Hahn Th. (ed.) International tables for crystallography. Vol. A. Space-group symmetry
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P3c1 C
3
3v
3m1 Trigonal
No. 158 P3c1
Patterson symmetry P
¯
3m1
Origin on 3c1
Asymmetric unit 0 ≤ x ≤
2
3
;0≤ y ≤
2
3
;0≤ z ≤
1
2
; x ≤ (1 + y)/2; y ≤ min(1 − x,(1 + x)/2)
Vertices 0,0, 0
1
2
,0, 0
2
3
,
1
3
,0
1
3
,
2
3
,00,
1
2
,0
0,0,
1
2
1
2
,0,
1
2
2
3
,
1
3
,
1
2
1
3
,
2
3
,
1
2
0,
1
2
,
1
2
Symmetry operations
(1) 1 (2) 3
+
0,0,z (3) 3
−
0,0,z
(4) cx, ¯x, z (5) cx,2x,z (6) c 2x,x,z
524
International Tables for Crystallography (2006). Vol. A, Space group 158, pp. 524–525.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 158 P3c1
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (4)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
General:
6 d 1(1)x,y,z (2) ¯y,x − y,z (3) ¯x + y, ¯x,z
(4) ¯y, ¯x,z +
1
2
(5) ¯x + y,y,z +
1
2
(6) x,x − y, z +
1
2
h
¯
h0l : l = 2n
000l : l = 2n
Special: as above, plus
2 c 3 ..
2
3
,
1
3
,z
2
3
,
1
3
,z +
1
2
hkil : l = 2n
2 b 3 ..
1
3
,
2
3
,z
1
3
,
2
3
,z +
1
2
hkil : l = 2n
2 a 3 .. 0,0,z 0,0,z +
1
2
hkil : l = 2n
Symmetry of special projections
Along [001] p3m1
a
= ab
= b
Origin at 0,0,z
Along [100] p1
a
=
1
2
(a + 2b) b
=
1
2
c
Origin at x,0,0
Along [210] p1g1
a
=
1
2
bb
= c
Origin at x,
1
2
x,0
Maximal non-isomorphic subgroups
I
[2] P311 (P3, 143) 1; 2; 3
[3] P1c1(Cc,9) 1; 4
[3] P1c1(Cc,9) 1; 5
[3] P1c1(Cc,9) 1; 6
IIa none
IIb [3] H 3c1(a
= 3a,b
= 3b)(P31c, 159)
Maximal isomorphic subgroups of lowest index
IIc
[3] P3c1(c
= 3c) (158); [4] P3c1(a
= 2a,b
= 2b) (158)
Minimal non-isomorphic supergroups
I
[2] P
¯
3c1 (165); [2] P6cc (184); [2] P6
3
cm(185); [2] P
¯
6c2 (188)
II [3] H 3c1(P31c, 159); [3] R3c (obverse) (161); [3] R3c (reverse) (161); [2] P3m1(c
=
1
2
c) (156)
525
P31cC
4
3v
31m Trigonal
No. 159 P31c
Patterson symmetry P
¯
31m
Origin on 31c
Asymmetric unit 0 ≤ x ≤
2
3
;0≤ y ≤
2
3
;0≤ z ≤
1
2
; x ≤ (1 + y)/2; y ≤ min(1 − x,(1 + x)/2)
Vertices 0,0, 0
1
2
,0, 0
2
3
,
1
3
,0
1
3
,
2
3
,00,
1
2
,0
0,0,
1
2
1
2
,0,
1
2
2
3
,
1
3
,
1
2
1
3
,
2
3
,
1
2
0,
1
2
,
1
2
Symmetry operations
(1) 1 (2) 3
+
0,0,z (3) 3
−
0,0,z
(4) cx,x,z (5) cx,0, z (6) c 0,y,z
526
International Tables for Crystallography (2006). Vol. A, Space group 159, pp. 526–527.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 159 P31c
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (4)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
General:
6 c 1(1)x,y,z (2) ¯y,x − y, z (3) ¯x + y, ¯x,z
(4) y,x,z +
1
2
(5) x − y, ¯y,z +
1
2
(6) ¯x, ¯x + y, z +
1
2
hh2hl : l = 2n
000l : l = 2n
Special: as above, plus
2 b 3 ..
1
3
,
2
3
,z
2
3
,
1
3
,z +
1
2
hkil : l = 2n
or h − k = 3n + 1
or h − k = 3n + 2
2 a 3 .. 0,0,z 0,0,z +
1
2
hkil : l = 2n
Symmetry of special projections
Along [001] p31m
a
= ab
= b
Origin at 0,0,z
Along [100] p1g1
a
=
1
2
(a + 2b) b
= c
Origin at x, 0,0
Along [210] p1
a
=
1
2
bb
=
1
2
c
Origin at x,
1
2
x,0
Maximal non-isomorphic subgroups
I
[2] P311 (P3, 143) 1; 2; 3
[3] P11c (Cc,9) 1; 4
[3] P11c (Cc,9) 1; 5
[3] P11c (Cc,9) 1; 6
IIa none
IIb [3] H 31c (a
= 3a,b
= 3b)(P3c1, 158); [3] R3c (a
= a − b,b
= a + 2b,c
= 3c) (161);
[3] R3c (a
= 2a + b,b
= −a + b,c
= 3c) (161)
Maximal isomorphic subgroups of lowest index
IIc
[3] P31c (c
= 3c) (159); [4] P31c (a
= 2a,b
= 2b) (159)
Minimal non-isomorphic supergroups
I
[2] P
¯
31c (163); [2] P6cc (184); [2] P6
3
mc(186); [2] P
¯
62c (190)
II [3] H 31c (P3c1, 158); [2] P31m (c
=
1
2
c) (157)
527
R3mC
5
3v
3m Trigonal
No. 160 R3m
Patterson symmetry R
¯
3m
HEXAGONAL AXES
Origin on 3m
Asymmetric unit 0 ≤ x ≤
2
3
;0≤ y ≤
2
3
;0≤ z ≤
1
3
; x ≤ 2y; y ≤ min(1 − x,2x)
Vertices 0,0, 0
2
3
,
1
3
,0
1
3
,
2
3
,0
0,0,
1
3
2
3
,
1
3
,
1
3
1
3
,
2
3
,
1
3
Symmetry operations
For (0,0,0)+ set
(1) 1 (2) 3
+
0,0,z (3) 3
−
0,0,z
(4) mx, ¯x,z (5) mx, 2x, z (6) m 2 x,x,z
For (
2
3
,
1
3
,
1
3
)+ set
(1) t(
2
3
,
1
3
,
1
3
) (2) 3
+
(0,0,
1
3
)
1
3
,
1
3
,z (3) 3
−
(0,0,
1
3
)
1
3
,0, z
(4) g(
1
6
,−
1
6
,
1
3
) x +
1
2
, ¯x, z (5) g(
1
6
,
1
3
,
1
3
) x +
1
4
,2x, z (6) g(
2
3
,
1
3
,
1
3
) 2x,x,z
For (
1
3
,
2
3
,
2
3
)+ set
(1) t(
1
3
,
2
3
,
2
3
) (2) 3
+
(0,0,
2
3
) 0,
1
3
,z (3) 3
−
(0,0,
2
3
)
1
3
,
1
3
,z
(4) g(−
1
6
,
1
6
,
2
3
) x +
1
2
, ¯x, z (5) g(
1
3
,
2
3
,
2
3
) x,2x,z (6) g(
1
3
,
1
6
,
2
3
) 2x −
1
2
,x,z
528
International Tables for Crystallography (2006). Vol. A, Space group 160, pp. 528–531.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 160 R3m
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); t(
2
3
,
1
3
,
1
3
); (2); (4)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates
(0,0, 0)+ (
2
3
,
1
3
,
1
3
)+ (
1
3
,
2
3
,
2
3
)+
Reflection conditions
General:
18 c 1(1)x, y,z (2) ¯y,x − y,z (3) ¯x + y, ¯x,z
(4) ¯y, ¯x,z (5) ¯x + y,y, z (6) x,x − y,z
hkil : −h + k + l = 3n
hki0:−h + k = 3n
hh
2hl : l = 3n
h
¯
h0l : h + l = 3n
000l : l = 3n
h
¯
h00 : h = 3n
Special: no extra conditions
9 b . mx, ¯x, zx,2x,z 2¯x, ¯x,z
3 a 3 m 0,0, z
Symmetry of special projections
Along [001] p31m
a
=
1
3
(2a + b) b
=
1
3
(−a + b)
Origin at 0,0,z
Along [100] p1
a
=
1
2
(a + 2b) b
=
1
3
(−a − 2b + c)
Origin at x,0,0
Along [210] p1m1
a
=
1
2
bb
=
1
3
c
Origin at x,
1
2
x,0
Maximal non-isomorphic subgroups
I
[2] R31 (R3, 146) (1; 2; 3)+
[3] R1m (Cm,8) (1; 4)+
[3] R1m (Cm,8) (1; 5)+
[3] R1m (Cm,8) (1; 6)+
IIa [3] P3m1 (156) 1; 2; 3; 4; 5; 6
IIb [2] R3c (a
= −a,b
= −b,c
= 2c) (161)
Maximal isomorphic subgroups of lowest index
IIc
[2] R3m (a
= −a,b
= −b,c
= 2c) (160); [4] R3m (a
= −2a, b
= −2b) (160)
Minimal non-isomorphic supergroups
I
[2] R
¯
3m (166); [4] P
¯
43m (215); [4] F
¯
43m (216); [4] I
¯
43m (217)
II [3] P31m (a
=
1
3
(2a + b),b
=
1
3
(−a + b),c
=
1
3
c) (157)
529
R3mC
5
3v
3m Trigonal
No. 160 R3m
Patterson symmetry R
¯
3m
RHOMBOHEDRAL AXES
Origin on 3m
Asymmetric unit 0 ≤ x ≤ 1; 0 ≤ y ≤ 1; 0 ≤ z ≤ 1; y ≤ x; z ≤ y
Vertices 0,0, 01,0, 01,1,01,1, 1
Symmetry operations
(1) 1 (2) 3
+
x,x, x (3) 3
−
x,x, x
(4) mx,y, x (5) mx,x,z (6) mx,y,y
530
CONTINUED No. 160 R3m
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (4)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
General:
6 c 1(1)x,y,z (2) z,x,y (3) y, z,x
(4) z,y, x (5) y,x,z (6) x,z,y
no conditions
Special: no extra conditions
3 b . mx,x,zz,x,xx,z,x
1 a 3 mx, x,x
Symmetry of special projections
Along [111] p31m
a
=
1
3
(2a − b− c) b
=
1
3
(−a + 2b − c)
Origin at x,x, x
Along [1
¯
10] p1
a
=
1
2
(a + b − 2c ) b
= c
Origin at x, ¯x,0
Along [2
¯
1
¯
1] p1m1
a
=
1
2
(b− c) b
=
1
3
(a + b + c)
Origin at 2x, ¯x, ¯x
Maximal non-isomorphic subgroups
I
[2] R31 (R3, 146) 1; 2; 3
[3] R1m (Cm,8) 1; 4
[3] R1m (Cm,8) 1; 5
[3] R1m (Cm,8) 1; 6
IIa none
IIb [2] F 3c (a
= 2a,b
= 2b,c
= 2c)(R3c, 161); [3] P3m1(a
= a − b,b
= b − c,c
= a + b + c) (156)
Maximal isomorphic subgroups of lowest index
IIc
[2] R3m (a
= b + c,b
= a + c,c
= a + b) (160); [4] R3m (a
= −a + b + c,b
= a − b + c,c
= a + b − c) (160)
Minimal non-isomorphic supergroups
I
[2] R
¯
3m (166); [4] P
¯
43m (215); [4] F
¯
43m (216); [4] I
¯
43m (217)
II [3] P31m (a
=
1
3
(2a − b− c), b
=
1
3
(−a + 2b − c),c
=
1
3
(a + b + c)) (157)
531
R3cC
6
3v
3m Trigonal
No. 161 R3c
Patterson symmetry R
¯
3m
HEXAGONAL AXES
Origin on 3c
Asymmetric unit 0 ≤ x ≤
2
3
;0≤ y ≤
2
3
;0≤ z ≤
1
6
; x ≤ (1 + y)/2; y ≤ min(1 − x,(1 + x)/2)
Vertices 0,0, 0
1
2
,0, 0
2
3
,
1
3
,0
1
3
,
2
3
,00,
1
2
,0
0,0,
1
6
1
2
,0,
1
6
2
3
,
1
3
,
1
6
1
3
,
2
3
,
1
6
0,
1
2
,
1
6
Symmetry operations
For (0,0,0)+ set
(1) 1 (2) 3
+
0,0,z (3) 3
−
0,0,z
(4) cx, ¯x,z (5) cx, 2x,z (6) c 2x,x, z
For (
2
3
,
1
3
,
1
3
)+ set
(1) t(
2
3
,
1
3
,
1
3
) (2) 3
+
(0,0,
1
3
)
1
3
,
1
3
,z (3) 3
−
(0,0,
1
3
)
1
3
,0, z
(4) g(
1
6
,−
1
6
,
5
6
) x +
1
2
, ¯x, z (5) g(
1
6
,
1
3
,
5
6
) x +
1
4
,2x, z (6) g(
2
3
,
1
3
,
5
6
) 2x,x,z
For (
1
3
,
2
3
,
2
3
)+ set
(1) t(
1
3
,
2
3
,
2
3
) (2) 3
+
(0,0,
2
3
) 0,
1
3
,z (3) 3
−
(0,0,
2
3
)
1
3
,
1
3
,z
(4) g(−
1
6
,
1
6
,
1
6
) x +
1
2
, ¯x, z (5) g(
1
3
,
2
3
,
1
6
) x,2x,z (6) g(
1
3
,
1
6
,
1
6
) 2x −
1
2
,x,z
532
International Tables for Crystallography (2006). Vol. A, Space group 161, pp. 532–535.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 161 R3c
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); t(
2
3
,
1
3
,
1
3
); (2); (4)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates
(0,0, 0)+ (
2
3
,
1
3
,
1
3
)+ (
1
3
,
2
3
,
2
3
)+
Reflection conditions
General:
18 b 1(1)x,y,z (2) ¯y,x − y,z (3) ¯x + y, ¯x,z
(4) ¯y, ¯x,z +
1
2
(5) ¯x + y,y,z +
1
2
(6) x,x − y, z +
1
2
hkil : −h + k + l = 3n
hki0:−h + k = 3n
hh
2hl : l = 3n
h
¯
h0l : h + l = 3n, l = 2n
000l : l = 6n
h
¯
h00 : h = 3 n
Special: as above, plus
6 a 3 . 0,0, z 0,0,z +
1
2
hkil : l = 2n
Symmetry of special projections
Along [001] p31m
a
=
1
3
(2a + b) b
=
1
3
(−a + b)
Origin at 0,0,z
Along [100] p1
a
=
1
6
(2a + 4b + c) b
=
1
6
(−a − 2b + c)
Origin at x, 0,0
Along [210] p1g1
a
=
1
2
bb
=
1
3
c
Origin at x,
1
2
x,0
Maximal non-isomorphic subgroups
I
[2] R31 (R3, 146) (1; 2; 3)+
[3] R1c (Cc,9) (1; 4)+
[3] R1c (Cc,9) (1; 5)+
[3] R1c (Cc,9) (1; 6)+
IIa [3] P3c1 (158) 1; 2; 3; 4; 5; 6
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[4] R3c (a
= −2a, b
= −2b) (161); [5] R3c (a
= −a,b
= −b,c
= 5c) (161)
Minimal non-isomorphic supergroups
I
[2] R
¯
3c (167); [4] P
¯
43n (218); [4] F
¯
43c (219); [4] I
¯
43d (220)
II [2] R3m (a
= −a,b
= −b,c
=
1
2
c) (160); [3] P31c (a
=
1
3
(2a + b),b
=
1
3
(−a + b),c
=
1
3
c) (159)
533