Hahn Th. (ed.) International tables for crystallography. Vol. A. Space-group symmetry

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 2

Origin at centre (2/m) at nn(2

1

/g,2/m),at

1

4

,−

1

4

,

1

4

from

¯

42m

Asymmetric unit −

1

4

≤ x ≤

1

4

; −

1

4

≤ y ≤

1

4

;0≤ z ≤

1

2

; x ≤−y

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,

1

4

(6) 2 x,

1

4

,

1

4

(7) 2 x,x,0(8)2x, ¯x +

1

2

,0

(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) n(

1

2

,0,

1

2

) x,0,z (14) n(0,

1

2

,

1

2

) 0,y, z (15) mx, ¯x,z (16) g(

1

2

,

1

2

,0) x, 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)

464

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 +

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 +

1

2

(6) x, ¯y +

1

2

, ¯z +

1

2

(7) y,x, ¯z (8) ¯y+

1

2

, ¯x +

1

2

, ¯z

(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 +

1

2

(14) ¯x,y +

1

2

,z +

1

2

(15) ¯y, ¯x,z (16) y +

1

2

,x +

1

2

,z

hk0: h + k = 2n

0kl : k + l = 2 n

00l : l = 2n

h00 : h = 2n

Special: as above, plus

8 m ..mx, ¯x,z ¯x +

1

2

,x +

1

2

,zx+

1

2

,x,z +

1

2

¯x, ¯x +

1

2

,z +

1

2

¯x+

1

2

, ¯x, ¯z +

1

2

x,x +

1

2

, ¯z +

1

2

¯x,x, ¯zx+

1

2

, ¯x +

1

2

, ¯z

no extra conditions

8 l ..2 x,x,

1

2

¯x +

1

2

, ¯x +

1

2

,

1

2

¯x +

1

2

,x,0 x, ¯x +

1

2

,0

¯x, ¯x,

1

2

x +

1

2

,x +

1

2

,

1

2

x +

1

2

, ¯x, 0¯x,x +

1

2

,0

hkl : h + k = 2n

8 k ..2 x, x,0¯x +

1

2

, ¯x +

1

2

,0¯x +

1

2

,x,

1

2

x, ¯x +

1

2

,

1

2

¯x, ¯x, 0 x +

1

2

,x +

1

2

,0 x +

1

2

, ¯x,

1

2

¯x,x +

1

2

,

1

2

hkl : h + k = 2n

8 j . 2 . x,

1

4

,

1

4

¯x +

1

2

,

1

4

,

1

4

1

4

,x,

3

4

1

4

, ¯x +

1

2

,

3

4

¯x,

3

4

,

3

4

x +

1

2

,

3

4

,

3

4

3

4

, ¯x,

1

4

3

4

,x +

1

2

,

1

4

hkl : h + k + l = 2n

8 i . 2 . x,

1

4

,

3

4

¯x +

1

2

,

1

4

,

3

4

1

4

,x,

1

4

1

4

, ¯x +

1

2

,

1

4

¯x,

3

4

,

1

4

x +

1

2

,

3

4

,

1

4

3

4

, ¯x,

3

4

3

4

,x +

1

2

,

3

4

hkl : h + k + l = 2n

8 h 2 ..

1

4

,

1

4

,z

1

4

,

1

4

,z +

1

2

1

4

,

1

4

, ¯z +

1

2

1

4

,

1

4

, ¯z

3

4

,

3

4

, ¯z

3

4

,

3

4

, ¯z +

1

2

3

4

,

3

4

,z +

1

2

3

4

,

3

4

,z

hkl : h + k,l = 2n

4 g 2 . mm

3

4

,

1

4

,z

1

4

,

3

4

,z +

1

2

3

4

,

1

4

, ¯z +

1

2

1

4

,

3

4

, ¯zhkl: h + k + l = 2n

4 f ..2/m 0,0,0

1

2

,

1

2

,0

1

2

,0,

1

2

0,

1

2

,

1

2

hkl : h + k,h + l,k + l = 2n

4 e ..2/m 0,0,

1

2

1

2

,

1

2

,

1

2

1

2

,0, 00,

1

2

,0 hkl : h + k,h + l, k + l = 2n

4 d 2 . 22

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

4 c 222.

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

2 b

¯

42m

3

4

,

1

4

,

1

4

1

4

,

3

4

,

3

4

hkl : h + k + l = 2n

2 a

¯

42m

1

4

,

3

4

,

1

4

3

4

,

1

4

,

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

1

4

,

1

4

,z

Along [100] c2mm

a



= bb



= c

Origin at x,

1

4

,

1

4

Along [110] p2mm

a



=

1

2

(−a + b) b



= c

Origin at x,x , 0

(Continued on preceding page)

465

P4

2

/mbc D

13

4h

4/mmm Tetragonal

No. 135 P 4

2

/m 2

1

/b 2/c Patterson symmetry P4/mmm

Origin at centre (2/m) at 4

2

/m1n

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,

1

2

,0)

1

4

,y,0(6)2(

1

2

,0, 0) x,

1

4

,0(7)2(

1

2

,

1

2

,0) x, x,

1

4

(8) 2 x, ¯x +

1

2

,

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) ax,

1

4

,z (14) b

1

4

,y,z (15) cx+

1

2

, ¯x, z (16) n(

1

2

,

1

2

,

1

2

) x,x,z

466

International Tables for Crystallography (2006). Vol. A, Space group 135, pp. 466–467.

CONTINUED No. 135 P4

2

/mbc

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 i 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

(9) ¯x, ¯y, ¯z (10) x,y, ¯z (11) y, ¯x, ¯z +

1

2

(12) ¯y,x, ¯z +

1

2

(13) x +

1

2

, ¯y +

1

2

,z (14) ¯x +

1

2

,y +

1

2

,z (15) ¯y +

1

2

, ¯x +

1

2

,z +

1

2

(16) 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

8 hm.. x,y,0¯x, ¯y,0¯y,x,

1

2

y, ¯x,

1

2

¯x +

1

2

,y +

1

2

,0 x +

1

2

, ¯y +

1

2

,0 y +

1

2

,x +

1

2

,

1

2

¯y +

1

2

, ¯x +

1

2

,

1

2

no extra conditions

8 g ..2 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, ¯x +

1

2

,

3

4

x,x +

1

2

,

3

4

x +

1

2

, ¯x,

1

4

¯x +

1

2

,x,

1

4

hkl : l = 2n

8 f 2 .. 0,

1

2

,z

1

2

,0, z +

1

2

1

2

,0, ¯z 0,

1

2

, ¯z +

1

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

8 e 2 .. 0, 0,z 0,0,z +

1

2

1

2

,

1

2

, ¯z

1

2

,

1

2

, ¯z +

1

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

4 d 2 . 22 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

4 c 2/m .. 0,

1

2

,0

1

2

,0,

1

2

1

2

,0, 00,

1

2

,

1

2

hkl : h + k,l = 2n

4 b

¯

4 .. 0,0,

1

4

0,0,

3

4

1

2

,

1

2

,

3

4

1

2

,

1

2

,

1

4

hkl : h + k,l = 2n

4 a 2/ m .. 0, 0,00, 0,

1

2

1

2

,

1

2

,0

1

2

,

1

2

,

1

2

hkl : h + k,l = 2n

Symmetry of special projections

Along [001] p4gm

a



= ab



= b

Origin at 0,0,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

¯

42

1

c (114) 1; 2; 5; 6; 11; 12; 15; 16

[2] P4

2

bc (106) 1; 2; 3; 4; 13; 14; 15; 16

[2] P4

2

1

2 (94) 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

1

/b1(Pbam, 55) 1; 2; 5; 6; 9; 10; 13; 14

IIa none

IIb none

Maximal isomorphic subgroups of lowest index

IIc

[3] P4

2

/mbc (c



= 3c) (135); [9] P4

2

/mbc(a



= 3a,b



= 3b) (135)

Minimal non-isomorphic supergroups

I

none

II [2] C 4

2

/mmc (P4

2

/mcm, 132); [2] I 4/mcm(140); [2] P4/mbm(c



=

1

2

c) (127)

467

P4

2

/mnm D

14

4h

4/mmm Tetragonal

No. 136 P 4

2

/m 2

1

/n 2/m Patterson symmetry P4/mmm

Origin at centre (mmm) at 2/m12/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,

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 x,x,0(8)2x, ¯x,0

(9)

¯

10, 0,0 (10) mx, y,0 (11)

¯

4

+

1

2

,0, z;

1

2

,0,

1

4

(12)

¯

4

−

0,

1

2

,z;0,

1

2

,

1

4

(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

468

International Tables for Crystallography (2006). Vol. A, Space group 136, pp. 468–469.

CONTINUED No. 136 P4

2

/mnm

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+

1

2

,y +

1

2

, ¯z +

1

2

(6) x +

1

2

, ¯y +

1

2

, ¯z +

1

2

(7) y,x, ¯z (8) ¯y, ¯x, ¯z

(9) ¯x, ¯y, ¯z (10) x,y, ¯z (11) y +

1

2

, ¯x +

1

2

, ¯z +

1

2

(12) ¯y +

1

2

,x +

1

2

, ¯z +

1

2

(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

0kl : k + l = 2n

00l : l = 2n

h00 : h = 2n

Special: as above, plus

8 j ..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 +

1

2

,x +

1

2

, ¯z +

1

2

x +

1

2

, ¯x +

1

2

, ¯z +

1

2

x,x, ¯z ¯x, ¯x, ¯z

no extra conditions

8 im.. x, y,0¯x, ¯y, 0¯y +

1

2

,x +

1

2

,

1

2

y +

1

2

, ¯x +

1

2

,

1

2

¯x +

1

2

,y +

1

2

,

1

2

x +

1

2

, ¯y +

1

2

,

1

2

y, x,0¯y, ¯x,0

no extra conditions

8 h 2 .. 0 ,

1

2

,z 0,

1

2

,z +

1

2

1

2

,0, ¯z +

1

2

1

2

,0, ¯z

0,

1

2

, ¯z 0,

1

2

, ¯z +

1

2

1

2

,0, z +

1

2

1

2

,0, z

hkl : h + k,l = 2n

4 gm. 2mx, ¯x,0¯x, x,0 x +

1

2

,x +

1

2

,

1

2

¯x+

1

2

, ¯x +

1

2

,

1

2

no extra conditions

4 fm. 2mx,x,0¯x, ¯x,0¯x +

1

2

,x +

1

2

,

1

2

x +

1

2

, ¯x +

1

2

,

1

2

no extra conditions

4 e 2 . mm 0,0,z

1

2

,

1

2

,z +

1

2

1

2

,

1

2

, ¯z +

1

2

0,0, ¯zhkl: h + k + l = 2n

4 d

¯

4 .. 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 2/m .. 0,

1

2

,00,

1

2

,

1

2

1

2

,0,

1

2

1

2

,0, 0 hkl : h + k,l = 2n

2 bm. mm 0,0,

1

2

1

2

,

1

2

,0 hkl : h + k + l = 2n

2 am. mm 0,0,0

1

2

,

1

2

,

1

2

hkl : h + k + l = 2n

Symmetry of special projections

Along [001] p4gm

a



= ab



= b

Origin at 0,

1

2

,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, 0

Maximal non-isomorphic subgroups

I

[2] P

¯

4n2 (118) 1; 2; 7; 8; 11; 12; 13; 14

[2] P

¯

42

1

m (113) 1; 2; 5; 6; 11; 12; 15; 16

[2] P4

2

nm (102) 1; 2; 3; 4; 13; 14; 15; 16

[2] P4

2

1

2 (94) 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

1

/n1(Pnnm, 58) 1; 2; 5; 6; 9; 10; 13; 14

IIa none

IIb none

Maximal isomorphic subgroups of lowest index

IIc

[3] P4

2

/mnm (c



= 3c) (136); [9] P4

2

/mnm (a



= 3a,b



= 3b) (136)

Minimal non-isomorphic supergroups

I

none

II [2] C 4

2

/mcm (P4

2

/mmc, 131); [2] I 4 /mmm(139); [2] P4/mbm(c



=

1

2

c) (127)

469

P4

2

/nmc D

15

4h

4/mmm Tetragonal

No. 137 P 4

2

/n 2

1

/m 2 /c Patterson symmetry P4/mmm

ORIGIN CHOICE 1

Origin at

¯

4m2/n,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,

1

2

,0)

1

4

,y,

1

4

(6) 2(

1

2

,0, 0) x,

1

4

,

1

4

(7) 2 x,x,0(8)2x, ¯x,0

(9)

¯

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) mx,0,z (14) m 0, y,z (15) cx+

1

2

, ¯x, z (16) n(

1

2

,

1

2

,

1

2

) x,x,z

470

International Tables for Crystallography (2006). Vol. A, Space group 137, pp. 470–473.

CONTINUED No. 137 P4

2

/nmc

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 h 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+

1

2

,y +

1

2

, ¯z +

1

2

(6) x +

1

2

, ¯y +

1

2

, ¯z +

1

2

(7) y,x, ¯z (8) ¯y, ¯x, ¯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, ¯y,z (14) ¯x,y,z (15) ¯y +

1

2

, ¯x +

1

2

,z +

1

2

(16) y +

1

2

,x +

1

2

,z +

1

2

hk0: h + k = 2n

hhl : l = 2n

00l : l = 2n

h00 : h = 2n

Special: as above, plus

8 g . m . 0,y,z 0, ¯y, z ¯y+

1

2

,

1

2

,z +

1

2

y +

1

2

,

1

2

,z +

1

2

1

2

,y +

1

2

, ¯z +

1

2

1

2

, ¯y +

1

2

, ¯z +

1

2

y, 0, ¯z ¯y,0, ¯z

no extra conditions

8 f ..2 x,x,0¯x, ¯x , 0¯x +

1

2

,x +

1

2

,

1

2

x +

1

2

, ¯x +

1

2

,

1

2

¯x +

1

2

, ¯x +

1

2

,

1

2

x +

1

2

,x +

1

2

,

1

2

x, ¯x,0¯x,x,0

hkl : h + k + l = 2n

8 e

¯

1

4

,

1

4

,

1

4

3

4

,

3

4

,

1

4

1

4

,

3

4

,

3

4

3

4

,

1

4

,

3

4

1

4

,

3

4

,

1

4

3

4

,

1

4

,

1

4

1

4

,

1

4

,

3

4

3

4

,

3

4

,

3

4

hkl : h,k,l = 2n

4 d 2 mm. 0,

1

2

,z 0,

1

2

,z +

1

2

1

2

,0, ¯z +

1

2

1

2

,0, ¯zhkl: l = 2n

4 c 2 mm. 0,0,z

1

2

,

1

2

,z +

1

2

1

2

,

1

2

, ¯z +

1

2

0,0, ¯zhkl: h + k + l = 2n

2 b

¯

4 m 20,0,

1

2

1

2

,

1

2

,0 hkl : h + k + l = 2n

2 a

¯

4 m 20,0,0

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] p2mg

a



= bb



= c

Origin at x,

1

4

,

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

¯

4m2 (115) 1; 2; 7; 8; 11; 12; 13; 14

[2] P

¯

42

1

c (114) 1; 2; 5; 6; 11; 12; 15; 16

[2] P4

2

mc (105) 1; 2; 3; 4; 13; 14; 15; 16

[2] P4

2

1

2 (94) 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

1

/m1(Pmmn, 59) 1; 2; 5; 6; 9; 10; 13; 14

IIa none

IIb none

Maximal isomorphic subgroups of lowest index

IIc

[3] P4

2

/nmc(c



= 3c) (137); [9] P4

2

/nmc(a



= 3a,b



= 3b) (137)

Minimal non-isomorphic supergroups

I

none

II [2] C 4

2

/mmc (P4

2

/mcm, 132); [2] I 4/mmm(139); [2] P4/nmm(c



=

1

2

c) (129)

471

P4

2

/nmc D

15

4h

4/mmm Tetragonal

No. 137 P 4

2

/n 2

1

/m 2 /c Patterson symmetry P4/mmm

ORIGIN CHOICE 2

Origin at

¯

1atn2

1

(c,n),at

1

4

,−

1

4

,

1

4

from

¯

4m2

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(0,

1

2

,0) 0,y,0(6)2(

1

2

,0, 0) x,0,0(7)2(

1

2

,

1

2

,0) x,x,

1

4

(8) 2 x , ¯x,

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) mx,

1

4

,z (14) m

1

4

,y,z (15) cx+

1

2

, ¯x,z (16) cx,x,z

472

CONTINUED No. 137 P4

2

/nmc

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 h 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,y +

1

2

, ¯z (6) x+

1

2

, ¯y, ¯z (7) y +

1

2

,x +

1

2

, ¯z +

1

2

(8) ¯y, ¯x, ¯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, ¯y +

1

2

,z (14) ¯x +

1

2

,y,z (15) ¯y+

1

2

, ¯x +

1

2

,z +

1

2

(16) y,x , z +

1

2

hk0: h + k = 2n

hhl : l = 2n

00l : l = 2n

h00 : h = 2n

Special: as above, plus

8 g . m .

1

4

,y,z

1

4

, ¯y +

1

2

,z ¯y+

1

2

,

1

4

,z +

1

2

y,

1

4

,z +

1

2

3

4

,y +

1

2

, ¯z

3

4

, ¯y, ¯zy+

1

2

,

3

4

, ¯z +

1

2

¯y,

3

4

, ¯z +

1

2

no extra conditions

8 f ..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 e

¯

10,0, 0

1

2

,

1

2

,0

1

2

,0,

1

2

0,

1

2

,

1

2

0,

1

2

,0

1

2

,0, 0

1

2

,

1

2

,

1

2

0,0,

1

2

hkl : h,k,l = 2n

4 d 2 mm.

1

4

,

1

4

,z

1

4

,

1

4

,z +

1

2

3

4

,

3

4

, ¯z

3

4

,

3

4

, ¯z +

1

2

hkl : l = 2n

4 c 2 mm.

3

4

,

1

4

,z

1

4

,

3

4

,z +

1

2

1

4

,

3

4

, ¯z

3

4

,

1

4

, ¯z +

1

2

hkl : h + k + l = 2n

2 b

¯

4 m 2

3

4

,

1

4

,

1

4

1

4

,

3

4

,

3

4

hkl : h + k + l = 2n

2 a

¯

4 m 2

3

4

,

1

4

,

3

4

1

4

,

3

4

,

1

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

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



=

1

2

c

Origin at x,x, 0

Maximal non-isomorphic subgroups

I

[2] P

¯

4m2 (115) 1; 2; 7; 8; 11; 12; 13; 14

[2] P

¯

42

1

c (114) 1; 2; 5; 6; 11; 12; 15; 16

[2] P4

2

mc (105) 1; 2; 3; 4; 13; 14; 15; 16

[2] P4

2

1

2 (94) 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

1

/m1(Pmmn, 59) 1; 2; 5; 6; 9; 10; 13; 14

IIa none

IIb none

Maximal isomorphic subgroups of lowest index

IIc

[3] P4

2

/nmc(c



= 3c) (137); [9] P4

2

/nmc(a



= 3a,b



= 3b) (137)

Minimal non-isomorphic supergroups

I

none

II [2] C 4

2

/mmc (P4

2

/mcm, 132); [2] I 4/mmm(139); [2] P4/nmm(c



=

1

2

c) (129)

473

Hahn Th. (ed.) International tables for crystallography. Vol. A. Space-group symmetry

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