mm.
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 mm. 0,0, z 0,0,z +
1
2
0,0, ¯z 0,0 , ¯z +
1
2
hkl : l = 2n
2 f
¯
4 m 2
1
2
,
1
2
,
1
4
1
2
,
1
2
,
3
4
hkl : l = 2n
2 e
¯
4 m 20,0,
1
4
0,0,
3
4
hkl : l = 2n
2 dmmm. 0,
1
2
,
1
2
1
2
,0, 0 hkl : h + k + l = 2n
2 cmmm. 0,
1
2
,0
1
2
,0,
1
2
hkl : h + k + l = 2n
2 bmmm.
1
2
,
1
2
,0
1
2
,
1
2
,
1
2
hkl : l = 2n
2 ammm. 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
=
1
2
c
Origin at x,x , 0
(Continued on preceding page)
455
P4
2
/mcm D
10
4h
4/mmm Tetragonal
No. 132 P 4
2
/m 2/c 2/m Patterson symmetry P4/mmm
Origin at centre (mmm) at 4
2
/mc2/m
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
2
;0≤ z ≤
1
2
; x ≤ y
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,
1
4
(6) 2 x,0,
1
4
(7) 2 x,x,0(8)2x, ¯x,0
(9)
¯
10, 0,0 (10) mx, y,0 (11)
¯
4
+
0,0,z;0,0,
1
4
(12)
¯
4
−
0,0,z;0,0,
1
4
(13) cx,0,z (14) c 0, y,z (15) mx, ¯x,z (16) mx,x, z
Maximal non-isomorphic subgroups
I
[2] P
¯
4c2 (116) 1; 2; 7; 8; 11; 12; 13; 14
[2] P
¯
42m (111) 1; 2; 5; 6; 11; 12; 15; 16
[2] P4
2
cm (101) 1; 2; 3; 4; 13; 14; 15; 16
[2] P4
2
22 (93) 1; 2; 3; 4; 5; 6; 7; 8
[2] P4
2
/m11 (P4
2
/m, 84) 1; 2; 3; 4; 9; 10; 11; 12
[2] P2/m12/m (Cmmm, 65) 1; 2; 7; 8; 9; 10; 15; 16
[2] P2/m2/c1(Pccm, 49) 1; 2; 5; 6; 9; 10; 13; 14
IIa none
IIb [2] C 4
2
/ecm (a
= 2a,b
= 2b)(P4
2
/nmc, 137); [2] C4
2
/mcd (a
= 2a,b
= 2b)(P4
2
/mbc, 135);
[2] C 4
2
/ecd (a
= 2a,b
= 2b)(P4
2
/nbc, 133); [2] C4
2
/mcm (a
= 2a,b
= 2b)(P4
2
/mmc, 131)
Maximal isomorphic subgroups of lowest index
IIc
[3] P4
2
/mcm (c
= 3c) (132); [9] P4
2
/mcm (a
= 3a,b
= 3b) (132)
Minimal non-isomorphic supergroups
I
none
II [2] C 4
2
/mcm (P4
2
/mmc, 131); [2] I 4 /mcm(140); [2] P4/mmm(c
=
1
2
c) (123)
456
International Tables for Crystallography (2006). Vol. A, Space group 132, pp. 456–457.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 132 P4
2
/mcm
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3); (5); (9)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
General:
16 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 +
1
2
(6) x, ¯y, ¯z +
1
2
(7) y,x, ¯z (8) ¯y, ¯x , ¯z
(9) ¯x, ¯y, ¯z (10) x,y, ¯z (11) y, ¯x, ¯z +
1
2
(12) ¯y,x, ¯z +
1
2
(13) x, ¯y,z +
1
2
(14) ¯x,y, z +
1
2
(15) ¯y, ¯x,z (16) y,x, z
0kl : l = 2n
00l : l = 2n
Special: as above, plus
8 o ..mx,x,z ¯x, ¯x,z ¯x, x,z +
1
2
x, ¯x,z +
1
2
¯x,x, ¯z +
1
2
x, ¯x, ¯z +
1
2
x,x, ¯z ¯x, ¯x, ¯z
no extra conditions
8 nm.. x,y,0¯x, ¯y,0¯y,x,
1
2
y, ¯x,
1
2
¯x,y,
1
2
x, ¯y,
1
2
y, x,0¯y, ¯x,0
no extra conditions
8 m . 2 . x,
1
2
,
1
4
¯x,
1
2
,
1
4
1
2
,x,
3
4
1
2
, ¯x,
3
4
¯x,
1
2
,
3
4
x,
1
2
,
3
4
1
2
, ¯x,
1
4
1
2
,x,
1
4
hkl : l = 2n
8 l . 2 . x,0,
1
4
¯x,0,
1
4
0,x,
3
4
0, ¯x,
3
4
¯x,0,
3
4
x,0,
3
4
0, ¯x,
1
4
0,x,
1
4
hkl : l = 2n
8 k 2 .. 0,
1
2
,z
1
2
,0, z +
1
2
0,
1
2
, ¯z +
1
2
1
2
,0, ¯z
0,
1
2
, ¯z
1
2
,0, ¯z +
1
2
0,
1
2
,z +
1
2
1
2
,0, z
hkl : h + k,l = 2n
4 jm. 2mx,x,
1
2
¯x, ¯x,
1
2
¯x,x,0 x, ¯x,0 no extra conditions
4 im. 2mx,x,0¯x, ¯x,0¯x,x,
1
2
x, ¯x,
1
2
no extra conditions
4 h 2 . mm
1
2
,
1
2
,z
1
2
,
1
2
,z +
1
2
1
2
,
1
2
, ¯z +
1
2
1
2
,
1
2
, ¯zhkl: l = 2n
4 g 2 . mm 0,0, z 0,0,z +
1
2
0,0, ¯z +
1
2
0,0, ¯zhkl: l = 2n
4 f 2/m .. 0,
1
2
,0
1
2
,0,
1
2
0,
1
2
,
1
2
1
2
,0, 0 hkl : h + k,l = 2n
4 e 222. 0 ,
1
2
,
1
4
1
2
,0,
3
4
0,
1
2
,
3
4
1
2
,0,
1
4
hkl : h + k,l = 2n
2 d
¯
42m
1
2
,
1
2
,
1
4
1
2
,
1
2
,
3
4
hkl : l = 2n
2 cm. mm
1
2
,
1
2
,0
1
2
,
1
2
,
1
2
hkl : l = 2n
2 b
¯
42m 0,0,
1
4
0,0,
3
4
hkl : l = 2n
2 am. mm 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
=
1
2
c
Origin at x,0,0
Along [110] p2mm
a
=
1
2
(−a + b) b
= c
Origin at x,x , 0
(Continued on preceding page)
457
P4
2
/nbc D
11
4h
4/mmm Tetragonal
No. 133 P 4
2
/n 2/b 2/c Patterson symmetry P4/mmm
ORIGIN CHOICE 1
Origin at
¯
412
1
/c,at−
1
4
,
1
4
,−
1
4
from
¯
1
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
2
;0≤ z ≤
1
4
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,y,
1
4
(6) 2 x,0,
1
4
(7) 2(
1
2
,
1
2
,0) x,x,0(8)2x, ¯x +
1
2
,0
(9)
¯
1
1
4
,
1
4
,
1
4
(10) n(
1
2
,
1
2
,0) x, y,
1
4
(11)
¯
4
+
0,0,z;0,0, 0 (12)
¯
4
−
0,0,z;0,0, 0
(13) ax,
1
4
,z (14) b
1
4
,y,z (15) cx, ¯x, z (16) cx,x, z
458
International Tables for Crystallography (2006). Vol. A, Space group 133, pp. 458–461.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 133 P4
2
/nbc
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3); (5); (9)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
General:
16 k 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,y, ¯z +
1
2
(6) x, ¯y, ¯z +
1
2
(7) y +
1
2
,x +
1
2
, ¯z (8) ¯y +
1
2
, ¯x +
1
2
, ¯z
(9) ¯x+
1
2
, ¯y +
1
2
, ¯z +
1
2
(10) x +
1
2
,y +
1
2
, ¯z +
1
2
(11) y, ¯x, ¯z (12) ¯y,x, ¯z
(13) x +
1
2
, ¯y +
1
2
,z (14) ¯x+
1
2
,y +
1
2
,z (15) ¯y, ¯x,z +
1
2
(16) y,x , z +
1
2
hk0: h + k = 2n
0kl : k = 2n
hhl : l = 2n
00l : l = 2n
h00 : h = 2n
Special: as above, plus
8 j ..2 x,x +
1
2
,0¯x, ¯x +
1
2
,0¯x,x +
1
2
,
1
2
x, ¯x +
1
2
,
1
2
¯x +
1
2
, ¯x,
1
2
x +
1
2
,x,
1
2
x +
1
2
, ¯x, 0¯x +
1
2
,x,0
hkl : h + k + l = 2n
8 i . 2 . x,0,
3
4
¯x,0,
3
4
1
2
,x +
1
2
,
1
4
1
2
, ¯x +
1
2
,
1
4
¯x +
1
2
,
1
2
,
3
4
x +
1
2
,
1
2
,
3
4
0, ¯x,
1
4
0,x,
1
4
hkl : h + k = 2n
8 h . 2 . x,0,
1
4
¯x,0,
1
4
1
2
,x +
1
2
,
3
4
1
2
, ¯x +
1
2
,
3
4
¯x +
1
2
,
1
2
,
1
4
x +
1
2
,
1
2
,
1
4
0, ¯x,
3
4
0,x,
3
4
hkl : h + k = 2n
8 g 2 .. 0,0,z
1
2
,
1
2
,z +
1
2
0,0, ¯z +
1
2
1
2
,
1
2
, ¯z
1
2
,
1
2
, ¯z +
1
2
0,0, ¯z
1
2
,
1
2
,z 0, 0,z +
1
2
hkl : h + k,l = 2n
8 f 2 .. 0,
1
2
,z 0,
1
2
,z +
1
2
0,
1
2
, ¯z +
1
2
0,
1
2
, ¯z
1
2
,0, ¯z +
1
2
1
2
,0, ¯z
1
2
,0, z
1
2
,0, z +
1
2
hkl : h + k,l = 2n
8 e
¯
1
1
4
,
1
4
,
1
4
3
4
,
3
4
,
1
4
1
4
,
3
4
,
3
4
3
4
,
1
4
,
3
4
3
4
,
1
4
,
1
4
1
4
,
3
4
,
1
4
3
4
,
3
4
,
3
4
1
4
,
1
4
,
3
4
hkl : h,k,l = 2n
4 d
¯
4 .. 0,0,0
1
2
,
1
2
,
1
2
0,0,
1
2
1
2
,
1
2
,0 hkl : h + k,l = 2n
4 c 2 . 22 0,
1
2
,00,
1
2
,
1
2
1
2
,0,
1
2
1
2
,0, 0 hkl : h + k, l = 2n
4 b 222. 0,0,
1
4
1
2
,
1
2
,
3
4
1
2
,
1
2
,
1
4
0,0,
3
4
hkl : h + k,l = 2n
4 a 222. 0,
1
2
,
1
4
0,
1
2
,
3
4
1
2
,0,
1
4
1
2
,0,
3
4
hkl : h + k,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] p2mm
a
=
1
2
bb
= c
Origin at x,0,
1
4
Along [110] p2mm
a
=
1
2
(−a + b) b
=
1
2
c
Origin at x, x,0
Maximal non-isomorphic subgroups
I
[2] P
¯
4b2 (117) 1; 2; 7; 8; 11; 12; 13; 14
[2] P
¯
42c (112) 1; 2; 5; 6; 11; 12; 15; 16
[2] P4
2
bc (106) 1; 2; 3; 4; 13; 14; 15; 16
[2] P4
2
22 (93) 1; 2; 3; 4; 5; 6; 7; 8
[2] P4
2
/n11(P4
2
/n, 86) 1; 2; 3; 4; 9; 10; 11; 12
[2] P2/n12/c (Ccce, 68) 1; 2; 7; 8; 9; 10; 15; 16
[2] P2/n2/b1(Pban, 50) 1; 2; 5; 6; 9; 10; 13; 14
IIa none
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] P4
2
/nbc(c
= 3c) (133); [9] P4
2
/nbc(a
= 3a,b
= 3b) (133)
Minimal non-isomorphic supergroups
I
none
II [2] C 4
2
/mmc (P4
2
/mcm, 132); [2] I 4/mcm(140); [2] P4/nbm(c
=
1
2
c) (125)
459
P4
2
/nbc D
11
4h
4/mmm Tetragonal
No. 133 P 4
2
/n 2/b 2/c Patterson symmetry P4/mmm
ORIGIN CHOICE 2
Origin at
¯
1atn(b,a)(n,c),at
1
4
,−
1
4
,
1
4
from
¯
4
Asymmetric unit −
1
4
≤ x ≤
1
4
; −
1
4
≤ y ≤
1
4
;0≤ z ≤
1
4
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) 2
1
4
,y,0(6)2x,
1
4
,0(7)2x,x,
1
4
(8) 2 x, ¯x +
1
2
,
1
4
(9)
¯
10, 0,0 (10) n(
1
2
,
1
2
,0) x,y,0 (11)
¯
4
+
1
4
,−
1
4
,z;
1
4
,−
1
4
,
1
4
(12)
¯
4
−
−
1
4
,
1
4
,z; −
1
4
,
1
4
,
1
4
(13) ax,0,z (14) b 0, y,z (15) cx, ¯x,z (16) n(
1
2
,
1
2
,
1
2
) x,x,z
460
CONTINUED No. 133 P4
2
/nbc
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3); (5); (9)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
General:
16 k 1(1)x,y, z (2) ¯x +
1
2
, ¯y +
1
2
,z (3) ¯y+
1
2
,x,z +
1
2
(4) y, ¯x +
1
2
,z +
1
2
(5) ¯x +
1
2
,y, ¯z (6) x, ¯y +
1
2
, ¯z (7) y,x, ¯z +
1
2
(8) ¯y +
1
2
, ¯x +
1
2
, ¯z +
1
2
(9) ¯x, ¯y, ¯z (10) x +
1
2
,y +
1
2
, ¯z (11) y +
1
2
, ¯x, ¯z +
1
2
(12) ¯y,x +
1
2
, ¯z +
1
2
(13) x +
1
2
, ¯y,z (14) ¯x,y +
1
2
,z (15) ¯y, ¯x,z +
1
2
(16) y +
1
2
,x +
1
2
,z +
1
2
hk0: h + k = 2n
0kl : k = 2n
hhl : l = 2n
00l : l = 2n
h00 : h = 2n
Special: as above, plus
8 j ..2 x,x,
1
4
¯x+
1
2
, ¯x +
1
2
,
1
4
¯x +
1
2
,x,
3
4
x, ¯x +
1
2
,
3
4
¯x, ¯x,
3
4
x +
1
2
,x +
1
2
,
3
4
x +
1
2
, ¯x,
1
4
¯x,x +
1
2
,
1
4
hkl : h + k + l = 2n
8 i . 2 . x,
1
4
,
1
2
¯x +
1
2
,
1
4
,
1
2
1
4
,x,0
1
4
, ¯x +
1
2
,0
¯x,
3
4
,
1
2
x +
1
2
,
3
4
,
1
2
3
4
, ¯x,0
3
4
,x +
1
2
,0
hkl : h + k = 2n
8 h . 2 . x,
1
4
,0¯x +
1
2
,
1
4
,0
1
4
,x,
1
2
1
4
, ¯x +
1
2
,
1
2
¯x,
3
4
,0 x +
1
2
,
3
4
,0
3
4
, ¯x,
1
2
3
4
,x +
1
2
,
1
2
hkl : h + k = 2n
8 g 2 ..
3
4
,
1
4
,z
1
4
,
3
4
,z +
1
2
3
4
,
1
4
, ¯z
1
4
,
3
4
, ¯z +
1
2
1
4
,
3
4
, ¯z
3
4
,
1
4
, ¯z +
1
2
1
4
,
3
4
,z
3
4
,
1
4
,z +
1
2
hkl : h + k,l = 2n
8 f 2 ..
1
4
,
1
4
,z
1
4
,
1
4
,z +
1
2
1
4
,
1
4
, ¯z
1
4
,
1
4
, ¯z +
1
2
3
4
,
3
4
, ¯z
3
4
,
3
4
, ¯z +
1
2
3
4
,
3
4
,z
3
4
,
3
4
,z +
1
2
hkl : h + k,l = 2n
8 e
¯
10, 0,0
1
2
,
1
2
,0
1
2
,0,
1
2
0,
1
2
,
1
2
1
2
,0, 00,
1
2
,00,0,
1
2
1
2
,
1
2
,
1
2
hkl : h,k,l = 2n
4 d
¯
4 ..
3
4
,
1
4
,
3
4
1
4
,
3
4
,
1
4
3
4
,
1
4
,
1
4
1
4
,
3
4
,
3
4
hkl : h + k,l = 2n
4 c 2 . 22
1
4
,
1
4
,
1
4
1
4
,
1
4
,
3
4
3
4
,
3
4
,
3
4
3
4
,
3
4
,
1
4
hkl : h + k,l = 2n
4 b 222.
3
4
,
1
4
,0
1
4
,
3
4
,
1
2
1
4
,
3
4
,0
3
4
,
1
4
,
1
2
hkl : h + k,l = 2n
4 a 222.
1
4
,
1
4
,0
1
4
,
1
4
,
1
2
3
4
,
3
4
,0
3
4
,
3
4
,
1
2
hkl : h + k,l = 2n
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] p2mm
a
=
1
2
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] P
¯
4b2 (117) 1; 2; 7; 8; 11; 12; 13; 14
[2] P
¯
42c (112) 1; 2; 5; 6; 11; 12; 15; 16
[2] P4
2
bc (106) 1; 2; 3; 4; 13; 14; 15; 16
[2] P4
2
22 (93) 1; 2; 3; 4; 5; 6; 7; 8
[2] P4
2
/n11(P4
2
/n, 86) 1; 2; 3; 4; 9; 10; 11; 12
[2] P2/n12/c (Ccce, 68) 1; 2; 7; 8; 9; 10; 15; 16
[2] P2/n2/b1(Pban, 50) 1; 2; 5; 6; 9; 10; 13; 14
IIa none
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] P4
2
/nbc(c
= 3c) (133); [9] P4
2
/nbc(a
= 3a,b
= 3b) (133)
Minimal non-isomorphic supergroups
I
none
II [2] C 4
2
/mmc (P4
2
/mcm, 132); [2] I 4/mcm(140); [2] P4/nbm(c
=
1
2
c) (125)
461
P4
2
/nnm D
12
4h
4/mmm Tetragonal
No. 134 P 4
2
/n 2/n 2/m Patterson symmetry P4/mmm
ORIGIN CHOICE 1
Origin at
¯
42m,at−
1
4
,
1
4
,−
1
4
from centre (2/m)
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤ 1; 0 ≤ z ≤
1
4
; x ≤ y; y ≤ 1 − x
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,y, 0(6)2x,0,0(7)2(
1
2
,
1
2
,0) x, x,
1
4
(8) 2 x, ¯x +
1
2
,
1
4
(9)
¯
1
1
4
,
1
4
,
1
4
(10) n(
1
2
,
1
2
,0) x, y,
1
4
(11)
¯
4
+
0,0,z;0,0, 0 (12)
¯
4
−
0,0,z;0,0, 0
(13) n(
1
2
,0,
1
2
) x,
1
4
,z (14) n(0,
1
2
,
1
2
)
1
4
,y,z (15) mx, ¯x,z (16) mx,x,z
Maximal non-isomorphic subgroups
I
[2] P
¯
4n2 (118) 1; 2; 7; 8; 11; 12; 13; 14
[2] P
¯
42m (111) 1; 2; 5; 6; 11; 12; 15; 16
[2] P4
2
nm (102) 1; 2; 3; 4; 13; 14; 15; 16
[2] P4
2
22 (93) 1; 2; 3; 4; 5; 6; 7; 8
[2] P4
2
/n11(P4
2
/n, 86) 1; 2; 3; 4; 9; 10; 11; 12
[2] P2/n12/m (Cmme, 67) 1; 2; 7; 8; 9; 10; 15; 16
[2] P2/n2/n1(Pnnn, 48) 1; 2; 5; 6; 9; 10; 13; 14
IIa none
IIb [2] F 4
1
/ddc(a
= 2a,b
= 2b,c
= 2c)(I 4
1
/acd, 142); [2] F 4
1
/ddm(a
= 2a,b
= 2b,c
= 2c)(I 4
1
/amd, 141)
Maximal isomorphic subgroups of lowest index
IIc
[3] P4
2
/nnm(c
= 3c) (134); [9] P4
2
/nnm(a
= 3a,b
= 3b) (134)
Minimal non-isomorphic supergroups
I
[3] Pn
¯
3m (224)
II [2] C 4
2
/mcm (P4
2
/mmc, 131); [2] I 4 /mmm(139); [2] P4/nbm( c
=
1
2
c) (125)
462
International Tables for Crystallography (2006). Vol. A, Space group 134, pp. 462–465.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 134 P4
2
/nnm
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3); (5); (9)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
General:
16 n 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,y, ¯z (6) x, ¯y, ¯z (7) y +
1
2
,x +
1
2
, ¯z +
1
2
(8) ¯y +
1
2
, ¯x +
1
2
, ¯z +
1
2
(9) ¯x+
1
2
, ¯y +
1
2
, ¯z +
1
2
(10) x +
1
2
,y +
1
2
, ¯z +
1
2
(11) y, ¯x, ¯z (12) ¯y,x, ¯z
(13) x +
1
2
, ¯y +
1
2
,z +
1
2
(14) ¯x +
1
2
,y +
1
2
,z +
1
2
(15) ¯y, ¯x,z (16) y,x,z
hk0: h + k = 2n
0kl : k + l = 2n
00l : l = 2n
h00 : h = 2n
Special: as above, plus
8 m ..mx,x, z ¯x, ¯x,z ¯x +
1
2
,x +
1
2
,z +
1
2
x +
1
2
, ¯x +
1
2
,z +
1
2
¯x,x, ¯zx, ¯x, ¯zx+
1
2
,x +
1
2
, ¯z +
1
2
¯x +
1
2
, ¯x +
1
2
, ¯z +
1
2
no extra conditions
8 l ..2 x,x +
1
2
,
3
4
¯x, ¯x +
1
2
,
3
4
¯x,x +
1
2
,
1
4
x, ¯x +
1
2
,
1
4
¯x+
1
2
, ¯x,
3
4
x +
1
2
,x,
3
4
x +
1
2
, ¯x,
1
4
¯x +
1
2
,x,
1
4
hkl : h + k = 2n
8 k ..2 x, x +
1
2
,
1
4
¯x, ¯x +
1
2
,
1
4
¯x,x +
1
2
,
3
4
x, ¯x +
1
2
,
3
4
¯x+
1
2
, ¯x,
1
4
x +
1
2
,x,
1
4
x +
1
2
, ¯x,
3
4
¯x +
1
2
,x,
3
4
hkl : h + k = 2n
8 j . 2 . x,0,
1
2
¯x,0,
1
2
1
2
,x +
1
2
,0
1
2
, ¯x +
1
2
,0
¯x+
1
2
,
1
2
,0 x +
1
2
,
1
2
,00, ¯x,
1
2
0,x,
1
2
hkl : h + k + l = 2n
8 i . 2 . x,0, 0¯x, 0,0
1
2
,x +
1
2
,
1
2
1
2
, ¯x +
1
2
,
1
2
¯x+
1
2
,
1
2
,
1
2
x +
1
2
,
1
2
,
1
2
0, ¯x,00,x,0
hkl : h + k + l = 2n
8 h 2 .. 0,
1
2
,z 0,
1
2
,z +
1
2
0,
1
2
, ¯z 0,
1
2
, ¯z +
1
2
1
2
,0, ¯z +
1
2
1
2
,0, ¯z
1
2
,0, z +
1
2
1
2
,0, z
hkl : h + k,l = 2n
4 g 2 . mm 0,0,z
1
2
,
1
2
,z +
1
2
0,0, ¯z
1
2
,
1
2
, ¯z +
1
2
hkl : h + k + l = 2n
4 f ..2/m
3
4
,
3
4
,
3
4
1
4
,
1
4
,
3
4
3
4
,
1
4
,
1
4
1
4
,
3
4
,
1
4
hkl : h + k,h + l,k + l = 2n
4 e ..2/m
1
4
,
1
4
,
1
4
3
4
,
3
4
,
1
4
1
4
,
3
4
,
3
4
3
4
,
1
4
,
3
4
hkl : h + k,h + l,k + l = 2n
4 d 2 . 22 0,
1
2
,
1
4
0,
1
2
,
3
4
1
2
,0,
1
4
1
2
,0,
3
4
hkl : h + k,l = 2n
4 c 222. 0,
1
2
,00,
1
2
,
1
2
1
2
,0,
1
2
1
2
,0, 0 hkl : h + k,l = 2n
2 b
¯
42m 0,0,
1
2
1
2
,
1
2
,0 hk l : h + k + l = 2n
2 a
¯
42m 0,0,0
1
2
,
1
2
,
1
2
hkl : h + k + 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] c2mm
a
= bb
= c
Origin at x, 0,0
Along [110] p2mm
a
=
1
2
(−a + b) b
= c
Origin at x,x ,
1
4
(Continued on preceding page)
463