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

P432 O

1

432 Cubic

No. 207 P432

Patterson symmetry Pm

¯

3m

Origin at 432

Asymmetric unit 0 ≤ x ≤ 1; 0 ≤ y ≤

1

2

;0≤ z ≤

1

2

; y ≤ min(x,1 − x); z ≤ y

Vertices 0,0, 01,0, 0

1

2

,

1

2

,0

1

2

,

1

2

,

1

2

Symmetry operations

(1) 1 (2) 2 0,0, z (3) 2 0,y, 0(4)2x , 0,0

(5) 3

+

x,x, x (6) 3

+

¯x,x, ¯x (7) 3

+

x, ¯x, ¯x (8) 3

+

¯x, ¯x, x

(9) 3

−

x,x, x (10) 3

−

x, ¯x, ¯x (11) 3

−

¯x, ¯x, x (12) 3

−

¯x,x, ¯x

(13) 2 x,x,0 (14) 2 x, ¯x,0 (15) 4

−

0,0,z (16) 4

+

0,0,z

(17) 4

−

x,0, 0 (18) 2 0, y,y (19) 2 0, y, ¯y (20) 4

+

x,0, 0

(21) 4

+

0,y,0 (22) 2 x,0,x (23) 4

−

0,y,0 (24) 2 ¯x,0,x

634

International Tables for Crystallography (2006). Vol. A, Space group 207, pp. 634–635.

CONTINUED No. 207 P432

Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3); (5); (13)

Positions

Multiplicity,

Wyckoff letter,

Site symmetry

Coordinates Reﬂection conditions

h,k,l permutable

General:

24 k 1(1)x,y, z (2) ¯x, ¯y,z (3) ¯x,y, ¯z (4) x, ¯y, ¯z

(5) z,x,y (6) z, ¯x, ¯y (7) ¯z, ¯x,y (8) ¯z,x, ¯y

(9) y,z,x (10) ¯y,z, ¯x (11) y, ¯z, ¯x (12) ¯y, ¯z,x

(13) y,x , ¯z

(14) ¯y, ¯x, ¯z (15) y, ¯x, z (16) ¯y,x,z

(17) x,z, ¯y (18) ¯x,z,y (19) ¯x, ¯z, ¯y (20) x , ¯z,y

(21) z,y, ¯x (22) z, ¯y,x (23) ¯z,y,x (24) ¯z, ¯y, ¯x

no conditions

Special: no extra conditions

12 j ..2

1

2

,y,y

1

2

, ¯y, y

1

2

,y, ¯y

1

2

, ¯y, ¯yy,

1

2

,yy,

1

2

, ¯y

¯y,

1

2

,y ¯y,

1

2

, ¯yy,y,

1

2

¯y,y,

1

2

y, ¯y,

1

2

¯y, ¯y,

1

2

12 i ..20,y, y 0, ¯y,y 0,y, ¯y 0, ¯y, ¯yy,0 , yy, 0, ¯y

¯y,0, y ¯y,0, ¯yy,y, 0¯y,y, 0 y, ¯y,0¯y, ¯y,0

12 h 2 .. x,

1

2

,0¯x,

1

2

,00, x,

1

2

0, ¯x,

1

2

1

2

,0, x

1

2

,0, ¯x

1

2

,x,0

1

2

, ¯x, 0 x,0,

1

2

¯x,0,

1

2

0,

1

2

, ¯x 0,

1

2

,x

8 g . 3 . x,x, x ¯x, ¯x,x ¯x,x, ¯xx, ¯x, ¯x

x,x, ¯x ¯x, ¯x, ¯xx, ¯x,x ¯x,x,x

6 f 4 .. x,

1

2

,

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

, ¯x

6 e 4 .. x,0,0¯x,0,00, x,00, ¯x,00,0,x 0,0, ¯x

3 d 42.2

1

2

,0, 00,

1

2

,00, 0,

1

2

3 c 42.20,

1

2

,

1

2

1

2

,0,

1

2

1

2

,

1

2

,0

1 b 432

1

2

,

1

2

,

1

2

1 a 432 0,0,0

Symmetry of special projections

Along [001] p4mm

a



= ab



= b

Origin at 0,0,z

Along [111] p3m1

a



=

1

3

(2a − b− c) b



=

1

3

(−a + 2b − c)

Origin at x,x , x

Along [110] p2mm

a



=

1

2

(−a + b) b



= c

Origin at x,x , 0

Maximal non-isomorphic subgroups

I

[2] P231 (P23, 195) 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12



[3] P412 (P422, 89) 1; 2; 3; 4; 13; 14; 15; 16

[3] P412 (P422, 89) 1; 2; 3; 4; 17; 18; 19; 20

[3] P412 (P422, 89) 1; 2; 3; 4; 21; 22; 23; 24

⎧

⎪

⎨

⎪

⎩

[4] P132 (R32, 155) 1; 5; 9; 14; 19; 24

[4] P132 (R32, 155) 1; 6; 12; 13; 18; 24

[4] P132 (R32, 155) 1; 7; 10; 13; 19; 22

[4] P132 (R32, 155) 1; 8; 11; 14; 18; 22

IIa none

IIb [2] F 432(a



= 2a,b



= 2b,c



= 2c) (209); [4] I 432 (a



= 2a,b



= 2b,c



= 2c) (211)

Maximal isomorphic subgroups of lowest index

IIc

[27] P432 (a



= 3a,b



= 3b,c



= 3c) (207)

Minimal non-isomorphic supergroups

I

[2] Pm

¯

3m (221); [2] Pn

¯

3n (222)

II [2] I 432 (211); [4] F 432 (209)

635

P4

2

32 O

2

432 Cubic

No. 208 P4

2

32 Patterson symmetry Pm

¯

3m

Origin at 23

Asymmetric unit 0 ≤ x ≤

1

2

;0≤ y ≤

1

2

; −

1

4

≤ z ≤

1

4

;max(−x,x −

1

2

,−y,y −

1

2

) ≤ z ≤ min(x,

1

2

− x,y,

1

2

− y)

Vertices 0,0, 0

1

2

,0, 0

1

2

,

1

2

,00,

1

2

,0

1

4

,

1

4

,

1

4

1

4

,

1

4

,−

1

4

Symmetry operations

(1) 1 (2) 2 0,0, z (3) 2 0,y,0(4)2x,0,0

(5) 3

+

x,x, x (6) 3

+

¯x,x, ¯x (7) 3

+

x, ¯x, ¯x (8) 3

+

¯x, ¯x, x

(9) 3

−

x,x, x (10) 3

−

x, ¯x, ¯x (11) 3

−

¯x, ¯x,x (12) 3

−

¯x,x, ¯x

(13) 2(

1

2

,

1

2

,0) x,x,

1

4

(14) 2 x, ¯x +

1

2

,

1

4

(15) 4

−

(0,0,

1

2

)

1

2

,0, z (16) 4

+

(0,0,

1

2

) 0,

1

2

,z

(17) 4

−

(

1

2

,0, 0) x,

1

2

,0 (18) 2(0 ,

1

2

,

1

2

)

1

4

,y,y (19) 2

1

4

,y +

1

2

, ¯y (20) 4

+

(

1

2

,0, 0) x,0,

1

2

(21) 4

+

(0,

1

2

,0)

1

2

,y,0 (22) 2(

1

2

,0,

1

2

) x,

1

4

,x (23) 4

−

(0,

1

2

,0) 0,y,

1

2

(24) 2 ¯x +

1

2

,

1

4

,x

636

International Tables for Crystallography (2006). Vol. A, Space group 208, pp. 636–638.

CONTINUED No. 208 P4

2

32

Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3); (5); (13)

Positions

Multiplicity,

Wyckoff letter,

Site symmetry

Coordinates Reﬂection conditions

h,k,l permutable

General:

24 m 1(1)x,y,z (2) ¯x, ¯y,z (3) ¯x,y, ¯z (4) x, ¯y, ¯z

(5) z,x,y (6) z, ¯x, ¯y (7) ¯z, ¯x,y (8) ¯z,x, ¯y

(9) y,z,x (10) ¯y,z, ¯x (11) y, ¯z, ¯x (12) ¯y, ¯z,x

(13) y +

1

2

,x +

1

2

, ¯z +

1

2

(14) ¯y +

1

2

, ¯x +

1

2

, ¯z +

1

2

(15) y +

1

2

, ¯x +

1

2

,z +

1

2

(16) ¯y +

1

2

,x +

1

2

,z +

1

2

(17) x +

1

2

,z +

1

2

, ¯y +

1

2

(18) ¯x +

1

2

,z +

1

2

,y +

1

2

(19) ¯x +

1

2

, ¯z +

1

2

, ¯y +

1

2

(20) x +

1

2

, ¯z +

1

2

,y +

1

2

(21) z +

1

2

,y +

1

2

, ¯x +

1

2

(22) z +

1

2

, ¯y +

1

2

,x +

1

2

(23) ¯z +

1

2

,y +

1

2

,x +

1

2

(24) ¯z+

1

2

, ¯y +

1

2

, ¯x +

1

2

h00 : h = 2n

Special: as above, plus

12 l ..2

1

4

,y,y +

1

2

3

4

, ¯y, y +

1

2

3

4

,y, ¯y +

1

2

1

4

, ¯y, ¯y +

1

2

y +

1

2

,

1

4

,yy+

1

2

,

3

4

, ¯y ¯y +

1

2

,

3

4

,y ¯y +

1

2

,

1

4

, ¯y

y, y +

1

2

,

1

4

¯y,y +

1

2

,

3

4

y, ¯y +

1

2

,

3

4

¯y, ¯y +

1

2

,

1

4

no extra conditions

12 k ..2

1

4

,y, ¯y +

1

2

3

4

, ¯y, ¯y +

1

2

3

4

,y,y +

1

2

1

4

, ¯y, y +

1

2

¯y +

1

2

,

1

4

,y ¯y +

1

2

,

3

4

, ¯yy+

1

2

,

3

4

,yy+

1

2

,

1

4

, ¯y

y, ¯y+

1

2

,

1

4

¯y, ¯y +

1

2

,

3

4

y, y +

1

2

,

3

4

¯y,y +

1

2

,

1

4

no extra conditions

12 j 2 .. x,

1

2

,0¯x,

1

2

,00, x,

1

2

0, ¯x,

1

2

1

2

,0, x

1

2

,0, ¯x

0,x +

1

2

,

1

2

0, ¯x+

1

2

,

1

2

x +

1

2

,

1

2

,0¯x +

1

2

,

1

2

,0

1

2

,0, ¯x +

1

2

1

2

,0, x +

1

2

hkl : h = 2n

hhl : l = 2n

12 i 2 .. x,0,

1

2

¯x,0,

1

2

1

2

,x,0

1

2

, ¯x, 00,

1

2

,x 0,

1

2

, ¯x

1

2

,x +

1

2

,0

1

2

, ¯x +

1

2

,0 x +

1

2

,0,

1

2

¯x +

1

2

,0,

1

2

0,

1

2

, ¯x +

1

2

0,

1

2

,x +

1

2

hkl : h = 2n

hhl : l = 2n

12 h 2 .. x, 0,0¯x,0,00,x,00, ¯x, 00,0,x 0,0, ¯x

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

1

2

,

1

2

, ¯x +

1

2

1

2

,

1

2

,x +

1

2

hkl : h + k + l = 2n

8 g . 3 . x,x , x ¯x, ¯x,x

¯x,x, ¯xx, ¯x , ¯x

x +

1

2

,x +

1

2

, ¯x +

1

2

¯x +

1

2

, ¯x +

1

2

, ¯x +

1

2

x +

1

2

, ¯x +

1

2

,x +

1

2

¯x +

1

2

,x +

1

2

,x +

1

2

0kl : k + l = 2 n

6 f 2 . 22

1

4

,

1

2

,0

3

4

,

1

2

,00,

1

4

,

1

2

0,

3

4

,

1

2

1

2

,0,

1

4

1

2

,0,

3

4

6 e 2 . 22

1

4

,0,

1

2

3

4

,0,

1

2

1

2

,

1

4

,0

1

2

,

3

4

,00,

1

2

,

1

4

0,

1

2

,

3

4



hkl : h + k + l = 2n

or h = 2n + 1,k = 4n and

l = 4n + 2

6 d 222.. 0,

1

2

,

1

2

1

2

,0,

1

2

1

2

,

1

2

,00,

1

2

,0

1

2

,0, 00,0,

1

2

hkl : h + k + l = 2n

4 c . 32

3

4

,

3

4

,

3

4

1

4

,

1

4

,

3

4

1

4

,

3

4

,

1

4

3

4

,

1

4

,

1

4

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

4 b . 32

1

4

,

1

4

,

1

4

3

4

,

3

4

,

1

4

3

4

,

1

4

,

3

4

1

4

,

3

4

,

3

4

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

2 a 23. 0,0,0

1

2

,

1

2

,

1

2

hkl : h + k + l = 2n

Symmetry of special projections

Along [001] p4mm

a



= ab



= b

Origin at 0,

1

2

,z

Along [111] p3m1

a



=

1

3

(2a − b− c) b



=

1

3

(−a + 2b − c)

Origin at x,x , x

Along [110] p2mm

a



=

1

2

(−a + b) b



= c

Origin at x,x ,

1

4

637

P4

2

32 No. 208 CONTINUED

Maximal non-isomorphic subgroups

I

[2] P231 (P23, 195) 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12



[3] P4

2

12(P4

2

22, 93) 1; 2; 3; 4; 13; 14; 15; 16

[3] P4

2

12(P4

2

22, 93) 1; 2; 3; 4; 17; 18; 19; 20

[3] P4

2

12(P4

2

22, 93) 1; 2; 3; 4; 21; 22; 23; 24

⎧

⎪

⎨

⎪

⎩

[4] P132 (R32, 155) 1; 5; 9; 14; 19; 24

[4] P132 (R32, 155) 1; 6; 12; 13; 18; 24

[4] P132 (R32, 155) 1; 7; 10; 13; 19; 22

[4] P132 (R32, 155) 1; 8; 11; 14; 18; 22

IIa none

IIb [2] F 4

1

32(a



= 2a,b



= 2b,c



= 2c) (210); [4] I 4

1

32(a



= 2a,b



= 2b,c



= 2c) (214)

Maximal isomorphic subgroups of lowest index

IIc

[27] P4

2

32(a



= 3a,b



= 3b,c



= 3c) (208)

Minimal non-isomorphic supergroups

I

[2] Pm

¯

3n (223); [2] Pn

¯

3m (224)

II [2] I 432 (211); [4] F 432 (209)

638

F432 O

3

432 Cubic

No. 209 F432

Patterson symmetry Fm

¯

3m

Origin at 432

Asymmetric unit 0 ≤ x ≤

1

2

;0≤ y ≤

1

4

; −

1

4

≤ z ≤

1

4

; y ≤ min(x,

1

2

− x); −y ≤ z ≤ y

Vertices 0,0, 0

1

2

,0, 0

1

4

,

1

4

,

1

4

1

4

,

1

4

,−

1

4

Symmetry operations

For (0,0,0)+ set

(1) 1 (2) 2 0,0,z (3) 2 0,y,0(4)2x,0,0

(5) 3

+

x,x,x (6) 3

+

¯x,x, ¯x (7) 3

+

x, ¯x, ¯x (8) 3

+

¯x, ¯x,x

(9) 3

−

x,x,x (10) 3

−

x, ¯x, ¯x (11) 3

−

¯x, ¯x,x (12) 3

−

¯x,x, ¯x

(13) 2 x,x,0 (14) 2 x, ¯x,0 (15) 4

−

0,0,z (16) 4

+

0,0,z

(17) 4

−

x,0,0 (18) 2 0,y,y (19) 2 0,y, ¯y (20) 4

+

x,0,0

(21) 4

+

0,y,0 (22) 2 x,0,x (23) 4

−

0,y,0 (24) 2 ¯x,0,x

640

International Tables for Crystallography (2006). Vol. A, Space group 209, pp. 639–641.

CONTINUED No. 209 F 432

Symmetry operations (continued)

For (0,

1

2

,

1

2

)+ set

(1) t(0,

1

2

,

1

2

) (2) 2(0,0,

1

2

) 0,

1

4

,z (3) 2(0,

1

2

,0) 0,y,

1

4

(4) 2 x,

1

4

,

1

4

(5) 3

+

(

1

3

,

1

3

,

1

3

) x−

1

3

,x−

1

6

,x (6) 3

+

¯x,x+

1

2

, ¯x (7) 3

+

(−

1

3

,

1

3

,

1

3

) x+

1

3

, ¯x−

1

6

, ¯x (8) 3

+

¯x, ¯x+

1

2

,x

(9) 3

−

(

1

3

,

1

3

,

1

3

) x−

1

6

,x+

1

6

,x (10) 3

−

(−

1

3

,

1

3

,

1

3

) x+

1

6

, ¯x+

1

6

, ¯x (11) 3

−

¯x+

1

2

, ¯x+

1

2

,x (12) 3

−

¯x−

1

2

,x+

1

2

, ¯x

(13) 2(

1

4

,

1

4

,0) x,x+

1

4

,

1

4

(14) 2(−

1

4

,

1

4

,0) x, ¯x+

1

4

,

1

4

(15) 4

−

(0,0,

1

2

)

1

4

,

1

4

,z (16) 4

+

(0,0,

1

2

) −

1

4

,

1

4

,z

(17) 4

−

x,

1

2

,0 (18) 2(0,

1

2

,

1

2

) 0,y,y (19) 2 0,y+

1

2

, ¯y (20) 4

+

x,0,

1

2

(21) 4

+

(0,

1

2

,0)

1

4

,y,

1

4

(22) 2(

1

4

,0,

1

4

) x−

1

4

,

1

4

,x (23) 4

−

(0,

1

2

,0) −

1

4

,y,

1

4

(24) 2(−

1

4

,0,

1

4

) ¯x+

1

4

,

1

4

,x

For (

1

2

,0,

1

2

)+ set

(1) t(

1

2

,0,

1

2

) (2) 2(0,0,

1

2

)

1

4

,0,z (3) 2

1

4

,y,

1

4

(4) 2(

1

2

,0,0) x,0,

1

4

(5) 3

+

(

1

3

,

1

3

,

1

3

) x+

1

6

,x−

1

6

,x (6) 3

+

(

1

3

,−

1

3

,

1

3

) ¯x+

1

6

,x+

1

6

, ¯x (7) 3

+

x+

1

2

, ¯x−

1

2

, ¯x (8) 3

+

¯x+

1

2

, ¯x+

1

2

,x

(9) 3

−

(

1

3

,

1

3

,

1

3

) x−

1

6

,x−

1

3

,x (10) 3

−

x+

1

2

, ¯x, ¯x (11) 3

−

¯x+

1

2

, ¯x,x (12) 3

−

(

1

3

,−

1

3

,

1

3

) ¯x−

1

6

,x+

1

3

, ¯x

(13) 2(

1

4

,

1

4

,0) x,x−

1

4

,

1

4

(14) 2(

1

4

,−

1

4

,0) x, ¯x+

1

4

,

1

4

(15) 4

−

(0,0,

1

2

)

1

4

,−

1

4

,z (16) 4

+

(0,0,

1

2

)

1

4

,

1

4

,z

(17) 4

−

(

1

2

,0,0) x,

1

4

,

1

4

(18) 2(0,

1

4

,

1

4

)

1

4

,y−

1

4

,y (19) 2(0,−

1

4

,

1

4

)

1

4

,y+

1

4

, ¯y (20) 4

+

(

1

2

,0,0) x,−

1

4

,

1

4

(21) 4

+

1

2

,y,0 (22) 2(

1

2

,0,

1

2

) x,0,x (23) 4

−

0,y,

1

2

(24) 2 ¯x+

1

2

,0,x

For (

1

2

,

1

2

,0)+ set

(1) t(

1

2

,

1

2

,0) (2) 2

1

4

,

1

4

,z (3) 2(0,

1

2

,0)

1

4

,y,0(4)2(

1

2

,0,0) x,

1

4

,0

(5) 3

+

(

1

3

,

1

3

,

1

3

) x+

1

6

,x+

1

3

,x (6) 3

+

¯x+

1

2

,x, ¯x (7) 3

+

x+

1

2

, ¯x, ¯x (8) 3

+

(

1

3

,

1

3

,−

1

3

) ¯x+

1

6

, ¯x+

1

3

,x

(9) 3

−

(

1

3

,

1

3

,

1

3

) x+

1

3

,x+

1

6

,x (10) 3

−

x, ¯x+

1

2

, ¯x (11) 3

−

(

1

3

,

1

3

,−

1

3

) ¯x+

1

3

, ¯x+

1

6

,x (12) 3

−

¯x,x+

1

2

, ¯x

(13) 2(

1

2

,

1

2

,0) x,x,0 (14) 2 x, ¯x+

1

2

,0 (15) 4

−

1

2

,0,z (16) 4

+

0,

1

2

,z

(17) 4

−

(

1

2

,0,0) x,

1

4

,−

1

4

(18) 2(0,

1

4

,

1

4

)

1

4

,y+

1

4

,y (19) 2(0,

1

4

,−

1

4

)

1

4

,y+

1

4

, ¯y (20) 4

+

(

1

2

,0,0) x,

1

4

,

1

4

(21) 4

+

(0,

1

2

,0)

1

4

,y,−

1

4

(22) 2(

1

4

,0,

1

4

) x+

1

4

,

1

4

,x (23) 4

−

(0,

1

2

,0)

1

4

,y,

1

4

(24) 2(

1

4

,0,−

1

4

) ¯x+

1

4

,

1

4

,x

Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); t(0,

1

2

,

1

2

); t(

1

2

,0,

1

2

); (2); (3); (5); (13)

Positions

Multiplicity,

Wyckoff letter,

Site symmetry

Coordinates

(0,0, 0)+ (0,

1

2

,

1

2

)+ (

1

2

,0,

1

2

)+ (

1

2

,

1

2

,0)+

Reﬂection conditions

h,k,l permutable

General:

96 j 1(1)x, y,z (2) ¯x, ¯y,z (3) ¯x,y, ¯z (4) x, ¯y, ¯z

(5) z,x,y (6) z, ¯x, ¯y (7) ¯z, ¯x,y (8) ¯z,x, ¯y

(9) y,z,x (10) ¯y,z, ¯x (11) y, ¯z, ¯x (12) ¯y, ¯z, x

(13) y,

x, ¯z (14) ¯y, ¯x, ¯z (15) y, ¯x,z (16) ¯y,x,z

(17) x,z, ¯y (18) ¯x,z,y (19) ¯x, ¯z, ¯y (20) x, ¯z,y

(21) z,y, ¯x (22) z, ¯y,x (23) ¯z,y,x (24) ¯z, ¯y, ¯x

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

0kl : k

,l = 2n

hhl : h + l = 2n

h00 : h = 2n

Special: as above, plus

48 i 2 .. x,

1

4

,

1

4

¯x,

3

4

,

1

4

1

4

,x,

1

4

1

4

, ¯x,

3

4

1

4

,

1

4

,x

3

4

,

1

4

, ¯x

1

4

,x,

3

4

3

4

, ¯x,

3

4

x,

1

4

,

3

4

¯x,

1

4

,

1

4

1

4

,

1

4

, ¯x

1

4

,

3

4

,x

hkl : h = 2n

48 h ..2

1

2

,y,y

1

2

, ¯y, y

1

2

,y, ¯y

1

2

, ¯y, ¯yy,

1

2

,yy,

1

2

, ¯y

¯y,

1

2

,y ¯y,

1

2

, ¯yy,y,

1

2

¯y,y,

1

2

y, ¯y,

1

2

¯y, ¯y,

1

2

no extra conditions

48 g ..20,y,y 0, ¯y,y 0,y, ¯y 0, ¯y, ¯yy,0 , yy, 0, ¯y

¯y,0, y ¯y,0, ¯yy,y, 0¯y,y, 0 y, ¯y,0¯y, ¯y,0

no extra conditions

32 f . 3 . x,x, x ¯x, ¯x,x ¯x,x, ¯xx, ¯x, ¯

x

x,x, ¯x ¯x, ¯x, ¯xx, ¯x,x ¯x,x,x

no extra conditions

24 e 4 .. x, 0,0¯x,0, 00, x,00, ¯x,00, 0,x 0,0, ¯x no extra conditions

24 d 2 . 22 0,

1

4

,

1

4

0,

3

4

,

1

4

1

4

,0,

1

4

1

4

,0,

3

4

1

4

,

1

4

,0

3

4

,

1

4

,0 hkl : h = 2n

8 c 23.

1

4

,

1

4

,

1

4

1

4

,

1

4

,

3

4

hkl : h = 2n

4 b 432

1

2

,

1

2

,

1

2

no extra conditions

4 a 432 0,0,0 no extra conditions

(Continued on page 639)

641

CONTINUED

(from page 641)

No. 209 F 432

Symmetry of special projections

Along [001] p4mm

a



=

1

2

ab



=

1

2

b

Origin at 0,0,z

Along [111] p3m1

a



=

1

6

(2a − b− c) b



=

1

6

(−a + 2b − c)

Origin at x,x , x

Along [110] c2mm

a



=

1

2

(−a + b) b



= c

Origin at x,x , 0

Maximal non-isomorphic subgroups

I

[2] F 231(F 23, 196) (1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12)+



[3] F 412(I 422, 97) (1; 2; 3; 4; 13; 14; 15; 16)+

[3] F 412(I 422, 97) (1; 2; 3; 4; 17; 18; 19; 20)+

[3] F 412(I 422, 97) (1; 2; 3; 4; 21; 22; 23; 24)+

⎧

⎪

⎨

⎪

⎩

[4] F 132(R32, 155) (1; 5; 9; 14; 19; 24)+

[4] F 132(R32, 155) (1; 6; 12; 13; 18; 24)+

[4] F 132(R32, 155) (1; 7; 10; 13; 19; 22)+

[4] F 132(R32, 155) (1; 8; 11; 14; 18; 22)+

IIa

⎧

⎪

⎨

⎪

⎩

[4] P4

2

32 (208) 1; 5; 9; 14; 19; 24; (4; 6; 11; 16; 18; 23)+(0,

1

2

,

1

2

); (3; 8; 10; 15; 20; 22)+(

1

2

,0,

1

2

); (2; 7; 12;

13; 17; 21)+(

1

2

,

1

2

,0)

[4] P4

2

32 (208) 1; 6; 12; 13; 18; 24; (4; 5; 10; 15; 19; 23)+(0,

1

2

,

1

2

); (3; 7; 11; 16; 17; 22)+(

1

2

,0,

1

2

); (2; 8; 9;

14; 20; 21)+(

1

2

,

1

2

,0)

[4] P4

2

32 (208) 1; 7; 10; 13; 19; 22; (4; 8; 12; 15; 18; 21)+(0,

1

2

,

1

2

); (3; 6; 9; 16; 20; 24)+(

1

2

,0,

1

2

); (2; 5; 11;

14; 17; 23)+(

1

2

,

1

2

,0)

[4] P4

2

32 (208) 1; 8; 11; 14; 18; 22; (4; 7; 9; 16; 19; 21)+(0,

1

2

,

1

2

); (3; 5; 12; 15; 17; 24)+(

1

2

,0,

1

2

); (2; 6; 10;

13; 20; 23)+(

1

2

,

1

2

,0)

⎧

⎪

⎨

⎪

⎩

[4] P432 (207) 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24

[4] P432 (207) 1; 2; 3; 4; 13; 14; 15; 16; (9; 10; 11; 12; 17; 18; 19; 20)+(0,

1

2

,

1

2

); (5; 6; 7; 8; 21; 22; 23; 24)+(

1

2

,0,

1

2

)

[4] P432 (207) 1; 2; 3; 4; 17; 18; 19; 20; (9; 10; 11; 12; 21; 22; 23; 24)+(

1

2

,0,

1

2

); (5; 6; 7; 8; 13; 14; 15; 16)+(

1

2

,

1

2

,0)

[4] P432 (207) 1; 2; 3; 4; 21; 22; 23; 24; (5; 6; 7; 8; 17; 18; 19; 20)+(0,

1

2

,

1

2

); (9; 10; 11; 12; 13; 14; 15; 16)+(

1

2

,

1

2

,0)

IIb none

Maximal isomorphic subgroups of lowest index

IIc

[27] F 432(a



= 3a,b



= 3b,c



= 3c) (209)

Minimal non-isomorphic supergroups

I

[2] Fm

¯

3m (225); [2] Fm

¯

3c (226)

II [2] P432(a



=

1

2

a,b



=

1

2

b,c



=

1

2

c) (207)

639

F4

1

32 O

4

432 Cubic

No. 210 F4

1

32 Patterson symmetry Fm

¯

3m

Origin at 23

Asymmetric unit 0 ≤ x ≤

1

2

; −

1

8

≤ y ≤

1

8

; −

1

8

≤ z ≤

1

8

; y ≤ min(x,

1

2

− x); −y ≤ z ≤ min(x,

1

2

− x)

Vertices 0,0, 0

1

8

,

1

8

,

1

8

1

8

,

1

8

,−

1

8

1

8

,−

1

8

,

1

8

1

2

,0, 0

3

8

,

1

8

,

1

8

3

8

,

1

8

,−

1

8

3

8

,−

1

8

,

1

8

Symmetry operations

For (0,0,0)+ set

(1) 1 (2) 2(0,0,

1

2

) 0,

1

4

,z (3) 2(0,

1

2

,0)

1

4

,y,0(4)2(

1

2

,0,0) x,0,

1

4

(5) 3

+

x,x,x (6) 3

+

(

1

3

,−

1

3

,

1

3

) ¯x+

1

6

,x+

1

6

, ¯x (7) 3

+

(−

1

3

,

1

3

,

1

3

) x+

1

3

, ¯x−

1

6

, ¯x (8) 3

+

(

1

3

,

1

3

,−

1

3

) ¯x+

1

6

, ¯x+

1

3

,x

(9) 3

−

x,x,x (10) 3

−

x, ¯x+

1

2

, ¯x (11) 3

−

¯x+

1

2

, ¯x,x (12) 3

−

¯x−

1

2

,x+

1

2

, ¯x

(13) 2(

1

2

,

1

2

,0) x,x−

1

4

,

3

8

(14) 2 x, ¯x+

1

4

,

1

8

(15) 4

−

(0,0,

3

4

)

1

2

,

1

4

,z (16) 4

+

(0,0,

1

4

) 0,

3

4

,z

(17) 4

−

(

3

4

,0,0) x,

1

2

,

1

4

(18) 2(0,

1

2

,

1

2

)

3

8

,y+

1

4

,y (19) 2

1

8

,y+

1

4

, ¯y (20) 4

+

(

1

4

,0,0) x,0,

3

4

(21) 4

+

(0,

1

4

,0)

3

4

,y,0 (22) 2(

1

2

,0,

1

2

) x−

1

4

,

3

8

,x (23) 4

−

(0,

3

4

,0)

1

4

,y,

1

2

(24) 2 ¯x+

1

4

,

1

8

,x

642

International Tables for Crystallography (2006). Vol. A, Space group 210, pp. 642–644.

CONTINUED No. 210 F 4

1

32

Symmetry operations (continued)

For (0,

1

2

,

1

2

)+ set

(1) t(0,

1

2

,

1

2

) (2) 2 0,0,z (3) 2

1

4

,y,

1

4

(4) 2(

1

2

,0,0) x,

1

4

,0

(5) 3

+

(

1

3

,

1

3

,

1

3

) x−

1

3

,x−

1

6

,x (6) 3

+

¯x+

1

2

,x, ¯x (7) 3

+

x, ¯x, ¯x (8) 3

+

¯x+

1

2

, ¯x+

1

2

,x

(9) 3

−

(

1

3

,

1

3

,

1

3

) x−

1

6

,x+

1

6

,x (10) 3

−

x+

1

2

, ¯x, ¯x (11) 3

−

(

1

3

,

1

3

,−

1

3

) ¯x+

1

3

, ¯x+

1

6

,x (12) 3

−

¯x,x, ¯x

(13) 2(

3

4

,

3

4

,0) x,x,

1

8

(14) 2(−

1

4

,

1

4

,0) x, ¯x+

1

2

,

3

8

(15) 4

−

(0,0,

1

4

)

1

4

,0,z (16) 4

+

(0,0,

3

4

)

1

4

,

1

2

,z

(17) 4

−

(

3

4

,0,0) x,

1

2

,−

1

4

(18) 2(0,

1

2

,

1

2

)

3

8

,y−

1

4

,y (19) 2

1

8

,y+

3

4

, ¯y (20) 4

+

(

1

4

,0,0) x,0,

1

4

(21) 4

+

(0,

3

4

,0)

1

2

,y,−

1

4

(22) 2(

1

4

,0,

1

4

) x,

1

8

,x (23) 4

−

(0,

1

4

,0) 0,y,

3

4

(24) 2(−

1

4

,0,

1

4

) ¯x+

1

2

,

3

8

,x

For (

1

2

,0,

1

2

)+ set

(1) t(

1

2

,0,

1

2

) (2) 2

1

4

,

1

4

,z (3) 2(0,

1

2

,0) 0,y,

1

4

(4) 2 x,0,0

(5) 3

+

(

1

3

,

1

3

,

1

3

) x+

1

6

,x−

1

6

,x (6) 3

+

¯x,x, ¯x (7) 3

+

x+

1

2

, ¯x, ¯x (8) 3

+

¯x, ¯x+

1

2

,x

(9) 3

−

(

1

3

,

1

3

,

1

3

) x−

1

6

,x−

1

3

,x (10) 3

−

(−

1

3

,

1

3

,

1

3

) x+

1

6

, ¯x+

1

6

, ¯x (11) 3

−

¯x, ¯x,x (12) 3

−

¯x,x+

1

2

, ¯x

(13) 2(

1

4

,

1

4

,0) x,x,

1

8

(14) 2(

1

4

,−

1

4

,0) x, ¯x+

1

2

,

3

8

(15) 4

−

(0,0,

1

4

)

3

4

,0,z (16) 4

+

(0,0,

3

4

) −

1

4

,

1

2

,z

(17) 4

−

(

1

4

,0,0) x,

1

4

,0 (18) 2(0,

3

4

,

3

4

)

1

8

,y,y (19) 2(0,−

1

4

,

1

4

)

3

8

,y+

1

2

, ¯y (20) 4

+

(

3

4

,0,0) x,

1

4

,

1

2

(21) 4

+

(0,

1

4

,0)

1

4

,y,0 (22) 2(

1

2

,0,

1

2

) x+

1

4

,

3

8

,x (23) 4

−

(0,

3

4

,0) −

1

4

,y,

1

2

(24) 2 ¯x+

3

4

,

1

8

,x

For (

1

2

,

1

2

,0)+ set

(1) t(

1

2

,

1

2

,0) (2) 2(0,0,

1

2

)

1

4

,0,z (3) 2 0,y,0(4)2x,

1

4

,

1

4

(5) 3

+

(

1

3

,

1

3

,

1

3

) x+

1

6

,x+

1

3

,x (6) 3

+

¯x,x+

1

2

, ¯x (7) 3

+

x+

1

2

, ¯x−

1

2

, ¯x (8) 3

+

¯x, ¯x,x

(9) 3

−

(

1

3

,

1

3

,

1

3

) x+

1

3

,x+

1

6

,x (10) 3

−

x, ¯x, ¯x (11) 3

−

¯x+

1

2

, ¯x+

1

2

,x (12) 3

−

(

1

3

,−

1

3

,

1

3

) ¯x−

1

6

,x+

1

3

, ¯x

(13) 2(

1

2

,

1

2

,0) x,x+

1

4

,

3

8

(14) 2 x, ¯x+

3

4

,

1

8

(15) 4

−

(0,0,

3

4

)

1

2

,−

1

4

,z (16) 4

+

(0,0,

1

4

) 0,

1

4

,z

(17) 4

−

(

1

4

,0,0) x,

3

4

,0 (18) 2(0,

1

4

,

1

4

)

1

8

,y,y (19) 2(0,

1

4

,−

1

4

)

3

8

,y+

1

2

, ¯y (20) 4

+

(

3

4

,0,0) x,−

1

4

,

1

2

(21) 4

+

(0,

3

4

,0)

1

2

,y,

1

4

(22) 2(

3

4

,0,

3

4

) x,

1

8

,x (23) 4

−

(0,

1

4

,0) 0,y,

1

4

(24) 2(

1

4

,0,−

1

4

) ¯x+

1

2

,

3

8

,x

Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); t(0,

1

2

,

1

2

); t(

1

2

,0,

1

2

); (2); (3); (5); (13)

Positions

Multiplicity,

Wyckoff letter,

Site symmetry

Coordinates

(0,0, 0)+ (0,

1

2

,

1

2

)+ (

1

2

,0,

1

2

)+ (

1

2

,

1

2

,0)+

Reﬂection conditions

h,k,l permutable

General:

96 h 1(1)x,y, z (2) ¯x, ¯y +

1

2

,z +

1

2

(3) ¯x +

1

2

,y +

1

2

, ¯z (4) x +

1

2

, ¯y, ¯z +

1

2

(5) z,x,y (6) z +

1

2

, ¯x, ¯y+

1

2

(7) ¯z, ¯x +

1

2

,y +

1

2

(8) ¯z +

1

2

,x +

1

2

, ¯y

(9) y,z,x (10) ¯y+

1

2

,z +

1

2

, ¯x (11) y +

1

2

, ¯z, ¯x +

1

2

(12) ¯y, ¯z +

1

2

,x +

1

2

(13) y +

3

4

,x +

1

4

, ¯z +

3

4

(14) ¯y +

1

4

, ¯x +

1

4

, ¯z +

1

4

(15) y +

1

4

, ¯x +

3

4

,z +

3

4

(16) ¯y +

3

4

,x +

3

4

,z +

1

4

(17) x +

3

4

,z +

1

4

, ¯y +

3

4

(18) ¯x +

3

4

,z +

3

4

,y +

1

4

(19) ¯x +

1

4

, ¯z +

1

4

, ¯y +

1

4

(20) x +

1

4

, ¯z +

3

4

,y +

3

4

(21) z +

3

4

,y +

1

4

, ¯x +

3

4

(22) z +

1

4

, ¯y +

3

4

,x +

3

4

(23) ¯z +

3

4

,y +

3

4

,x +

1

4

(24) ¯z+

1

4

, ¯y +

1

4

, ¯x +

1

4

hkl : h + k = 2n and

h + l,k + l = 2n

0kl : k, l = 2n

hhl : h + l = 2n

h00 : h = 4n

Special: as above, plus

48 g ..2

1

8

,y, ¯y +

1

4

7

8

, ¯y +

1

2

, ¯y +

3

4

3

8

,y +

1

2

,y +

3

4

5

8

, ¯y, y +

1

4

¯y+

1

4

,

1

8

,y ¯y +

3

4

,

7

8

, ¯y +

1

2

y +

3

4

,

3

8

,y +

1

2

y +

1

4

,

5

8

, ¯y

y, ¯y +

1

4

,

1

8

¯y+

1

2

, ¯y +

3

4

,

7

8

y +

1

2

,y +

3

4

,

3

8

¯y,y +

1

4

,

5

8

no extra conditions

48 f 2 .. x,0,0¯x ,

1

2

,

1

2

0,x, 0

1

2

, ¯x,

1

2

0,0,x

1

2

,

1

2

, ¯x

3

4

,x +

1

4

,

3

4

1

4

, ¯x +

1

4

,

1

4

x +

3

4

,

1

4

,

3

4

¯x +

3

4

,

3

4

,

1

4

3

4

,

1

4

, ¯x +

3

4

1

4

,

3

4

,x +

3

4

hkl : h = 2n + 1

or h + k + l = 4n

32 e . 3 . x,x, x ¯x, ¯x+

1

2

,x +

1

2

¯x +

1

2

,x +

1

2

, ¯xx+

1

2

, ¯x, ¯x +

1

2

x +

3

4

,x +

1

4

, ¯x +

3

4

¯x +

1

4

, ¯x +

1

4

, ¯x +

1

4

x +

1

4

, ¯x +

3

4

,x +

3

4

¯x +

3

4

,x +

3

4

,x +

1

4

0kl : k + l = 4 n

16 d . 32

5

8

,

5

8

,

5

8

3

8

,

7

8

,

1

8

7

8

,

1

8

,

3

8

1

8

,

3

8

,

7

8

16 c . 32

1

8

,

1

8

,

1

8

7

8

,

3

8

,

5

8

3

8

,

5

8

,

7

8

5

8

,

7

8

,

3

8



hkl : h = 2n + 1

or h, k, l = 4n + 2

or h, k, l = 4n

8 b 23.

1

2

,

1

2

,

1

2

1

4

,

3

4

,

1

4

8 a 23. 0,0,0

3

4

,

1

4

,

3

4



hkl : h = 2n + 1

or h + k + l = 4n

Symmetry of special projections

Along [001] p4mm

a



=

1

2

ab



=

1

2

b

Origin at

1

4

,0, z

Along [111] p3m1

a



=

1

6

(2a − b− c) b



=

1

6

(−a + 2b − c)

Origin at x,x , x

Along [110] c2mm

a



=

1

2

(−a + b) b



= c

Origin at x,x ,

1

8

643

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

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