Hahn Th. (ed.) International tables for crystallography. Vol. A. Space-group symmetry
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P6
1
C
2
6
6 Hexagonal
No. 169 P6
1
Patterson symmetry P6/m
Origin on 6
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(0,0,
1
2
) 0,0,z (5) 6
−
(0,0,
5
6
) 0,0,z (6) 6
+
(0,0,
1
6
) 0,0, z
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 a 1(1)x, y,z (2) ¯y,x − y,z +
1
3
(3) ¯x+ y, ¯x,z +
2
3
(4) ¯x, ¯y, z +
1
2
(5) y, ¯x + y, z +
5
6
(6) x − y,x,z +
1
6
000l : l = 6n
Symmetry of special projections
Along [001] p6
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] p1g1
a
=
1
2
bb
= c
Origin at x,
1
2
x,0
Maximal non-isomorphic subgroups
I
[2] P3
1
(144) 1; 2; 3
[3] P2
1
(4) 1; 4
IIa none
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] H 6
1
(a
= 3a,b
= 3b)(P6
1
, 169); [5] P6
5
(c
= 5c) (170); [7] P6
1
(c
= 7c) (169)
Minimal non-isomorphic supergroups
I
[2] P6
1
22 (178)
II [2] P6
2
(c
=
1
2
c) (171); [3] P6
3
(c
=
1
3
c) (173)
554
International Tables for Crystallography (2006). Vol. A, Space group 169, p. 554.
Copyright © 2006 International Union of Crystallography
Hexagonal 6 C
3
6
P6
5
Patterson symmetry P6/m P6
5
No. 170
Origin on 6
5
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(0,0,
1
2
) 0,0,z (5) 6
−
(0,0,
1
6
) 0,0,z (6) 6
+
(0,0,
5
6
) 0,0, z
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 a 1(1)x, y,z (2) ¯y,x − y,z +
2
3
(3) ¯x+ y, ¯x,z +
1
3
(4) ¯x, ¯y, z +
1
2
(5) y, ¯x + y, z +
1
6
(6) x − y,x,z +
5
6
000l : l = 6n
Symmetry of special projections
Along [001] p6
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] p1g1
a
=
1
2
bb
= c
Origin at x,
1
2
x,0
Maximal non-isomorphic subgroups
I
[2] P3
2
(145) 1; 2; 3
[3] P2
1
(4) 1; 4
IIa none
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] H 6
5
(a
= 3a,b
= 3b)(P6
5
, 170); [5] P6
1
(c
= 5c) (169); [7] P6
5
(c
= 7c) (170)
Minimal non-isomorphic supergroups
I
[2] P6
5
22 (179)
II [2] P6
4
(c
=
1
2
c) (172); [3] P6
3
(c
=
1
3
c) (173)
555
International Tables for Crystallography (2006). Vol. A, Space group 170, p. 555.
Copyright © 2006 International Union of Crystallography
P6
2
C
4
6
6 Hexagonal
No. 171 P6
2
Patterson symmetry P6/m
Origin on 2 on 6
2
Asymmetric unit 0 ≤ x ≤ 1; 0 ≤ y ≤ 1; 0 ≤ z ≤
1
3
; y ≤ x
Vertices 0,0, 01,0, 01,1,0
0,0,
1
3
1,0,
1
3
1,1,
1
3
Symmetry operations
(1) 1 (2) 3
+
(0,0,
2
3
) 0,0,z (3) 3
−
(0,0,
1
3
) 0,0, z
(4) 2 0,0,z (5) 6
−
(0,0,
2
3
) 0,0,z (6) 6
+
(0,0,
1
3
) 0,0, z
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) ¯x, ¯y, z (5) y, ¯x + y,z +
2
3
(6) x − y,x, z +
1
3
000l : l = 3n
Special: as above, plus
3 b 2 ..
1
2
,
1
2
,z
1
2
,0, z +
2
3
0,
1
2
,z +
1
3
hkil : h = 2n + 1
or k = 2 n + 1
or l = 3n
3 a 2 .. 0,0,z 0,0,z +
2
3
0,0,z +
1
3
hkil : l = 3n
Symmetry of special projections
Along [001] p6
a
= ab
= b
Origin at 0,0,z
Along [100] p1m1
a
=
1
2
(a + 2b) b
= c
Origin at x,0,0
Along [210] p1m1
a
=
1
2
bb
= c
Origin at x,
1
2
x,0
Maximal non-isomorphic subgroups
I
[2] P3
2
(145) 1; 2; 3
[3] P2 (3) 1; 4
IIa none
IIb [2] P6
1
(c
= 2c) (169)
Maximal isomorphic subgroups of lowest index
IIc
[2] P6
4
(c
= 2c) (172); [3] H 6
2
(a
= 3a,b
= 3b)(P6
2
, 171); [7] P6
2
(c
= 7c) (171)
Minimal non-isomorphic supergroups
I
[2] P6
2
22 (180)
II [3] P6(c
=
1
3
c) (168)
556
International Tables for Crystallography (2006). Vol. A, Space group 171, p. 556.
Copyright © 2006 International Union of Crystallography
Hexagonal 6 C
5
6
P6
4
Patterson symmetry P6/m P6
4
No. 172
Origin on 2 on 6
4
Asymmetric unit 0 ≤ x ≤ 1; 0 ≤ y ≤ 1; 0 ≤ z ≤
1
3
; y ≤ x
Vertices 0,0, 01,0, 01,1,0
0,0,
1
3
1,0,
1
3
1,1,
1
3
Symmetry operations
(1) 1 (2) 3
+
(0,0,
1
3
) 0,0,z (3) 3
−
(0,0,
2
3
) 0,0, z
(4) 2 0,0,z (5) 6
−
(0,0,
1
3
) 0,0,z (6) 6
+
(0,0,
2
3
) 0,0, z
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) ¯x, ¯y, z (5) y, ¯x + y,z +
1
3
(6) x − y,x, z +
2
3
000l : l = 3n
Special: as above, plus
3 b 2 ..
1
2
,
1
2
,z
1
2
,0, z +
1
3
0,
1
2
,z +
2
3
hkil : h = 2n + 1
or k = 2 n + 1
or l = 3n
3 a 2 .. 0,0,z 0,0,z +
1
3
0,0,z +
2
3
hkil : l = 3n
Symmetry of special projections
Along [001] p6
a
= ab
= b
Origin at 0,0,z
Along [100] p1m1
a
=
1
2
(a + 2b) b
= c
Origin at x,0,0
Along [210] p1m1
a
=
1
2
bb
= c
Origin at x,
1
2
x,0
Maximal non-isomorphic subgroups
I
[2] P3
1
(144) 1; 2; 3
[3] P2 (3) 1; 4
IIa none
IIb [2] P6
5
(c
= 2c) (170)
Maximal isomorphic subgroups of lowest index
IIc
[2] P6
2
(c
= 2c) (171); [3] H 6
4
(a
= 3a,b
= 3b)(P6
4
, 172); [7] P6
4
(c
= 7c) (172)
Minimal non-isomorphic supergroups
I
[2] P6
4
22 (181)
II [3] P6(c
=
1
3
c) (168)
557
International Tables for Crystallography (2006). Vol. A, Space group 172, p. 557.
Copyright © 2006 International Union of Crystallography
P6
3
C
6
6
6 Hexagonal
No. 173 P6
3
Patterson symmetry P6/m
Origin on 3 on 6
3
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(0,0,
1
2
) 0,0,z (5) 6
−
(0,0,
1
2
) 0,0,z (6) 6
+
(0,0,
1
2
) 0,0, z
558
International Tables for Crystallography (2006). Vol. A, Space group 173, pp. 558–559.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 173 P6
3
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) ¯x, ¯y,z +
1
2
(5) y, ¯x + y, z +
1
2
(6) x − y,x, z +
1
2
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] p6
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] p1g1
a
=
1
2
bb
= c
Origin at x,
1
2
x,0
Maximal non-isomorphic subgroups
I
[2] P3 (143) 1; 2; 3
[3] P2
1
(4) 1; 4
IIa none
IIb [3] P6
5
(c
= 3c) (170); [3] P6
1
(c
= 3c) (169)
Maximal isomorphic subgroups of lowest index
IIc
[3] P6
3
(c
= 3c) (173); [3] H 6
3
(a
= 3a,b
= 3b)(P6
3
, 173)
Minimal non-isomorphic supergroups
I
[2] P6
3
/m (176); [2] P6
3
22 (182); [2] P6
3
cm (185); [2] P6
3
mc(186)
II [2] P6(c
=
1
2
c) (168)
559
P
¯
6 C
1
3h
¯
6 Hexagonal
No. 174 P
¯
6
Patterson symmetry P6/m
Origin at
¯
6
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) mx,y, 0(5)
¯
6
−
0,0,z;0,0, 0(6)
¯
6
+
0,0,z;0,0, 0
560
International Tables for Crystallography (2006). Vol. A, Space group 174, pp. 560–561.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 174 P
¯
6
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) x,y, ¯z (5) ¯y,x − y, ¯z (6) ¯x + y, ¯x, ¯z
no conditions
Special: no extra conditions
3 km.. x,y,
1
2
¯y,x − y,
1
2
¯x + y, ¯x,
1
2
3 jm.. x,y, 0¯y,x − y,0¯x + y, ¯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
¯
6 ..
2
3
,
1
3
,
1
2
1 e
¯
6 ..
2
3
,
1
3
,0
1 d
¯
6 ..
1
3
,
2
3
,
1
2
1 c
¯
6 ..
1
3
,
2
3
,0
1 b
¯
6 .. 0,0,
1
2
1 a
¯
6 .. 0,0,0
Symmetry of special projections
Along [001] p3
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] p11m
a
=
1
2
bb
= c
Origin at x,
1
2
x,0
Maximal non-isomorphic subgroups
I
[2] P3 (143) 1; 2; 3
[3] Pm(6) 1; 4
IIa none
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[2] P
¯
6(c
= 2c) (174); [3] H
¯
6(a
= 3a,b
= 3b)(P
¯
6, 174)
Minimal non-isomorphic supergroups
I
[2] P6/m (175); [2] P6
3
/m (176); [2] P
¯
6m2 (187); [2] P
¯
6c2 (188); [2] P
¯
62m (189); [2] P
¯
62c (190)
II none
561
P6/mC
1
6h
6/m Hexagonal
No. 175 P6/m
Patterson symmetry P6/m
Origin at centre (6/m)
Asymmetric unit 0 ≤ x ≤
2
3
;0≤ y ≤
1
2
;0≤ z ≤
1
2
; x ≤ (1 + y)/2; y ≤ min(1 − x,x)
Vertices 0,0, 0
1
2
,0, 0
2
3
,
1
3
,0
1
2
,
1
2
,0
0,0,
1
2
1
2
,0,
1
2
2
3
,
1
3
,
1
2
1
2
,
1
2
,
1
2
Symmetry operations
(1) 1 (2) 3
+
0,0,z (3) 3
−
0,0,z
(4) 2 0,0,z (5) 6
−
0,0,z (6) 6
+
0,0,z
(7)
¯
10, 0,0(8)
¯
3
+
0,0,z;0,0,0(9)
¯
3
−
0,0,z;0,0,0
(10) mx,y, 0 (11)
¯
6
−
0,0,z;0,0,0 (12)
¯
6
+
0,0,z;0,0,0
562
International Tables for Crystallography (2006). Vol. A, Space group 175, pp. 562–563.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 175 P6/m
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (4); (7)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
General:
12 l 1(1)x, y,z (2) ¯y,x − y,z (3) ¯x+ y, ¯x,z
(4) ¯x, ¯y, z (5) y, ¯x + y,z (6) x − y,x , z
(7) ¯x, ¯y, ¯z (8) y, ¯x + y, ¯z (9) x − y,x, ¯z
(10) x,y, ¯z (11) ¯y,x − y, ¯z (12) ¯x+ y, ¯x
, ¯z
no conditions
Special: no extra conditions
6 km.. x,y,
1
2
¯y,x − y,
1
2
¯x + y, ¯x,
1
2
¯x, ¯y,
1
2
y, ¯x + y,
1
2
x − y, x,
1
2
6 jm.. x,y,0¯y, x − y,0¯x + y, ¯x,0¯x, ¯y,0 y, ¯x + y,0 x− y,x,0
6 i 2 ..
1
2
,0, z 0,
1
2
,z
1
2
,
1
2
,z
1
2
,0, ¯z 0,
1
2
, ¯z
1
2
,
1
2
, ¯z
4 h 3 ..
1
3
,
2
3
,z
2
3
,
1
3
,z
2
3
,
1
3
, ¯z
1
3
,
2
3
, ¯z
3 g 2/ m ..
1
2
,0,
1
2
0,
1
2
,
1
2
1
2
,
1
2
,
1
2
3 f 2/m ..
1
2
,0, 00,
1
2
,0
1
2
,
1
2
,0
2 e 6 .. 0, 0,z 0,0, ¯z
2 d
¯
6 ..
1
3
,
2
3
,
1
2
2
3
,
1
3
,
1
2
2 c
¯
6 ..
1
3
,
2
3
,0
2
3
,
1
3
,0
1 b 6/ m .. 0, 0,
1
2
1 a 6/ m .. 0, 0,0
Symmetry of special projections
Along [001] p6
a
= ab
= b
Origin at 0,0,z
Along [100] p2mm
a
=
1
2
(a + 2b) b
= c
Origin at x,0,0
Along [210] p2mm
a
=
1
2
bb
= c
Origin at x,
1
2
x,0
Maximal non-isomorphic subgroups
I
[2] P
¯
6 (174) 1; 2; 3; 10; 11; 12
[2] P6 (168) 1; 2; 3; 4; 5; 6
[2] P
¯
3 (147) 1; 2; 3; 7; 8; 9
[3] P2/m (10) 1; 4; 7; 10
IIa none
IIb [2] P6
3
/m (c
= 2c) (176)
Maximal isomorphic subgroups of lowest index
IIc
[2] P6/m (c
= 2c) (175); [3] H 6/m (a
= 3a,b
= 3b)(P6/m, 175)
Minimal non-isomorphic supergroups
I
[2] P6/mmm(191); [2] P6/mcc (192)
II none
563