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

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c2mm 2mm Rectangular

No. 9 c2mm

Patterson symmetry c2mm

Origin at 2mm

Asymmetric unit 0 ≤ x ≤

;0≤ y ≤

Symmetry operations

For (0,0)+ set

(1) 1 (2) 2 0,0(3)m 0,y (4) mx,0

For (

)+ set

(1) t(

) (2) 2

(3) b

,y (4) ax,

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

); (2); (3)

Positions

Multiplicity,

Wyckoff letter,

Site symmetry

Coordinates

(0,0)+ (

Reﬂection conditions

General:

8 f 1(1)x, y (2) ¯x, ¯y (3) ¯x,y (4) x, ¯yhk: h + k = 2n

h0: h = 2n

0k: k = 2n

Special: as above, plus

4 e . m . 0, y 0, ¯y no extra conditions

4 d ..mx,0¯x,0 no extra conditions

4 c 2 ..

hk: h = 2n

2 b 2 mm 0,

no extra conditions

2 a 2 mm 0,0 no extra conditions

Maximal non-isomorphic subgroups

[2] c1m 1(cm,5) (1; 3)+

[2] c11m (cm,5) (1; 4)+

[2] c211(p2, 2) (1; 2)+

IIa [2] p2gg(8) 1; 2; (3; 4)+(

)

[2] p 2gm(p2mg,7) 1; 4; (2; 3)+(

)

[2] p 2mg(7) 1; 3; (2; 4)+(

)

[2] p 2mm(6) 1;2;3;4

IIb none

Maximal isomorphic subgroups of lowest index

IIc

[3] c2mm (a



= 3a or b



= 3b)(9)

Minimal non-isomorphic supergroups

[2] p 4mm(11); [2] p4gm (12); [3] p6mm(17)

II [2] p2mm(a



a,b



b)(6)

100

International Tables for Crystallography (2006). Vol. A, Plane group 9, p. 100.

Square 4 p4

Patterson symmetry p4 p4 No. 10

Origin at 4

Asymmetric unit 0 ≤ x ≤

;0≤ y ≤

Symmetry operations

(1) 1 (2) 2 0,0(3)4

0,0(4)4

−

0,0

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

Positions

Multiplicity,

Wyckoff letter,

Site symmetry

Coordinates Reﬂection conditions

General:

4 d 1(1)x,y (2) ¯x, ¯y (3) ¯y,x (4) y, ¯x no conditions

Special:

2 c 2 ..

,00,

hk: h + k = 2n

1 b 4 ..

no extra conditions

1 a 4 .. 0,0 no extra conditions

Maximal non-isomorphic subgroups

[2] p 2 (2) 1; 2

IIa none

IIb none

Maximal isomorphic subgroups of lowest index

IIc

[2] c4(a



= 2a,b



= 2b)(p 4, 10)

Minimal non-isomorphic supergroups

[2] p 4mm(11); [2] p4gm (12)

II none

101

International Tables for Crystallography (2006). Vol. A, Plane group 10, p. 101.

p4mm 4mm Square

No. 11 p4mm

Patterson symmetry p4mm

Origin at 4mm

Asymmetric unit 0 ≤ x ≤

;0≤ y ≤

; x ≤ y

Symmetry operations

(1) 1 (2) 2 0, 0(3)4

0,0(4)4

−

0,0

(5) m 0,y (6) mx, 0(7)mx,x (8) mx, ¯x

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

Positions

Multiplicity,

Wyckoff letter,

Site symmetry

Coordinates Reﬂection conditions

General:

8 g 1(1)x,y (2) ¯x, ¯y (3) ¯y,x (4) y, ¯x

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

no conditions

Special:

4 f ..mx, x ¯x, ¯x ¯x,xx, ¯x no extra conditions

4 e . m . x,

¯x,

, ¯x no extra conditions

4 d . m . x,0¯x,00,x 0, ¯x no extra conditions

2 c 2 mm.

,00,

hk: h + k = 2n

1 b 4 mm

no extra conditions

1 a 4 mm 0, 0 no extra conditions

Maximal non-isomorphic subgroups

[2] p 411 (p4, 10) 1; 2; 3; 4

[2] p 21m (c 2mm,9) 1;2;7;8

[2] p 2m1(p2mm,6) 1;2;5;6

IIa none

IIb [2] c4mg(a



= 2a,b



= 2b)(p 4gm, 12)

Maximal isomorphic subgroups of lowest index

IIc

[2] c4mm (a



= 2a,b



= 2b)(p 4mm, 11)

Minimal non-isomorphic supergroups

none

II none

102

International Tables for Crystallography (2006). Vol. A, Plane group 11, p. 102.

Square 4mm p4gm

Patterson symmetry p4mm p4gm No. 12

Origin at 41g

Asymmetric unit 0 ≤ x ≤

;0≤ y ≤

; y ≤

− x

Symmetry operations

(1) 1 (2) 2 0,0(3)4

0,0(4)4

−

0,0

(5) b

,y (6) ax,

(7) g(

) x, x (8) mx+

, ¯x

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

Positions

Multiplicity,

Wyckoff letter,

Site symmetry

Coordinates Reﬂection conditions

General:

8 d 1(1)x,y (2) ¯x, ¯y (3) ¯y, x (4) y, ¯x

(5) ¯x +

,y +

(6) x +

, ¯y+

(7) y+

,x +

(8) ¯y+

, ¯x +

h0: h = 2n

0k: k = 2n

Special: as above, plus

4 c ..mx, x +

¯x, ¯x+

¯x +

,xx+

, ¯x no extra conditions

2 b 2 . mm

,00,

hk: h + k = 2n

2 a 4 .. 0, 0

hk: h + k = 2n

Maximal non-isomorphic subgroups

[2] p 411 (p4, 10) 1; 2; 3; 4

[2] p 21m (c 2mm,9) 1;2;7;8

[2] p 2g1(p2gg,8) 1;2;5;6

IIa none

IIb none

Maximal isomorphic subgroups of lowest index

IIc

[9] p 4gm( a



= 3a,b



= 3b) (12)

Minimal non-isomorphic supergroups

none

II [2] c4gm (p4mm, 11)

103

International Tables for Crystallography (2006). Vol. A, Plane group 12, p. 103.

p3 3 Hexagonal

No. 13 p3

Patterson symmetry p6

Origin at 3

Asymmetric unit 0 ≤ x ≤

;0≤ y ≤

; x ≤ (1 + y)/2; y ≤ min(1 − x,(1 + x)/2)

Vertices 0,0

Symmetry operations

(1) 1 (2) 3

0,0(3)3

−

0,0

Generators selected (1); t(1, 0); t(0,1); (2)

Positions

Multiplicity,

Wyckoff letter,

Site symmetry

Coordinates Reﬂection conditions

General:

3 d 1(1)x,y (2) ¯y,x − y (3) ¯x+ y, ¯x no conditions

Special: no extra conditions

1 c 3 ..

1 b 3 ..

1 a 3 .. 0,0

Maximal non-isomorphic subgroups

[3] p 1(1) 1

IIa none

IIb none

Maximal isomorphic subgroups of lowest index

IIc

[3] h3(a



= 3a,b



= 3b)(p 3, 13)

Minimal non-isomorphic supergroups

[2] p 3m1 (14); [2] p 31m (15); [2] p6 (16)

II none

104

International Tables for Crystallography (2006). Vol. A, Plane group 13, p. 104.

Hexagonal 3m p3m1

Patterson symmetry p6mm p3m1 No. 14

Origin at 3m1

Asymmetric unit 0 ≤ x ≤

;0≤ y ≤

; x ≤ 2y; y ≤ min(1 − x, 2x)

Vertices 0,0

Symmetry operations

(1) 1 (2) 3

0,0(3)3

−

0,0

(4) mx, ¯x (5) mx,2 x (6) m 2x, x

Generators selected (1); t(1, 0); t(0,1); (2); (4)

Positions

Multiplicity,

Wyckoff letter,

Site symmetry

Coordinates Reﬂection conditions

General:

6 e 1(1)x,y (2) ¯y,x − y (3) ¯x+ y, ¯x

(4) ¯y, ¯x (5) ¯x + y, y (6) x, x − y

no conditions

Special: no extra conditions

3 d . m . x, ¯xx, 2x 2¯x, ¯x

1 c 3 m .

1 b 3 m .

1 a 3 m . 0,0

Maximal non-isomorphic subgroups

[2] p 311 (p3, 13) 1; 2; 3



[3] p 1m1(cm,5) 1; 4

[3] p 1m1(cm,5) 1; 5

[3] p 1m1(cm,5) 1; 6

IIa none

IIb [3] h3m1(a



= 3a,b



= 3b)(p31m, 15)

Maximal isomorphic subgroups of lowest index

IIc

[4] p 3m1(a



= 2a,b



= 2b) (14)

Minimal non-isomorphic supergroups

[2] p 6mm(17)

II [3] h3m1(p31m, 15)

105

International Tables for Crystallography (2006). Vol. A, Plane group 14, p. 105.

p31m 3m Hexagonal

No. 15 p31m

Patterson symmetry p6mm

Origin at 31m

Asymmetric unit 0 ≤ x ≤

;0≤ y ≤

; x ≤ (1 + y)/2; y ≤ min(1 − x,x)

Vertices 0,0

Symmetry operations

(1) 1 (2) 3

0,0(3)3

−

0,0

(4) mx,x (5) mx,0(6)m 0, y

Generators selected (1); t(1, 0); t(0,1); (2); (4)

Positions

Multiplicity,

Wyckoff letter,

Site symmetry

Coordinates Reﬂection conditions

General:

6 d 1(1)x,y (2) ¯y,x − y (3) ¯x+ y, ¯x

(4) y, x (5) x − y, ¯y (6) ¯x, ¯x + y

no conditions

Special: no extra conditions

3 c ..mx, 00, x ¯x, ¯x

2 b 3 ..

1 a 3 . m 0,0

Maximal non-isomorphic subgroups

[2] p 311 (p3, 13) 1; 2; 3



[3] p 11m (cm,5) 1; 4

[3] p 11m (cm,5) 1; 5

[3] p 11m (cm,5) 1; 6

IIa none

IIb [3] h31m (a



= 3a,b



= 3b)(p3m1, 14)

Maximal isomorphic subgroups of lowest index

IIc

[4] p 31m (a



= 2a,b



= 2b) (15)

Minimal non-isomorphic supergroups

[2] p 6mm(17)

II [3] h31m (p3m1, 14)

106

International Tables for Crystallography (2006). Vol. A, Plane group 15, p. 106.

Hexagonal 6 p6

Patterson symmetry p6 p6 No. 16

Origin at 6

Asymmetric unit 0 ≤ x ≤

;0≤ y ≤

; x ≤ (1 + y)/2; y ≤ min(1 − x,x)

Vertices 0,0

Symmetry operations

(1) 1 (2) 3

0,0(3)3

−

0,0

(4) 2 0, 0(5)6

−

0,0(6)6

0,0

Generators selected (1); t(1, 0); t(0,1); (2); (4)

Positions

Multiplicity,

Wyckoff letter,

Site symmetry

Coordinates Reﬂection conditions

General:

6 d 1(1)x,y (2) ¯y,x − y (3) ¯x+ y, ¯x

(4) ¯x, ¯y (5) y, ¯x + y (6) x − y,x

no conditions

Special: no extra conditions

3 c 2 ..

,00,

2 b 3 ..

1 a 6 .. 0,0

Maximal non-isomorphic subgroups

[2] p 3 (13) 1; 2; 3

[3] p 2 (2) 1; 4

IIa none

IIb none

Maximal isomorphic subgroups of lowest index

IIc

[3] h6(a



= 3a,b



= 3b)(p 6, 16)

Minimal non-isomorphic supergroups

[2] p 6mm(17)

II none

107

International Tables for Crystallography (2006). Vol. A, Plane group 16, p. 107.

p6mm 6mm Hexagonal

No. 17 p6mm

Patterson symmetry p6mm

Origin at 6mm

Asymmetric unit 0 ≤ x ≤

;0≤ y ≤

; x ≤ (1 + y)/2; y ≤ x/2

Vertices 0,0

Symmetry operations

(1) 1 (2) 3

0,0(3)3

−

0,0

(4) 2 0, 0(5)6

−

0,0(6)6

0,0

(7) mx, ¯x (8) mx, 2x (9) m 2x,x

(10) mx,x (11) mx,0 (12) m 0,y

108

International Tables for Crystallography (2006). Vol. A, Plane group 17, pp. 108–109.

CONTINUED No. 17 p6mm

Generators selected (1); t(1, 0); t(0,1); (2); (4); (7)

Positions

Multiplicity,

Wyckoff letter,

Site symmetry

Coordinates Reﬂection conditions

General:

12 f 1(1)x, y (2) ¯y,x − y (3) ¯x + y, ¯x

(4) ¯x, ¯y (5) y, ¯x + y (6) x − y, x

(7) ¯y, ¯x (8) ¯x + y, y (9) x, x − y

(10) y, x (11) x − y, ¯y (12) ¯x, ¯x + y

no conditions

Special: no extra conditions

6 e . m . x, ¯xx, 2x 2¯x, ¯x ¯x,x ¯x, 2¯

x 2x,x

6 d ..mx, 00, x ¯x, ¯x ¯x,00, ¯xx,x

3 c 2 mm

,00,

2 b 3 m .

1 a 6 mm 0,0

Maximal non-isomorphic subgroups

[2] p 611 (p6, 16) 1; 2; 3; 4; 5; 6

[2] p 31m (15) 1; 2; 3; 10; 11; 12

[2] p 3m1 (14) 1; 2; 3; 7; 8; 9



[3] p 2mm( c 2mm,9) 1;4;7;10

[3] p 2mm( c 2mm,9) 1;4;8;11

[3] p 2mm( c 2mm,9) 1;4;9;12

IIa none

IIb none

Maximal isomorphic subgroups of lowest index

IIc

[3] h6mm(a



= 3a,b



= 3b)(p6mm, 17)

Minimal non-isomorphic supergroups

none

II none

109