Hahn Th. (ed.) International tables for crystallography. Vol. A. Space-group symmetry
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C2/mC
3
2h
2/m Monoclinic
No. 12 A112/ m
Patterson symmetry A112/m
UNIQUE AXIS c, CELL CHOICE 1
Origin at centre (2/m)
Asymmetric unit 0 ≤ x ≤ 1; 0 ≤ y ≤
1
2
;0≤ z ≤
1
4
Symmetry operations
For (0,0,0)+ set
(1) 1 (2) 2 0, 0,z (3)
¯
10, 0,0(4)mx,y,0
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)
¯
10,
1
4
,
1
4
(4) bx,y,
1
4
172
CONTINUED No. 12 C2/m
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
hk0: k = 2n
0kl : k + l = 2n
h0l : l = 2n
00l : l = 2n
0k0: k = 2n
Special: as above, plus
4 im x,y, 0¯x, ¯y,0 no extra conditions
4 h 2
1
2
,0, z
1
2
,0, ¯z no extra conditions
4 g 20,0, z 0,0, ¯z no extra conditions
4 f
¯
1
1
2
,
1
4
,
1
4
1
2
,
3
4
,
1
4
hkl : k = 2n
4 e
¯
10,
1
4
,
1
4
0,
3
4
,
1
4
hkl : k = 2n
2 d 2/m
1
2
,0,
1
2
no extra conditions
2 c 2/m
1
2
,0, 0 no extra conditions
2 b 2/ m 0,0,
1
2
no extra conditions
2 a 2/ m 0,0,0 no extra conditions
Symmetry of special projections
Along [001] p2
a
= ab
=
1
2
b
Origin at 0,0,z
Along [100] c2mm
a
= b
p
b
= c
Origin at x,0,0
Along [010] p2mm
a
=
1
2
cb
= a
p
Origin at 0,y, 0
Maximal non-isomorphic subgroups
I
[2] A11m (Cm,8) (1; 4)+
[2] A112 (C2, 5) (1; 2)+
[2] A
¯
1(P
¯
1, 2) (1; 3)+
IIa [2] P112
1
/b (P2
1
/c, 14) 1; 3; (2; 4)+(0,
1
2
,
1
2
)
[2] P112/b (P2/ c, 13) 1; 2; (3; 4)+(0,
1
2
,
1
2
)
[2] P112
1
/m (P2
1
/m, 11) 1; 4; (2; 3)+(0,
1
2
,
1
2
)
[2] P112/m (P2/m, 10) 1; 2; 3; 4
IIb [2] A112/a (a
= 2a)(C 2/c, 15); [2] I 112/a (a
= 2a)(C 2/c, 15)
Maximal isomorphic subgroups of lowest index
IIc
[2] A112/m (a
= 2a or a
= 2a,b
= 2a + b)(C 2/m, 12); [3] A112/m (c
= 3c)(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] P112/m (b
=
1
2
b,c
=
1
2
c)(P2/m, 10)
173
C2/mC
3
2h
2/m Monoclinic
No. 12
UNIQUE AXIS c, DIFFERENT CELL CHOICES
A112/m
UNIQUE AXIS c, CELL CHOICE 1
Origin at centre (2/m)
Asymmetric unit 0 ≤ x ≤ 1; 0 ≤ y ≤
1
2
;0≤ z ≤
1
4
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
hk0: k = 2n
0kl : k + l = 2n
h0l : l = 2n
00l : l = 2n
0k0: k = 2n
Special: as above, plus
4 im x,y, 0¯x, ¯y,0 no extra conditions
4 h 2
1
2
,0, z
1
2
,0, ¯z 4 g 20,0 , z 0,0, ¯z no extra conditions
4 f
¯
1
1
2
,
1
4
,
1
4
1
2
,
3
4
,
1
4
4 e
¯
10,
1
4
,
1
4
0,
3
4
,
1
4
hkl : k = 2 n
2 d 2/m
1
2
,0,
1
2
2 c 2/m
1
2
,0, 0 no extra conditions
2 b 2/ m 0,0,
1
2
2 a 2/ m 0,0,0 no extra conditions
174
CONTINUED No. 12 C2/m
B112/m
UNIQUE AXIS c, CELL CHOICE 2
Origin at centre (2/m)
Asymmetric unit 0 ≤ x ≤ 1; 0 ≤ y ≤
1
2
;0≤ z ≤
1
4
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); t(
1
2
,0,
1
2
); (2); (3)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates
(0,0, 0)+ (
1
2
,0,
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 + l = 2n
hk0: h = 2n
0kl : l = 2n
h0l : h + l = 2n
00l : l = 2n
h00 : h = 2n
Special: as above, plus
4 im x,y, 0¯x, ¯y,0 no extra conditions
4 h 2
1
2
,
1
2
,z
1
2
,
1
2
, ¯z 4 g 20,0,z 0,0, ¯z no extra conditions
4 f
¯
1
3
4
,
1
2
,
1
4
1
4
,
1
2
,
1
4
4 e
¯
1
1
4
,0,
1
4
3
4
,0,
1
4
hkl : h = 2n
2 d 2/m
1
2
,
1
2
,
1
2
2 c 2/m
1
2
,
1
2
,0 no extra conditions
2 b 2/ m 0,0,
1
2
2 a 2/ m 0,0,0 no extra conditions
I 112/m
UNIQUE AXIS c, CELL CHOICE 3
Origin at centre (2/m)
Asymmetric unit 0 ≤ x ≤ 1; 0 ≤ y ≤
1
2
;0≤ z ≤
1
4
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
hk0: h + k = 2n
0kl : k + l = 2n
h0l : h + l = 2n
00l : l = 2n
h00 : h = 2n
0k0: k = 2n
Special: as above, plus
4 im x,y, 0¯x, ¯y
,0 no extra conditions
4 h 20,
1
2
,z 0,
1
2
, ¯z 4 g 20,0,z 0,0, ¯z no extra conditions
4 f
¯
1
3
4
,
1
4
,
1
4
1
4
,
3
4
,
1
4
4 e
¯
1
3
4
,
3
4
,
1
4
1
4
,
1
4
,
1
4
hkl : l = 2n
2 d 2/m 0,
1
2
,
1
2
2 c 2/m 0,
1
2
,0 no extra conditions
2 b 2/ m 0,0,
1
2
2 a 2/ m 0,0,0 no extra conditions
175
P2/cC
4
2h
2/m Monoclinic
No. 13 P12/ c1
Patterson symmetry P12/m1
UNIQUE AXIS b, CELL CHOICE 1
Origin at
¯
1 on glide plane c
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤ 1; 0 ≤ z ≤
1
2
Symmetry operations
(1) 1 (2) 2 0,y,
1
4
(3)
¯
10,0,0(4)cx,0,z
176
International Tables for Crystallography (2006). Vol. A, Space group 13, pp. 176–183.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 13 P2/c
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 g 1(1)x, y,z (2) ¯x,y, ¯z +
1
2
(3) ¯x, ¯y, ¯z (4) x, ¯y,z +
1
2
h0l : l = 2n
00l : l = 2n
Special: as above, plus
2 f 2
1
2
,y,
1
4
1
2
, ¯y,
3
4
no extra conditions
2 e 20,y,
1
4
0, ¯y,
3
4
no extra conditions
2 d
¯
1
1
2
,0, 0
1
2
,0,
1
2
hkl : l = 2n
2 c
¯
10,
1
2
,00,
1
2
,
1
2
hkl : l = 2n
2 b
¯
1
1
2
,
1
2
,0
1
2
,
1
2
,
1
2
hkl : l = 2n
2 a
¯
10, 0,00, 0,
1
2
hkl : l = 2n
Symmetry of special projections
Along [001] p2mm
a
= a
p
b
= b
Origin at 0,0,z
Along [100] p2gm
a
= bb
= c
p
Origin at x,0,0
Along [010] p2
a
=
1
2
cb
= a
Origin at 0,y, 0
Maximal non-isomorphic subgroups
I
[2] P1c1(Pc,7) 1; 4
[2] P121 (P2, 3) 1; 2
[2] P
¯
1 (2) 1; 3
IIa none
IIb [2] P12
1
/c1(b
= 2b)(P2
1
/c, 14); [2] C12/c1(a
= 2a,b
= 2b)(C2/c, 15)
Maximal isomorphic subgroups of lowest index
IIc
[2] P12/c1(b
= 2b)(P2/c, 13); [2] P12/c1(a
= 2a or a
= 2a,c
= 2a + c)(P2/c, 13)
Minimal non-isomorphic supergroups
I
[2] Pnnn(48); [2] Pccm (49); [2] Pban(50); [2] Pmma (51); [2] Pnna(52); [2] Pmna (53); [2] Pcca (54); [2] Pccn(56);
[2] Pbcm (57); [2] Pmmn (59); [2] Pbcn (60); [2]Cmme (67); [2] Ccce(68); [2] P4/n (85); [2] P4
2
/n (86)
II [2] A12/m1(C 2/m, 12); [2] C12/c1(C2/c , 15); [2] I 12/c1(C2/c , 15); [2] P12/m1(c
=
1
2
c)(P2/m, 10)
177
P2/cC
4
2h
2/m Monoclinic
No. 13
UNIQUE AXIS b, DIFFERENT CELL CHOICES
P12/c1
UNIQUE AXIS b, CELL CHOICE 1
Origin at
¯
1 on glide plane c
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤ 1; 0 ≤ z ≤
1
2
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 g 1(1)x, y,z (2) ¯x,y, ¯z +
1
2
(3) ¯x, ¯y, ¯z (4) x, ¯y,z +
1
2
h0l : l = 2n
00l : l = 2n
Special: as above, plus
2 f 2
1
2
,y,
1
4
1
2
, ¯y,
3
4
no extra conditions
2 e 20,y,
1
4
0, ¯y,
3
4
no extra conditions
2 d
¯
1
1
2
,0, 0
1
2
,0,
1
2
2 c
¯
10,
1
2
,00,
1
2
,
1
2
hkl : l = 2n
2 b
¯
1
1
2
,
1
2
,0
1
2
,
1
2
,
1
2
2 a
¯
10, 0,00, 0,
1
2
hkl : l = 2n
178
CONTINUED No. 13 P2/c
P12/n1
UNIQUE AXIS b, CELL CHOICE 2
Origin at
¯
1 on glide plane n
Asymmetric unit 0 ≤ x ≤ 1; 0 ≤ y ≤ 1; 0 ≤ z ≤
1
4
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 g 1(1)x, y,z (2) ¯x +
1
2
,y, ¯z +
1
2
(3) ¯x, ¯y, ¯z (4) x +
1
2
, ¯y, z +
1
2
h0l : h + l = 2n
h00 : h = 2n
00l : l = 2n
Special: as above, plus
2 f 2
3
4
,y,
1
4
1
4
, ¯y,
3
4
no extra conditions
2 e 2
3
4
,y,
3
4
1
4
, ¯y,
1
4
no extra conditions
2 d
¯
10, 0,
1
2
1
2
,0, 02c
¯
10,
1
2
,0
1
2
,
1
2
,
1
2
hkl : h + l = 2n
2 b
¯
10,
1
2
,
1
2
1
2
,
1
2
,02a
¯
10, 0,0
1
2
,0,
1
2
hkl : h + l = 2n
P12/a1
UNIQUE AXIS b, CELL CHOICE 3
Origin at
¯
1 on glide plane a
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤ 1; 0 ≤ z ≤
1
2
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 g 1(1)x, y,z (2) ¯x +
1
2
,y, ¯z (3) ¯x, ¯y, ¯z (4) x +
1
2
, ¯y, zh0l : h = 2n
h00 : h = 2n
Special: as above, plus
2 f 2
3
4
,y,
1
2
1
4
, ¯y,
1
2
no extra conditions
2 e 2
1
4
,y,0
3
4
, ¯y, 0 no extra conditions
2 d
¯
1
1
2
,0,
1
2
0,0,
1
2
2 c
¯
10,
1
2
,0
1
2
,
1
2
,0 hkl : h = 2n
2 b
¯
1
1
2
,
1
2
,
1
2
0,
1
2
,
1
2
2 a
¯
10, 0,0
1
2
,0, 0 hkl : h = 2n
179
P2/cC
4
2h
2/m Monoclinic
No. 13 P112/ a
Patterson symmetry P112/m
UNIQUE AXIS c, CELL CHOICE 1
Origin at
¯
1 on glide plane a
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
2
;0≤ z ≤ 1
Symmetry operations
(1) 1 (2) 2
1
4
,0, z (3)
¯
10,0,0(4)ax,y,0
180
CONTINUED No. 13 P2/c
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 g 1(1)x, y,z (2) ¯x +
1
2
, ¯y, z (3) ¯x, ¯y, ¯z (4) x +
1
2
,y, ¯zhk0: h = 2n
h00 : h = 2n
Special: as above, plus
2 f 2
1
4
,
1
2
,z
3
4
,
1
2
, ¯z no extra conditions
2 e 2
1
4
,0, z
3
4
,0, ¯z no extra conditions
2 d
¯
10,
1
2
,0
1
2
,
1
2
,0 hkl : h = 2n
2 c
¯
10, 0,
1
2
1
2
,0,
1
2
hkl : h = 2n
2 b
¯
10,
1
2
,
1
2
1
2
,
1
2
,
1
2
hkl : h = 2n
2 a
¯
10, 0,0
1
2
,0, 0 hkl : h = 2n
Symmetry of special projections
Along [001] p2
a
=
1
2
ab
= b
Origin at 0,0,z
Along [100] p2mm
a
= b
p
b
= c
Origin at x,0,0
Along [010] p2gm
a
= cb
= a
p
Origin at 0,y, 0
Maximal non-isomorphic subgroups
I
[2] P11a (Pc,7) 1; 4
[2] P112 (P2, 3) 1; 2
[2] P
¯
1 (2) 1; 3
IIa none
IIb [2] P112
1
/a (c
= 2c)(P2
1
/c, 14); [2] A112/a (b
= 2b,c
= 2c)(C2/c, 15)
Maximal isomorphic subgroups of lowest index
IIc
[2] P112/a (c
= 2c)(P2/c, 13); [2] P112/ a (b
= 2b or a
= a + 2b,b
= 2b)(P2/c, 13)
Minimal non-isomorphic supergroups
I
[2] Pnnn(48); [2] Pccm (49); [2] Pban(50); [2] Pmma (51); [2] Pnna(52); [2] Pmna (53); [2] Pcca (54); [2] Pccn(56);
[2] Pbcm (57); [2] Pmmn (59); [2] Pbcn (60); [2]Cmme (67); [2] Ccce(68); [2] P4/n (85); [2] P4
2
/n (86)
II [2] A112/a (C2/c, 15); [2] B112/m (C 2/m, 12); [2] I 112/a (C2/c, 15); [2] P112/m (a
=
1
2
a)(P2/m, 10)
181