Hahn Th. (ed.) International tables for crystallography. Vol. A. Space-group symmetry
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P312 D
1
3
312 Trigonal
No. 149 P312
Patterson symmetry P
¯
31m
Origin at 312
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) 2 x, ¯x, 0(5)2x, 2x,0(6)22x, x,0
504
International Tables for Crystallography (2006). Vol. A, Space group 149, pp. 504–505.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 149 P312
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 l 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
no conditions
Special: no extra conditions
3 k ..2 x, ¯x,
1
2
x,2x,
1
2
2¯x, ¯x,
1
2
3 j ..2 x, ¯x,0 x,2x,02¯x, ¯x, 0
2 i 3 ..
2
3
,
1
3
,z
2
3
,
1
3
, ¯z
2 h 3 ..
1
3
,
2
3
,z
1
3
,
2
3
, ¯z
2 g 3 .. 0,0,z 0,0, ¯z
1 f 3 . 2
2
3
,
1
3
,
1
2
1 e 3 . 2
2
3
,
1
3
,0
1 d 3 . 2
1
3
,
2
3
,
1
2
1 c 3 . 2
1
3
,
2
3
,0
1 b 3 . 20,0,
1
2
1 a 3 . 20,0,0
Symmetry of special projections
Along [001] p3m1
a
= ab
= b
Origin at 0,0,z
Along [100] p11m
a
=
1
2
(a + 2b) b
= c
Origin at x,0,0
Along [210] p2
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] P112 (C2, 5) 1; 4
[3] P112 (C2, 5) 1; 5
[3] P112 (C2, 5) 1; 6
IIa none
IIb [3] P3
2
12(c
= 3c) (153); [3] P3
1
12(c
= 3c) (151); [3] H 312(a
= 3a,b
= 3b)(P321, 150);
[3] R32 (a
= a − b,b
= a + 2b,c
= 3c) (155); [3] R32(a
= 2a + b,b
= −a + b,c
= 3c) (155)
Maximal isomorphic subgroups of lowest index
IIc
[2] P312 (c
= 2c) (149); [4] P312 (a
= 2a,b
= 2b) (149)
Minimal non-isomorphic supergroups
I
[2] P
¯
31m (162); [2] P
¯
31c (163); [2] P622 (177); [2] P6
3
22 (182); [2] P
¯
6m2 (187); [2] P
¯
6c2 (188)
II [3] H 312(P321, 150)
505
P321 D
2
3
321 Trigonal
No. 150 P321
Patterson symmetry P
¯
3m1
Origin at 321
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) 2 x,x,0(5)2x,0,0(6)20, y,0
506
International Tables for Crystallography (2006). Vol. A, Space group 150, pp. 506–507.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 150 P321
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 g 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
no conditions
Special: no extra conditions
3 f . 2 . x, 0,
1
2
0,x,
1
2
¯x, ¯x,
1
2
3 e . 2 . x,0, 00,x,0¯x, ¯x,0
2 d 3 ..
1
3
,
2
3
,z
2
3
,
1
3
, ¯z
2 c 3 .. 0,0, z 0,0, ¯z
1 b 32. 0,0,
1
2
1 a 32. 0,0,0
Symmetry of special projections
Along [001] p31m
a
= ab
= b
Origin at 0,0,z
Along [100] p2
a
=
1
2
(a + 2b) b
= c
Origin at x,0,0
Along [210] p11m
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] P121 (C2, 5) 1; 4
[3] P121 (C2, 5) 1; 5
[3] P121 (C2, 5) 1; 6
IIa none
IIb [3] P3
2
21(c
= 3c) (154); [3] P3
1
21(c
= 3c) (152); [3] H 321(a
= 3a,b
= 3b)(P312, 149)
Maximal isomorphic subgroups of lowest index
IIc
[2] P321 (c
= 2c) (150); [4] P321 (a
= 2a,b
= 2b) (150)
Minimal non-isomorphic supergroups
I
[2] P
¯
3m1 (164); [2] P
¯
3c1 (165); [2] P622 (177); [2] P6
3
22 (182); [2] P
¯
62m (189); [2] P
¯
62c (190)
II [3] H 321(P312, 149); [3] R32 (obverse) (155); [3] R32 (reverse) (155)
507
P3
1
12 D
3
3
312 Trigonal
No. 151 P3
1
12 Patterson symmetry P
¯
31m
Origin on 2[210] at 3
1
1(1, 1,2)
Asymmetric unit 0 ≤ x ≤ 1; 0 ≤ y ≤ 1; 0 ≤ z ≤
1
6
Vertices 0,0, 01,0, 01,1,00,1, 0
0,0,
1
6
1,0,
1
6
1,1,
1
6
0,1,
1
6
Symmetry operations
(1) 1 (2) 3
+
(0,0,
1
3
) 0,0, z (3) 3
−
(0,0,
2
3
) 0,0,z
(4) 2 x, ¯x,
1
3
(5) 2 x,2x,
1
6
(6) 2 2x,x,0
508
International Tables for Crystallography (2006). Vol. A, Space group 151, pp. 508–509.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 151 P3
1
12
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 +
1
3
(3) ¯x+ y, ¯x,z +
2
3
(4) ¯y, ¯x, ¯z +
2
3
(5) ¯x + y,y, ¯z +
1
3
(6) x,x − y, ¯z
000l : l = 3n
Special: no extra conditions
3 b ..2 x, ¯x ,
5
6
x,2x,
1
6
2¯x, ¯x,
1
2
3 a ..2 x, ¯x ,
1
3
x,2x,
2
3
2¯x, ¯x,0
Symmetry of special projections
Along [001] p3m1
a
= ab
= b
Origin at 0,0,z
Along [100] p11m
a
=
1
2
(a + 2b) b
= c
Origin at x,0,
1
6
Along [210] p2
a
=
1
2
bb
= c
Origin at x,
1
2
x,0
Maximal non-isomorphic subgroups
I
[2] P3
1
11(P3
1
, 144) 1; 2; 3
[3] P112 (C2, 5) 1; 4
[3] P112 (C2, 5) 1; 5
[3] P112 (C2, 5) 1; 6
IIa none
IIb [3] H 3
1
12(a
= 3a,b
= 3b)(P3
1
21, 152)
Maximal isomorphic subgroups of lowest index
IIc
[2] P3
2
12(c
= 2c) (153); [4] P3
1
12(a
= 2a,b
= 2b) (151); [7] P3
1
12(c
= 7c) (151)
Minimal non-isomorphic supergroups
I
[2] P6
1
22 (178); [2] P6
4
22 (181)
II [3] H 3
1
12(P3
1
21, 152); [3] P312(c
=
1
3
c) (149)
509
P3
1
21 D
4
3
321 Trigonal
No. 152 P3
1
21 Patterson symmetry P
¯
3m1
Origin on 2[110] at 3
1
(1,1, 2)1
Asymmetric unit 0 ≤ x ≤ 1; 0 ≤ y ≤ 1; 0 ≤ z ≤
1
6
Vertices 0,0, 01,0, 01,1,00,1, 0
0,0,
1
6
1,0,
1
6
1,1,
1
6
0,1,
1
6
Symmetry operations
(1) 1 (2) 3
+
(0,0,
1
3
) 0,0,z (3) 3
−
(0,0,
2
3
) 0,0,z
(4) 2 x,x,0(5)2x,0,
1
3
(6) 2 0, y,
1
6
510
International Tables for Crystallography (2006). Vol. A, Space group 152, pp. 510–511.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 152 P3
1
21
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 +
1
3
(3) ¯x+ y, ¯x,z +
2
3
(4) y,x, ¯z (5) x − y, ¯y, ¯z +
2
3
(6) ¯x, ¯x + y, ¯z+
1
3
000l : l = 3n
Special: no extra conditions
3 b . 2 . x,0,
5
6
0,x,
1
6
¯x, ¯x,
1
2
3 a . 2 . x,0,
1
3
0,x,
2
3
¯x, ¯x, 0
Symmetry of special projections
Along [001] p31m
a
= ab
= b
Origin at 0,0,z
Along [100] p2
a
=
1
2
(a + 2b) b
= c
Origin at x,0,
1
3
Along [210] p11m
a
=
1
2
bb
= c
Origin at x,
1
2
x,
1
6
Maximal non-isomorphic subgroups
I
[2] P3
1
11(P3
1
, 144) 1; 2; 3
[3] P121 (C2, 5) 1; 4
[3] P121 (C2, 5) 1; 5
[3] P121 (C2, 5) 1; 6
IIa none
IIb [3] H 3
1
21(a
= 3a,b
= 3b)(P3
1
12, 151)
Maximal isomorphic subgroups of lowest index
IIc
[2] P3
2
21(c
= 2c) (154); [4] P3
1
21(a
= 2a,b
= 2b) (152); [7] P3
1
21(c
= 7c) (152)
Minimal non-isomorphic supergroups
I
[2] P6
1
22 (178); [2] P6
4
22 (181)
II [3] H 3
1
21(P3
1
12, 151); [3] R32 (obverse) (155); [3] R32 (reverse) (155); [3] P321(c
=
1
3
c) (150)
511
P3
2
12 D
5
3
312 Trigonal
No. 153 P3
2
12 Patterson symmetry P
¯
31m
Origin on 2[210] at 3
2
1(1, 1,2)
Asymmetric unit 0 ≤ x ≤ 1; 0 ≤ y ≤ 1; 0 ≤ z ≤
1
6
Vertices 0,0, 01,0, 01,1,00,1, 0
0,0,
1
6
1,0,
1
6
1,1,
1
6
0,1,
1
6
Symmetry operations
(1) 1 (2) 3
+
(0,0,
2
3
) 0,0, z (3) 3
−
(0,0,
1
3
) 0,0,z
(4) 2 x, ¯x,
1
6
(5) 2 x,2x,
1
3
(6) 2 2x,x,0
512
International Tables for Crystallography (2006). Vol. A, Space group 153, pp. 512–513.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 153 P3
2
12
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 +
2
3
(3) ¯x+ y, ¯x,z +
1
3
(4) ¯y, ¯x, ¯z +
1
3
(5) ¯x + y,y, ¯z +
2
3
(6) x,x − y, ¯z
000l : l = 3n
Special: no extra conditions
3 b ..2 x, ¯x ,
1
6
x,2x,
5
6
2¯x, ¯x,
1
2
3 a ..2 x, ¯x ,
2
3
x,2x,
1
3
2¯x, ¯x,0
Symmetry of special projections
Along [001] p3m1
a
= ab
= b
Origin at 0,0,z
Along [100] p11m
a
=
1
2
(a + 2b) b
= c
Origin at x,0,
1
3
Along [210] p2
a
=
1
2
bb
= c
Origin at x,
1
2
x,0
Maximal non-isomorphic subgroups
I
[2] P3
2
11(P3
2
, 145) 1; 2; 3
[3] P112 (C2, 5) 1; 4
[3] P112 (C2, 5) 1; 5
[3] P112 (C2, 5) 1; 6
IIa none
IIb [3] H 3
2
12(a
= 3a,b
= 3b)(P3
2
21, 154)
Maximal isomorphic subgroups of lowest index
IIc
[2] P3
1
12(c
= 2c) (151); [4] P3
2
12(a
= 2a,b
= 2b) (153); [7] P3
2
12(c
= 7c) (153)
Minimal non-isomorphic supergroups
I
[2] P6
5
22 (179); [2] P6
2
22 (180)
II [3] H 3
2
12(P3
2
21, 154); [3] P312(c
=
1
3
c) (149)
513