Hahn Th. (ed.) International tables for crystallography. Vol. A. Space-group symmetry
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C2/cC
6
2h
2/m Monoclinic
No. 15 C12/c1
Patterson symmetry C 12/m1
UNIQUE AXIS b, CELL CHOICE 1
Origin at
¯
1 on glide plane c
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
2
;0≤ z ≤
1
2
Symmetry operations
For (0,0,0)+ set
(1) 1 (2) 2 0, y,
1
4
(3)
¯
10, 0,0(4)cx,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,
1
4
(3)
¯
1
1
4
,
1
4
,0(4)n(
1
2
,0,
1
2
) x,
1
4
,z
192
International Tables for Crystallography (2006). Vol. A, Space group 15, pp. 192–199.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 15 C2/c
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 f 1(1)x,y,z (2) ¯x,y, ¯z +
1
2
(3) ¯x, ¯y, ¯z (4) x, ¯y,z +
1
2
hkl : h + k = 2n
h0l : h, l = 2n
0kl : k = 2n
hk0: h + k = 2n
0k0: k = 2n
h00 : h = 2n
00l : l = 2n
Special: as above, plus
4 e 20,y,
1
4
0, ¯y,
3
4
no extra conditions
4 d
¯
1
1
4
,
1
4
,
1
2
3
4
,
1
4
,0 hkl : k + l = 2n
4 c
¯
1
1
4
,
1
4
,0
3
4
,
1
4
,
1
2
hkl : k + l = 2n
4 b
¯
10,
1
2
,00,
1
2
,
1
2
hkl : l = 2n
4 a
¯
10, 0,00, 0,
1
2
hkl : l = 2n
Symmetry of special projections
Along [001] c2mm
a
= a
p
b
= b
Origin at 0,0,z
Along [100] p2gm
a
=
1
2
bb
= c
p
Origin at x,0,0
Along [010] p2
a
=
1
2
cb
=
1
2
a
Origin at 0,y, 0
Maximal non-isomorphic subgroups
I
[2] C 1c1(Cc,9) (1; 4 )+
[2] C 121(C2, 5) (1; 2)+
[2] C
¯
1(P
¯
1, 2) (1; 3)+
IIa [2] P12
1
/n1(P2
1
/c, 14) 1; 3; (2; 4)+(
1
2
,
1
2
,0)
[2] P12
1
/c1(P2
1
/c, 14) 1; 4; (2; 3)+(
1
2
,
1
2
,0)
[2] P12/c1(P2/c, 13) 1; 2; 3; 4
[2] P12/n1(P2/c, 13) 1; 2; (3; 4)+(
1
2
,
1
2
,0)
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] C 12/c1(b
= 3b)(C2/c, 15); [3] C12/c1(c
= 3c)(C 2/c, 15);
[3] C 12/c1(a
= 3a or a
= 3a,c
= −a + c or a
= 3a,c
= a + c)(C 2/c, 15)
Minimal non-isomorphic supergroups
I
[2] Cmcm(63); [2]Cmce(64); [2]Cccm (66); [2] Ccce (68); [2] Fddd(70); [2] Ibam (72); [2] Ibca(73); [2] Imma (74);
[2] I 4
1
/a (88); [3] P
¯
31c (163); [3] P
¯
3c1 (165); [3] R
¯
3c (167)
II [2] F 12/m1(C 2/m, 12); [2] C12/m1(c
=
1
2
c)(C 2/m, 12); [2] P12/c1(a
=
1
2
a,b
=
1
2
b)(P2/c, 13)
193
C2/cC
6
2h
2/m Monoclinic
No. 15
UNIQUE AXIS b, DIFFERENT CELL CHOICES
C12/c1
UNIQUE AXIS b, CELL CHOICE 1
Origin at
¯
1 on glide plane c
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
2
;0≤ z ≤
1
2
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 f 1(1)x,y,z (2) ¯x,y, ¯z +
1
2
(3) ¯x, ¯y, ¯z (4) x, ¯y,z +
1
2
hkl : h + k = 2n
h0l : h,l = 2n
0kl : k = 2n
hk0: h + k = 2n
0k0: k = 2n
h00 : h = 2n
00l : l = 2n
Special: as above, plus
4 e 20,y,
1
4
0, ¯y,
3
4
no extra conditions
4 d
¯
1
1
4
,
1
4
,
1
2
3
4
,
1
4
,04c
¯
1
1
4
,
1
4
,0
3
4
,
1
4
,
1
2
hkl : k + l = 2n
4 b
¯
10,
1
2
,00,
1
2
,
1
2
4 a
¯
10, 0,00, 0,
1
2
hkl : l = 2n
194
CONTINUED No. 15 C2/c
A12/n1
UNIQUE AXIS b, CELL CHOICE 2
Origin at
¯
1 on glide plane n
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤ 1; 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 f 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
hkl : k + l = 2n
h0l : h,l = 2n
0kl : k + l = 2n
hk0: k = 2n
0k0: k = 2n
h00 : h = 2n
00l : l = 2n
Special: as above, plus
4 e 2
3
4
,y,
3
4
1
4
, ¯y,
1
4
no extra conditions
4 d
¯
1
1
2
,
1
4
,
3
4
0,
1
4
,
3
4
4 c
¯
10,
1
4
,
1
4
1
2
,
1
4
,
1
4
hkl : h = 2n
4 b
¯
10,
1
2
,0
1
2
,
1
2
,
1
2
4 a
¯
10, 0,0
1
2
,0,
1
2
hkl : h + k = 2n
I 12/ a1
UNIQUE AXIS b, CELL CHOICE 3
Origin at
¯
1 on glide plane a
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 f 1(1)x,y,z (2) ¯x +
1
2
,y, ¯z (3) ¯x, ¯y, ¯z (4) x +
1
2
, ¯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 e 2
1
4
,y,0
3
4
, ¯y, 0 no extra conditions
4 d
¯
1
1
4
,
1
4
,
3
4
1
4
,
1
4
,
1
4
4 c
¯
1
3
4
,
1
4
,
3
4
3
4
,
1
4
,
1
4
hkl : l = 2n
4 b
¯
10,
1
2
,0
1
2
,
1
2
,04a
¯
10, 0,0
1
2
,0, 0 hkl : h = 2n
195
C2/cC
6
2h
2/m Monoclinic
No. 15 A112/ a
Patterson symmetry A112/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
2
Symmetry operations
For (0,0,0)+ set
(1) 1 (2) 2
1
4
,0, z (3)
¯
10, 0,0(4)ax,y,0
For (0,
1
2
,
1
2
)+ set
(1) t(0,
1
2
,
1
2
) (2) 2(0, 0,
1
2
)
1
4
,
1
4
,z (3)
¯
10,
1
4
,
1
4
(4) n(
1
2
,
1
2
,0) x,y,
1
4
196
CONTINUED No. 15 C2/c
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 f 1(1)x,y,z (2) ¯x +
1
2
, ¯y, z (3) ¯x, ¯y, ¯z (4) x +
1
2
,y, ¯zhkl: k + l = 2n
hk0: h, k = 2n
0kl : k + l = 2 n
h0l : l = 2n
00l : l = 2n
h00 : h = 2n
0k0: k = 2n
Special: as above, plus
4 e 2
1
4
,0, z
3
4
,0, ¯z no extra conditions
4 d
¯
1
1
2
,
1
4
,
1
4
0,
3
4
,
1
4
hkl : h + k = 2n
4 c
¯
10,
1
4
,
1
4
1
2
,
3
4
,
1
4
hkl : h + k = 2n
4 b
¯
10, 0,
1
2
1
2
,0,
1
2
hkl : h = 2n
4 a
¯
10, 0,0
1
2
,0, 0 hkl : h = 2n
Symmetry of special projections
Along [001] p2
a
=
1
2
ab
=
1
2
b
Origin at 0,0,z
Along [100] c2mm
a
= b
p
b
= c
Origin at x, 0,0
Along [010] p2gm
a
=
1
2
cb
= a
p
Origin at 0,y, 0
Maximal non-isomorphic subgroups
I
[2] A11a (Cc,9) (1; 4)+
[2] A112 (C2, 5) (1; 2)+
[2] A
¯
1(P
¯
1, 2) (1; 3)+
IIa [2] P112
1
/n (P2
1
/c, 14) 1; 3; (2; 4)+(0,
1
2
,
1
2
)
[2] P112
1
/a (P2
1
/c, 14) 1; 4; (2; 3)+(0,
1
2
,
1
2
)
[2] P112/a (P2/c, 13) 1; 2; 3; 4
[2] P112/n (P2/ c, 13) 1; 2; (3; 4)+(0,
1
2
,
1
2
)
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] A112/a (c
= 3c)(C 2/c, 15); [3] A112/a (a
= 3a)(C2/c, 15);
[3] A112/a (b
= 3b or a
= a − b,b
= 3b or a
= a + b,b
= 3b)(C2/c, 15)
Minimal non-isomorphic supergroups
I
[2] Cmcm(63); [2]Cmce(64); [2]Cccm (66); [2] Ccce (68); [2] Fddd(70); [2] Ibam (72); [2] Ibca(73); [2] Imma (74);
[2] I 4
1
/a (88); [3] P
¯
31c (163); [3] P
¯
3c1 (165); [3] R
¯
3c (167)
II [2] F 112/m (C2/m, 12); [2] A112/m (a
=
1
2
a)(C 2/m, 12); [2] P112/a (b
=
1
2
b,c
=
1
2
c)(P2/c, 13)
197
C2/cC
6
2h
2/m Monoclinic
No. 15
UNIQUE AXIS c, DIFFERENT CELL CHOICES
A112/a
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
2
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 f 1(1)x,y,z (2) ¯x +
1
2
, ¯y, z (3) ¯x, ¯y, ¯z (4) x +
1
2
,y, ¯zhkl: k + l = 2n
hk0: h,k = 2 n
0kl : k + l = 2n
h0l : l = 2n
00l : l = 2n
h00 : h = 2n
0k0: k = 2n
Special: as above, plus
4 e 2
1
4
,0, z
3
4
,0, ¯z no extra conditions
4 d
¯
1
1
2
,
1
4
,
1
4
0,
3
4
,
1
4
4 c
¯
10,
1
4
,
1
4
1
2
,
3
4
,
1
4
hkl : h + k = 2n
4 b
¯
10, 0,
1
2
1
2
,0,
1
2
4 a
¯
10, 0,0
1
2
,0, 0 hkl : h = 2n
198
CONTINUED No. 15 C2/c
B112/n
UNIQUE AXIS c, CELL CHOICE 2
Origin at
¯
1 on glide plane n
Asymmetric unit 0 ≤ x ≤
1
4
;0≤ y ≤
1
2
;0≤ z ≤ 1
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 f 1(1)x,y,z (2) ¯x +
1
2
, ¯y +
1
2
,z (3) ¯x, ¯y, ¯z (4) x +
1
2
,y +
1
2
, ¯zhkl: h + l = 2n
hk0: h,k = 2 n
0kl : l = 2n
h0l : h + l = 2n
00l : l = 2n
h00 : h = 2n
0k0: k = 2n
Special: as above, plus
4 e 2
3
4
,
3
4
,z
1
4
,
1
4
, ¯z no extra conditions
4 d
¯
1
3
4
,
1
2
,
1
4
3
4
,0,
1
4
4 c
¯
1
1
4
,0,
1
4
1
4
,
1
2
,
1
4
hkl : k = 2 n
4 b
¯
10, 0,
1
2
1
2
,
1
2
,
1
2
4 a
¯
10, 0,0
1
2
,
1
2
,0 hk l : h + k = 2n
I 112/b
UNIQUE AXIS c, CELL CHOICE 3
Origin at
¯
1 on glide plane b
Asymmetric unit 0 ≤ x ≤
1
4
;0≤ y ≤ 1; 0 ≤ z ≤
1
2
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 f 1(1)x,y,z (2) ¯x, ¯y +
1
2
,z (3) ¯x, ¯y, ¯z (4) x,y +
1
2
, ¯zhkl: h + k + l = 2n
hk0: h,k = 2 n
0kl : k + l = 2n
h0l : h + l = 2n
00l : l = 2n
h00 : h = 2n
0k0: k = 2n
Special: as above, plus
4 e 20,
1
4
,z 0,
3
4
, ¯z no extra conditions
4 d
¯
1
3
4
,
1
4
,
1
4
1
4
,
1
4
,
1
4
4 c
¯
1
3
4
,
3
4
,
1
4
1
4
,
3
4
,
1
4
hkl : h = 2n
4 b
¯
10, 0,
1
2
0,
1
2
,
1
2
4 a
¯
10, 0,00,
1
2
,0 hkl : k = 2n
199
P222 D
1
2
222 Orthorhombic
No. 16 P222
Patterson symmetry Pmmm
Origin at 222
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
2
;0≤ z ≤ 1
Symmetry operations
(1) 1 (2) 2 0,0,z (3) 2 0,y,0(4)2x , 0,0
Maximal non-isomorphic subgroups
I
[2] P112 (P2, 3) 1; 2
[2] P121 (P2, 3) 1; 3
[2] P211 (P2, 3) 1; 4
IIa none
IIb [2] P2
1
22(a
= 2a)(P222
1
, 17); [2] P22
1
2(b
= 2b)(P222
1
, 17); [2] P222
1
(c
= 2c) (17);
[2] A222 (b
= 2b,c
= 2c)(C 222, 21); [2] B222 (a
= 2a,c
= 2c)(C 222, 21); [2] C222 (a
= 2a,b
= 2b) (21);
[2] F 222(a
= 2a,b
= 2b,c
= 2c) (22)
Maximal isomorphic subgroups of lowest index
IIc
[2] P222 (a
= 2a or b
= 2b or c
= 2c) (16)
Minimal non-isomorphic supergroups
I
[2] Pmmm (47); [2] Pnnn (48); [2] Pccm (49); [2] Pban (50); [2] P422 (89); [2] P4
2
22 (93); [2] P
¯
42c (112); [2] P
¯
42m (111);
[3] P23 (195)
II [2] A222(C 222, 21); [2] B222 (C222, 21); [2]C 222 (21); [2] I 222 (23)
200
International Tables for Crystallography (2006). Vol. A, Space group 16, pp. 200–201.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 16 P222
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 u 1(1)x, y,z (2) ¯x, ¯y,z (3) ¯x,y, ¯z (4) x, ¯y, ¯z no conditions
Special: no extra conditions
2 t ..2
1
2
,
1
2
,z
1
2
,
1
2
, ¯z
2 s ..20,
1
2
,z 0,
1
2
, ¯z
2 r ..2
1
2
,0, z
1
2
,0, ¯z
2 q ..20,0, z 0,0, ¯z
2 p . 2 .
1
2
,y,
1
2
1
2
, ¯y,
1
2
2 o . 2 .
1
2
,y,0
1
2
, ¯y, 0
2 n . 2 . 0, y,
1
2
0, ¯y,
1
2
2 m . 2 . 0,y,00, ¯y,0
2 l 2 .. x,
1
2
,
1
2
¯x,
1
2
,
1
2
2 k 2 .. x,
1
2
,0¯x,
1
2
,0
2 j 2 .. x,0,
1
2
¯x,0,
1
2
2 i 2 .. x,0,0¯x, 0,0
1 h 222
1
2
,
1
2
,
1
2
1 g 222 0,
1
2
,
1
2
1 f 222
1
2
,0,
1
2
1 e 222
1
2
,
1
2
,0
1 d 222 0,0,
1
2
1 c 222 0,
1
2
,0
1 b 222
1
2
,0, 0
1 a 222 0,0,0
Symmetry of special projections
Along [001] p2mm
a
= ab
= b
Origin at 0,0,z
Along [100] p2mm
a
= bb
= c
Origin at x, 0,0
Along [010] p2mm
a
= cb
= a
Origin at 0,y, 0
(Continued on preceding page)
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