Hahn Th. (ed.) International tables for crystallography. Vol. A. Space-group symmetry
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P2/mC
1
2h
2/m Monoclinic
No. 10 P112/ m
Patterson symmetry P112/m
UNIQUE AXIS c
Origin at centre (2/m)
Asymmetric unit 0 ≤ x ≤ 1; 0 ≤ y ≤
1
2
;0≤ z ≤
1
2
Symmetry operations
(1) 1 (2) 2 0,0,z (3)
¯
10, 0,0(4)mx,y,0
Maximal isomorphic subgroups of lowest index
IIc
[2] P112/m (c
= 2c)(P2/m, 10); [2] P112/m (a
= 2a or b
= 2b or a
= a − b,b
= a + b)(P2/m, 10)
Minimal non-isomorphic supergroups
I
[2] Pmmm (47); [2] Pccm (49); [2] Pmma (51); [2] Pmna (53); [2] Pbam(55); [2] Pnnm(58); [2] Cmmm (65); [2] Cccm (66);
[2] P4/m (83); [2] P4
2
/m (84); [3] P6/m (175)
II [2] A112/m (C2/m, 12); [2] B112/m (C2/m, 12); [2] I 112/ m (C 2/m, 12)
162
CONTINUED No. 10 P2/m
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
General:
4 o 1(1)x, y,z (2) ¯x, ¯y,z (3) ¯x, ¯y, ¯z (4) x,y, ¯z no conditions
Special: no extra conditions
2 nm x,y,
1
2
¯x, ¯y,
1
2
2 mm x,y,0¯x, ¯y,0
2 l 2
1
2
,
1
2
,z
1
2
,
1
2
, ¯z
2 k 2
1
2
,0, z
1
2
,0, ¯z
2 j 20,
1
2
,z 0,
1
2
, ¯z
2 i 20,0,z 0,0, ¯z
1 h 2/ m
1
2
,
1
2
,
1
2
1 g 2/ m
1
2
,
1
2
,0
1 f 2/m
1
2
,0,
1
2
1 e 2/m 0,
1
2
,
1
2
1 d 2/m 0,
1
2
,0
1 c 2/m
1
2
,0, 0
1 b 2/ m 0,0,
1
2
1 a 2/ m 0,0,0
Symmetry of special projections
Along [001] p2
a
= ab
= b
Origin at 0,0,z
Along [100] p2mm
a
= b
p
b
= c
Origin at x, 0,0
Along [010] p2mm
a
= cb
= a
p
Origin at 0,y, 0
Maximal non-isomorphic subgroups
I
[2] P11m (Pm,6) 1; 4
[2] P112 (P2, 3) 1; 2
[2] P
¯
1 (2) 1; 3
IIa none
IIb [2] P112
1
/m (c
= 2c)(P2
1
/m, 11); [2] P112/a (a
= 2a)(P2/c, 13); [2] P112/b (b
= 2b)(P2/c, 13);
[2] C 112/e (a
= 2a,b
= 2b)(P2/c, 13); [2] A112/m (b
= 2b,c
= 2c)(C 2/m, 12); [2] B112/m ( a
= 2a,c
= 2c)(C2/m , 12);
[2] F 112/m (a
= 2a,b
= 2b,c
= 2c)(C2/m , 12)
(Continued on preceding page)
163
P2
1
/mC
2
2h
2/m Monoclinic
No. 11 P12
1
/m1 Patterson symmetry P12/m1
UNIQUE AXIS b
Origin at
¯
1on2
1
Asymmetric unit 0 ≤ x ≤ 1; 0 ≤ y ≤
1
4
;0≤ z ≤ 1
Symmetry operations
(1) 1 (2) 2(0,
1
2
,0) 0,y,0(3)
¯
10, 0,0(4)mx,
1
4
,z
164
International Tables for Crystallography (2006). Vol. A, Space group 11, pp. 164–167.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 11 P2
1
/m
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
General:
4 f 1(1)x,y,z (2) ¯x,y +
1
2
, ¯z (3) ¯x, ¯y, ¯z (4) x, ¯y +
1
2
,z 0k0: k = 2n
Special: as above, plus
2 em x,
1
4
,z ¯x,
3
4
, ¯z no extra conditions
2 d
¯
1
1
2
,0,
1
2
1
2
,
1
2
,
1
2
hkl : k = 2n
2 c
¯
10,0,
1
2
0,
1
2
,
1
2
hkl : k = 2n
2 b
¯
1
1
2
,0, 0
1
2
,
1
2
,0 hkl : k = 2n
2 a
¯
10,0,00,
1
2
,0 hkl : k = 2n
Symmetry of special projections
Along [001] p2gm
a
= a
p
b
= b
Origin at 0,0,z
Along [100] p2mg
a
= bb
= c
p
Origin at x, 0,0
Along [010] p2
a
= cb
= a
Origin at 0,y, 0
Maximal non-isomorphic subgroups
I
[2] P1m1(Pm,6) 1; 4
[2] P12
1
1(P2
1
,4) 1; 2
[2] P
¯
1 (2) 1; 3
IIa none
IIb [2] P12
1
/c1(c
= 2c)(P2
1
/c, 14); [2] P12
1
/a1(a
= 2a)(P2
1
/c, 14); [2] B12
1
/e1(a
= 2a,c
= 2c)(P2
1
/c, 14)
Maximal isomorphic subgroups of lowest index
IIc
[2] P12
1
/m1(c
= 2c or a
= 2a or a
= a + c,c
= −a + c)(P2
1
/m, 11); [3] P12
1
/m1(b
= 3b)(P2
1
/m, 11)
Minimal non-isomorphic supergroups
I
[2] Pmma (51); [2] Pbcm (57); [2] Pmmn (59); [2] Pnma (62); [2] Cmcm(63); [3] P6
3
/m (176)
II [2] C 12/m1(C2/m, 12); [2] A12/m1(C 2/m, 12); [2] I 12/m1(C 2 /m, 12); [2] P12/m1(b
=
1
2
b)(P2/m, 10)
165
P2
1
/mC
2
2h
2/m Monoclinic
No. 11 P112
1
/m Patterson symmetry P112/m
UNIQUE AXIS c
Origin at
¯
1on2
1
Asymmetric unit 0 ≤ x ≤ 1; 0 ≤ y ≤ 1; 0 ≤ z ≤
1
4
Symmetry operations
(1) 1 (2) 2(0, 0,
1
2
) 0,0,z (3)
¯
10, 0,0(4)mx, y,
1
4
166
CONTINUED No. 11 P2
1
/m
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
General:
4 f 1(1)x,y,z (2) ¯x, ¯y,z +
1
2
(3) ¯x, ¯y, ¯z (4) x, y, ¯z +
1
2
00l : l = 2n
Special: as above, plus
2 em x,y,
1
4
¯x, ¯y,
3
4
no extra conditions
2 d
¯
1
1
2
,
1
2
,0
1
2
,
1
2
,
1
2
hkl : l = 2n
2 c
¯
1
1
2
,0, 0
1
2
,0,
1
2
hkl : l = 2n
2 b
¯
10,
1
2
,00,
1
2
,
1
2
hkl : l = 2n
2 a
¯
10,0,00,0,
1
2
hkl : l = 2n
Symmetry of special projections
Along [001] p2
a
= ab
= b
Origin at 0,0,z
Along [100] p2gm
a
= b
p
b
= c
Origin at x,0,0
Along [010] p2mg
a
= cb
= a
p
Origin at 0,y, 0
Maximal non-isomorphic subgroups
I
[2] P11m (Pm,6) 1; 4
[2] P112
1
(P2
1
,4) 1; 2
[2] P
¯
1 (2) 1; 3
IIa none
IIb [2] P112
1
/a (a
= 2a)(P2
1
/c, 14); [2] P112
1
/b (b
= 2b)(P2
1
/c, 14); [2]C 112
1
/e (a
= 2a,b
= 2b)(P2
1
/c, 14)
Maximal isomorphic subgroups of lowest index
IIc
[2] P112
1
/m (a
= 2a or b
= 2b or a
= a − b,b
= a + b)(P2
1
/m, 11); [3] P112
1
/m (c
= 3c)(P2
1
/m, 11)
Minimal non-isomorphic supergroups
I
[2] Pmma (51); [2] Pbcm (57); [2] Pmmn (59); [2] Pnma (62); [2] Cmcm(63); [3] P6
3
/m (176)
II [2] A112/m (C2/m, 12); [2] B112/m (C2/m, 12); [2] I 112/ m (C 2/m, 12); [2] P112/m (c
=
1
2
c)(P2/m, 10)
167
C2/mC
3
2h
2/m Monoclinic
No. 12 C12/m1
Patterson symmetry C 12/m1
UNIQUE AXIS b, CELL CHOICE 1
Origin at centre (2/m)
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
4
;0≤ z ≤ 1
Symmetry operations
For (0,0,0)+ set
(1) 1 (2) 2 0, y,0(3)
¯
10, 0,0(4)mx,0, z
For (
1
2
,
1
2
,0)+ set
(1) t(
1
2
,
1
2
,0) (2) 2(0,
1
2
,0)
1
4
,y,0(3)
¯
1
1
4
,
1
4
,0(4)ax,
1
4
,z
168
International Tables for Crystallography (2006). Vol. A, Space group 12, pp. 168–175.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 12 C2/m
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); t(
1
2
,
1
2
,0); (2); (3)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates
(0,0, 0)+ (
1
2
,
1
2
,0)+
Reflection conditions
General:
8 j 1(1)x,y, z (2) ¯x,y, ¯z (3) ¯x, ¯y, ¯z (4) x, ¯y,zhkl: h + k = 2n
h0l : h = 2n
0kl : k = 2n
hk0: h + k = 2n
0k0: k = 2n
h00 : h = 2n
Special: as above, plus
4 im x,0,z ¯x,0, ¯z no extra conditions
4 h 20,y,
1
2
0, ¯y,
1
2
no extra conditions
4 g 20,y,00, ¯y,0 no extra conditions
4 f
¯
1
1
4
,
1
4
,
1
2
3
4
,
1
4
,
1
2
hkl : h = 2n
4 e
¯
1
1
4
,
1
4
,0
3
4
,
1
4
,0 hkl : h = 2n
2 d 2/m 0,
1
2
,
1
2
no extra conditions
2 c 2/m 0,0 ,
1
2
no extra conditions
2 b 2/ m 0,
1
2
,0 no extra conditions
2 a 2/ m 0,0,0 no extra conditions
Symmetry of special projections
Along [001] c2mm
a
= a
p
b
= b
Origin at 0,0,z
Along [100] p2mm
a
=
1
2
bb
= c
p
Origin at x,0,0
Along [010] p2
a
= cb
=
1
2
a
Origin at 0, y,0
Maximal non-isomorphic subgroups
I
[2] C 1m1(Cm,8) (1; 4)+
[2] C 121(C2, 5) (1; 2)+
[2] C
¯
1(P
¯
1, 2) (1; 3)+
IIa [2] P12
1
/a1(P2
1
/c, 14) 1; 3; (2; 4)+(
1
2
,
1
2
,0)
[2] P12/a1(P2/c, 13) 1; 2; (3; 4)+(
1
2
,
1
2
,0)
[2] P12
1
/m1(P2
1
/m, 11) 1; 4; (2; 3)+(
1
2
,
1
2
,0)
[2] P12/m1(P2/m, 10) 1; 2; 3; 4
IIb [2] C 12/c1(c
= 2c)(C 2/c, 15); [2] I 12/c1(c
= 2c)(C 2/c, 15)
Maximal isomorphic subgroups of lowest index
IIc
[2] C 12/m1(c
= 2c or a
= a + 2c,c
= 2c)(C 2/m, 12); [3] C 12/m1(b
= 3b)(C2/m, 12)
Minimal non-isomorphic supergroups
I
[2] Cmcm(63); [2]Cmce(64); [2]Cmmm (65); [2] Cmme (67); [2] F mmm(69); [2] Immm (71); [2] Ibam (72); [2] Imma (74);
[2] I 4/m (87); [3] P
¯
31m (162); [3] P
¯
3m1 (164); [3] R
¯
3m (166)
II [2] P12/m1(a
=
1
2
a,b
=
1
2
b)(P2/m, 10)
169
C2/mC
3
2h
2/m Monoclinic
No. 12
UNIQUE AXIS b, DIFFERENT CELL CHOICES
C12/m1
UNIQUE AXIS b, CELL CHOICE 1
Origin at centre (2/m)
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
4
;0≤ z ≤ 1
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); t(
1
2
,
1
2
,0); (2); (3)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates
(0,0, 0)+ (
1
2
,
1
2
,0)+
Reflection conditions
General:
8 j 1(1)x,y, z (2) ¯x,y, ¯z (3) ¯x, ¯y, ¯z (4) x, ¯y,zhkl: h + k = 2n
h0l : h = 2n
0kl : k = 2n
hk0: h + k = 2n
0k0: k = 2n
h00 : h = 2n
Special: as above, plus
4 im x,0,z ¯x,0, ¯z no extra conditions
4 h 20,y,
1
2
0, ¯y,
1
2
4 g 20,y,00, ¯y,0 no extra conditions
4 f
¯
1
1
4
,
1
4
,
1
2
3
4
,
1
4
,
1
2
4 e
¯
1
1
4
,
1
4
,0
3
4
,
1
4
,0 hkl : h = 2n
2 d 2/m 0,
1
2
,
1
2
2 c 2/m 0,0 ,
1
2
no extra conditions
2 b 2/ m 0,
1
2
,02a 2/m 0, 0,0 no extra conditions
170
CONTINUED No. 12 C2/m
A12/m1
UNIQUE AXIS b, CELL CHOICE 2
Origin at centre (2/m)
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
4
;0≤ z ≤ 1
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); t(0,
1
2
,
1
2
); (2); (3)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates
(0,0, 0)+ (0,
1
2
,
1
2
)+
Reflection conditions
General:
8 j 1(1)x,y, z (2) ¯x,y, ¯z (3) ¯x, ¯y, ¯z (4) x, ¯y,zhkl: k + l = 2n
h0l : l = 2n
0kl : k + l = 2n
hk0: k = 2n
0k0: k = 2n
00l : l = 2n
Special: as above, plus
4 im x,0,z ¯x,0, ¯z no extra conditions
4 h 2
1
2
,y,
1
2
1
2
, ¯y,
1
2
4 g 20,y,00, ¯y,0 no extra conditions
4 f
¯
1
1
2
,
1
4
,
3
4
1
2
,
1
4
,
1
4
4 e
¯
10,
1
4
,
1
4
0,
1
4
,
3
4
hkl : k = 2 n
2 d 2/m
1
2
,
1
2
,
1
2
2 c 2/m
1
2
,0,
1
2
no extra conditions
2 b 2/ m 0,
1
2
,02a 2/m 0, 0,0 no extra conditions
I 12/ m1
UNIQUE AXIS b, CELL CHOICE 3
Origin at centre (2/m)
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
4
;0≤ z ≤ 1
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); t(
1
2
,
1
2
,
1
2
); (2); (3)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates
(0,0, 0)+ (
1
2
,
1
2
,
1
2
)+
Reflection conditions
General:
8 j 1(1)x,y, z (2) ¯x,y, ¯z (3) ¯x, ¯y, ¯z (4) x, ¯y,zhkl: h + k + l = 2n
h0l : h + l = 2n
0kl : k + l = 2n
hk0: h + k = 2n
0k0: k = 2n
h00 : h = 2n
00l : l = 2n
Special: as above, plus
4 im x,0,z ¯x,0
, ¯z no extra conditions
4 h 2
1
2
,y,0
1
2
, ¯y, 04g 20,y, 00, ¯y,0 no extra conditions
4 f
¯
1
1
4
,
1
4
,
3
4
3
4
,
1
4
,
1
4
4 e
¯
1
3
4
,
1
4
,
3
4
1
4
,
1
4
,
1
4
hkl : k = 2 n
2 d 2/m
1
2
,
1
2
,02c 2/m
1
2
,0, 0 no extra conditions
2 b 2/ m 0,
1
2
,02a 2/m 0, 0,0 no extra conditions
171