Hahn Th. (ed.) International tables for crystallography. Vol. A. Space-group symmetry
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P4/mnc D
6
4h
4/mmm Tetragonal
No. 128 P 4/m 2
1
/n 2/c Patterson symmetry P4/mmm
Origin at centre (4/m) at 4/m1n
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
2
;0≤ z ≤
1
4
Symmetry operations
(1) 1 (2) 2 0,0,z (3) 4
+
0,0,z (4) 4
−
0,0,z
(5) 2(0,
1
2
,0)
1
4
,y,
1
4
(6) 2(
1
2
,0, 0) x,
1
4
,
1
4
(7) 2(
1
2
,
1
2
,0) x,x,
1
4
(8) 2 x, ¯x +
1
2
,
1
4
(9)
¯
10, 0,0 (10) mx, y,0 (11)
¯
4
+
0,0,z;0,0,0 (12)
¯
4
−
0,0,z;0,0, 0
(13) n(
1
2
,0,
1
2
) x,
1
4
,z (14) n(0,
1
2
,
1
2
)
1
4
,y,z (15) cx+
1
2
, ¯x, z (16) n(
1
2
,
1
2
,
1
2
) x,x,z
444
International Tables for Crystallography (2006). Vol. A, Space group 128, pp. 444–445.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 128 P4/mnc
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3); (5); (9)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
General:
16 i 1(1)x, y,z (2) ¯x, ¯y, z (3) ¯y,x,z (4) y, ¯x,z
(5) ¯x+
1
2
,y +
1
2
, ¯z +
1
2
(6) x +
1
2
, ¯y +
1
2
, ¯z +
1
2
(7) y +
1
2
,x +
1
2
, ¯z +
1
2
(8) ¯y +
1
2
, ¯x +
1
2
, ¯z +
1
2
(9) ¯x, ¯y, ¯z (10) x,y, ¯z (11) y, ¯x, ¯z (12) ¯y,x, ¯z
(13) x +
1
2
, ¯y +
1
2
,z +
1
2
(14) ¯x +
1
2
,y +
1
2
,z +
1
2
(15) ¯y +
1
2
, ¯x +
1
2
,z +
1
2
(16) y +
1
2
,x +
1
2
,z +
1
2
0kl : k + l = 2n
hhl : l = 2n
00l : l = 2n
h00 : h = 2n
Special: as above, plus
8 hm.. x,y,0¯x, ¯y,0¯y,x, 0 y, ¯x,0
¯x +
1
2
,y +
1
2
,
1
2
x +
1
2
, ¯y +
1
2
,
1
2
y +
1
2
,x +
1
2
,
1
2
¯y +
1
2
, ¯x +
1
2
,
1
2
no extra conditions
8 g ..2 x,x +
1
2
,
1
4
¯x, ¯x +
1
2
,
1
4
¯x+
1
2
,x,
1
4
x +
1
2
, ¯x,
1
4
¯x, ¯x +
1
2
,
3
4
x,x +
1
2
,
3
4
x +
1
2
, ¯x,
3
4
¯x +
1
2
,x,
3
4
hkl : l = 2n
8 f 2 .. 0,
1
2
,z
1
2
,0, z
1
2
,0, ¯z +
1
2
0,
1
2
, ¯z +
1
2
0,
1
2
, ¯z
1
2
,0, ¯z
1
2
,0, z +
1
2
0,
1
2
,z +
1
2
hkl : h + k,l = 2n
4 e 4 .. 0, 0,z
1
2
,
1
2
, ¯z +
1
2
0,0, ¯z
1
2
,
1
2
,z +
1
2
hkl : h + k + l = 2n
4 d 2 . 22 0,
1
2
,
1
4
1
2
,0,
1
4
0,
1
2
,
3
4
1
2
,0,
3
4
hkl : h + k,l = 2n
4 c 2/m .. 0,
1
2
,0
1
2
,0, 0
1
2
,0,
1
2
0,
1
2
,
1
2
hkl : h + k,l = 2n
2 b 4/ m .. 0, 0,
1
2
1
2
,
1
2
,0 hkl : h + k + l = 2n
2 a 4/ m .. 0, 0,0
1
2
,
1
2
,
1
2
hkl : h + k + l = 2n
Symmetry of special projections
Along [001] p4gm
a
= ab
= b
Origin at 0,0,z
Along [100] c2mm
a
= bb
= c
Origin at x, 0,0
Along [110] p2mm
a
=
1
2
(−a + b) b
=
1
2
c
Origin at x,x , 0
Maximal non-isomorphic subgroups
I
[2] P
¯
4n2 (118) 1; 2; 7; 8; 11; 12; 13; 14
[2] P
¯
42
1
c (114) 1; 2; 5; 6; 11; 12; 15; 16
[2] P4nc (104) 1; 2; 3; 4; 13; 14; 15; 16
[2] P42
1
2 (90) 1; 2; 3; 4; 5; 6; 7; 8
[2] P4/m11(P4/m, 83) 1; 2; 3; 4; 9; 10; 11; 12
[2] P2/m12/c (Cccm, 66) 1; 2; 7; 8; 9; 10; 15; 16
[2] P2/m2
1
/n1(Pnnm, 58) 1; 2; 5; 6; 9; 10; 13; 14
IIa none
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] P4/mnc(c
= 3c) (128); [9] P4/mnc(a
= 3a,b
= 3b) (128)
Minimal non-isomorphic supergroups
I
none
II [2] C 4/mcc (P4/mcc, 124); [2] I 4/mmm(139); [2] P4/mbm (c
=
1
2
c) (127)
445
P4/nmm D
7
4h
4/mmm Tetragonal
No. 129 P 4/n 2
1
/m 2 /m Patterson symmetry P4/mmm
ORIGIN CHOICE 1
Origin at
¯
4m2at
¯
4/nm2/g,at−
1
4
,
1
4
,0 from centre (2/m)
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
2
;0≤ z ≤
1
2
; y ≤
1
2
− x
Symmetry operations
(1) 1 (2) 2 0, 0,z (3) 4
+
0,
1
2
,z (4) 4
−
1
2
,0, z
(5) 2(0,
1
2
,0)
1
4
,y,0(6)2(
1
2
,0, 0) x,
1
4
,0(7)2x,x, 0(8)2x, ¯x,0
(9)
¯
1
1
4
,
1
4
,0 (10) n(
1
2
,
1
2
,0) x,y,0 (11)
¯
4
+
0,0,z;0,0, 0 (12)
¯
4
−
0,0,z;0,0, 0
(13) mx,0,z (14) m 0, y,z (15) mx+
1
2
, ¯x, z (16) g(
1
2
,
1
2
,0) x,x,z
446
International Tables for Crystallography (2006). Vol. A, Space group 129, pp. 446–449.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 129 P4/nmm
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3); (5); (9)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
General:
16 k 1(1)x,y, z (2) ¯x, ¯y,z (3) ¯y +
1
2
,x +
1
2
,z (4) y +
1
2
, ¯x +
1
2
,z
(5) ¯x +
1
2
,y +
1
2
, ¯z (6) x +
1
2
, ¯y +
1
2
, ¯z (7) y, x, ¯z (8) ¯y, ¯x, ¯z
(9) ¯x +
1
2
, ¯y +
1
2
, ¯z (10) x +
1
2
,y +
1
2
, ¯z (11) y, ¯x, ¯z (12) ¯y,x, ¯z
(13) x, ¯y,z (14) ¯x,y,z (15) ¯y +
1
2
, ¯x +
1
2
,z (16) y +
1
2
,x +
1
2
,z
hk0: h + k = 2n
h00 : h = 2n
Special: as above, plus
8 j ..mx,x +
1
2
,z ¯x, ¯x +
1
2
,z ¯x,x +
1
2
,zx, ¯x +
1
2
,z
¯x+
1
2
,x, ¯zx+
1
2
, ¯x, ¯zx+
1
2
,x, ¯z ¯x +
1
2
, ¯x, ¯z
no extra conditions
8 i . m . 0,y, z 0, ¯y,z ¯y +
1
2
,
1
2
,zy+
1
2
,
1
2
,z
1
2
,y +
1
2
, ¯z
1
2
, ¯y +
1
2
, ¯zy,0, ¯z ¯y,0, ¯z
no extra conditions
8 h ..2 x,x,
1
2
¯x, ¯x,
1
2
¯x+
1
2
,x +
1
2
,
1
2
x +
1
2
, ¯x +
1
2
,
1
2
¯x+
1
2
, ¯x +
1
2
,
1
2
x +
1
2
,x +
1
2
,
1
2
x, ¯x,
1
2
¯x,x,
1
2
hkl : h + k = 2n
8 g ..2 x,x,0¯x, ¯x,0¯x +
1
2
,x +
1
2
,0 x +
1
2
, ¯x +
1
2
,0
¯x+
1
2
, ¯x +
1
2
,0 x +
1
2
,x +
1
2
,0 x, ¯x,0¯x,x, 0
hkl : h + k = 2n
4 f 2 mm. 0, 0,z
1
2
,
1
2
,z
1
2
,
1
2
, ¯z 0,0, ¯zhkl: h + k = 2n
4 e ..2/m
1
4
,
1
4
,
1
2
3
4
,
3
4
,
1
2
1
4
,
3
4
,
1
2
3
4
,
1
4
,
1
2
hkl : h,k = 2n
4 d ..2/m
1
4
,
1
4
,0
3
4
,
3
4
,0
1
4
,
3
4
,0
3
4
,
1
4
,0 hkl : h, k = 2n
2 c 4 mm 0,
1
2
,z
1
2
,0, ¯z no extra conditions
2 b
¯
4 m 20,0,
1
2
1
2
,
1
2
,
1
2
hkl : h + k = 2n
2 a
¯
4 m 20,0,0
1
2
,
1
2
,0 hkl : h + k = 2n
Symmetry of special projections
Along [001] p4mm
a
=
1
2
(a − b) b
=
1
2
(a + b)
Origin at 0,0,z
Along [100] p2mg
a
= bb
= c
Origin at x,
1
4
,0
Along [110] p2mm
a
=
1
2
(−a + b) b
= c
Origin at x, x,0
Maximal non-isomorphic subgroups
I
[2] P
¯
4m2 (115) 1; 2; 7; 8; 11; 12; 13; 14
[2] P
¯
42
1
m (113) 1; 2; 5; 6; 11; 12; 15; 16
[2] P4mm (99) 1; 2; 3; 4; 13; 14; 15; 16
[2] P42
1
2 (90) 1; 2; 3; 4; 5; 6; 7; 8
[2] P4/n11(P4/n, 85) 1; 2; 3; 4; 9; 10; 11; 12
[2] P2/n12/m (Cmme, 67) 1; 2; 7; 8; 9; 10; 15; 16
[2] P2/n2
1
/m1(Pmmn, 59) 1; 2; 5; 6; 9; 10; 13; 14
IIa none
IIb [2] P4
2
/ncm( c
= 2c) (138); [2] P4
2
/nmc(c
= 2c) (137); [2] P4/ncc(c
= 2c) (130)
Maximal isomorphic subgroups of lowest index
IIc
[2] P4/nmm(c
= 2c) (129); [9] P4/nmm(a
= 3a,b
= 3b) (129)
Minimal non-isomorphic supergroups
I
none
II [2] C 4/mmm (P4/mmm, 123); [2] I 4/mmm(139)
447
P4/nmm D
7
4h
4/mmm Tetragonal
No. 129 P 4/n 2
1
/m 2 /m Patterson symmetry P4/mmm
ORIGIN CHOICE 2
Origin at centre (2/m) at n2
1
(2/m,2
1
/g),at
1
4
,−
1
4
,0 from
¯
4m2
Asymmetric unit −
1
4
≤ x ≤
1
4
; −
1
4
≤ y ≤
1
4
;0≤ z ≤
1
2
; x ≤ y
Symmetry operations
(1) 1 (2) 2
1
4
,
1
4
,z (3) 4
+
1
4
,
1
4
,z (4) 4
−
1
4
,
1
4
,z
(5) 2(0,
1
2
,0) 0,y,0(6)2(
1
2
,0, 0) x,0,0(7)2(
1
2
,
1
2
,0) x,x,0(8)2x, ¯x,0
(9)
¯
10, 0,0 (10) n(
1
2
,
1
2
,0) x, y,0 (11)
¯
4
+
1
4
,−
1
4
,z;
1
4
,−
1
4
,0 (12)
¯
4
−
−
1
4
,
1
4
,z; −
1
4
,
1
4
,0
(13) mx,
1
4
,z (14) m
1
4
,y,z (15) mx+
1
2
, ¯x,z (16) mx,x , z
448
CONTINUED No. 129 P4/nmm
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3); (5); (9)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
General:
16 k 1(1)x,y, z (2) ¯x +
1
2
, ¯y +
1
2
,z (3) ¯y +
1
2
,x,z (4) y, ¯x +
1
2
,z
(5) ¯x,y +
1
2
, ¯z (6) x +
1
2
, ¯y, ¯z (7) y +
1
2
,x +
1
2
, ¯z (8) ¯y, ¯x, ¯z
(9) ¯x, ¯y, ¯z (10) x +
1
2
,y +
1
2
, ¯z (11) y +
1
2
, ¯x, ¯z (12) ¯y,x +
1
2
, ¯z
(13) x, ¯y +
1
2
,z (14) ¯x +
1
2
,y,z (15) ¯y +
1
2
, ¯x +
1
2
,z (16) y,x,z
hk0: h + k = 2n
h00 : h = 2n
Special: as above, plus
8 j ..mx,x,z ¯x +
1
2
, ¯x +
1
2
,z ¯x +
1
2
,x,zx, ¯x +
1
2
,z
¯x,x +
1
2
, ¯zx+
1
2
, ¯x, ¯zx+
1
2
,x +
1
2
, ¯z ¯x, ¯x, ¯z
no extra conditions
8 i . m .
1
4
,y,z
1
4
, ¯y +
1
2
,z ¯y +
1
2
,
1
4
,zy,
1
4
,z
3
4
,y +
1
2
, ¯z
3
4
, ¯y, ¯zy+
1
2
,
3
4
, ¯z ¯y,
3
4
, ¯z
no extra conditions
8 h ..2 x, ¯x,
1
2
¯x +
1
2
,x +
1
2
,
1
2
x +
1
2
,x,
1
2
¯x, ¯x +
1
2
,
1
2
¯x,x,
1
2
x +
1
2
, ¯x +
1
2
,
1
2
¯x +
1
2
, ¯x,
1
2
x,x +
1
2
,
1
2
hkl : h + k = 2n
8 g ..2 x, ¯x, 0¯x +
1
2
,x +
1
2
,0 x +
1
2
,x,0¯x, ¯x +
1
2
,0
¯x,x,0 x+
1
2
, ¯x +
1
2
,0¯x +
1
2
, ¯x,0 x,x +
1
2
,0
hkl : h + k = 2n
4 f 2 mm.
3
4
,
1
4
,z
1
4
,
3
4
,z
1
4
,
3
4
, ¯z
3
4
,
1
4
, ¯zhkl: h + k = 2n
4 e ..2/m 0,0,
1
2
1
2
,
1
2
,
1
2
1
2
,0,
1
2
0,
1
2
,
1
2
hkl : h,k = 2n
4 d ..2/m 0,0, 0
1
2
,
1
2
,0
1
2
,0, 00,
1
2
,0 hkl : h,k = 2n
2 c 4 mm
1
4
,
1
4
,z
3
4
,
3
4
, ¯z no extra conditions
2 b
¯
4 m 2
3
4
,
1
4
,
1
2
1
4
,
3
4
,
1
2
hkl : h + k = 2n
2 a
¯
4 m 2
3
4
,
1
4
,0
1
4
,
3
4
,0 hkl : h + k = 2n
Symmetry of special projections
Along [001] p4mm
a
=
1
2
(a − b) b
=
1
2
(a + b)
Origin at
1
4
,
1
4
,z
Along [100] p2mg
a
= bb
= c
Origin at x,0,0
Along [110] p2mm
a
=
1
2
(−a + b) b
= c
Origin at x, x,0
Maximal non-isomorphic subgroups
I
[2] P
¯
4m2 (115) 1; 2; 7; 8; 11; 12; 13; 14
[2] P
¯
42
1
m (113) 1; 2; 5; 6; 11; 12; 15; 16
[2] P4mm (99) 1; 2; 3; 4; 13; 14; 15; 16
[2] P42
1
2 (90) 1; 2; 3; 4; 5; 6; 7; 8
[2] P4/n11(P4/n, 85) 1; 2; 3; 4; 9; 10; 11; 12
[2] P2/n12/m (Cmme, 67) 1; 2; 7; 8; 9; 10; 15; 16
[2] P2/n2
1
/m1(Pmmn, 59) 1; 2; 5; 6; 9; 10; 13; 14
IIa none
IIb [2] P4
2
/ncm( c
= 2c) (138); [2] P4
2
/nmc(c
= 2c) (137); [2] P4/ncc(c
= 2c) (130)
Maximal isomorphic subgroups of lowest index
IIc
[2] P4/nmm(c
= 2c) (129); [9] P4/nmm(a
= 3a,b
= 3b) (129)
Minimal non-isomorphic supergroups
I
none
II [2] C 4/mmm (P4/mmm, 123); [2] I 4/mmm(139)
449
P4/ncc D
8
4h
4/mmm Tetragonal
No. 130 P 4/n 2
1
/c 2/c Patterson symmetry P4/mmm
ORIGIN CHOICE 1
Origin at
¯
4/ncn,at−
1
4
,
1
4
,0 from
¯
1
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
2
;0≤ z ≤
1
4
Symmetry operations
(1) 1 (2) 2 0,0,z (3) 4
+
0,
1
2
,z (4) 4
−
1
2
,0, z
(5) 2(0,
1
2
,0)
1
4
,y,
1
4
(6) 2(
1
2
,0, 0) x,
1
4
,
1
4
(7) 2 x,x,
1
4
(8) 2 x, ¯x,
1
4
(9)
¯
1
1
4
,
1
4
,0 (10) n(
1
2
,
1
2
,0) x,y,0 (11)
¯
4
+
0,0,z;0,0,0 (12)
¯
4
−
0,0,z;0,0, 0
(13) cx,0,z (14) c 0,y,z (15) cx+
1
2
, ¯x, z (16) n(
1
2
,
1
2
,
1
2
) x,x,z
450
International Tables for Crystallography (2006). Vol. A, Space group 130, pp. 450–453.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 130 P4/ncc
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3); (5); (9)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
General:
16 g 1(1)x,y, z (2) ¯x, ¯y,z (3) ¯y +
1
2
,x +
1
2
,z (4) y +
1
2
, ¯x +
1
2
,z
(5) ¯x+
1
2
,y +
1
2
, ¯z +
1
2
(6) x +
1
2
, ¯y +
1
2
, ¯z +
1
2
(7) y,x, ¯z +
1
2
(8) ¯y, ¯x, ¯z +
1
2
(9) ¯x+
1
2
, ¯y +
1
2
, ¯z (10) x +
1
2
,y +
1
2
, ¯z (11) y, ¯x, ¯z (12) ¯y,x, ¯z
(13) x, ¯y,z +
1
2
(14) ¯x,y, z +
1
2
(15) ¯y +
1
2
, ¯x +
1
2
,z +
1
2
(16) y +
1
2
,x +
1
2
,z +
1
2
hk0: h + k = 2n
0kl : l = 2n
hhl : l = 2n
00l : l = 2n
h00 : h = 2n
Special: as above, plus
8 f ..2 x,x,
1
4
¯x, ¯x,
1
4
¯x +
1
2
,x +
1
2
,
1
4
x +
1
2
, ¯x +
1
2
,
1
4
¯x +
1
2
, ¯x +
1
2
,
3
4
x +
1
2
,x +
1
2
,
3
4
x, ¯x,
3
4
¯x,x,
3
4
hkl : h + k + l = 2n
8 e 2 .. 0, 0,z
1
2
,
1
2
,z
1
2
,
1
2
, ¯z +
1
2
0,0, ¯z +
1
2
1
2
,
1
2
, ¯z 0,0, ¯z 0, 0,z +
1
2
1
2
,
1
2
,z +
1
2
hkl : h + k,l = 2n
8 d
¯
1
1
4
,
1
4
,0
3
4
,
3
4
,0
1
4
,
3
4
,0
3
4
,
1
4
,0
1
4
,
3
4
,
1
2
3
4
,
1
4
,
1
2
1
4
,
1
4
,
1
2
3
4
,
3
4
,
1
2
hkl : h,k,l = 2n
4 c 4 .. 0,
1
2
,z
1
2
,0, ¯z +
1
2
1
2
,0, ¯z 0,
1
2
,z +
1
2
hkl : l = 2n
4 b
¯
4 .. 0,0,0
1
2
,
1
2
,0
1
2
,
1
2
,
1
2
0,0,
1
2
hkl : h + k,l = 2n
4 a 2 . 22 0, 0,
1
4
1
2
,
1
2
,
1
4
1
2
,
1
2
,
3
4
0,0,
3
4
hkl : h + k,l = 2n
Symmetry of special projections
Along [001] p4mm
a
=
1
2
(a − b) b
=
1
2
(a + b)
Origin at 0,0,z
Along [100] p2mg
a
= bb
=
1
2
c
Origin at x,
1
4
,0
Along [110] p2mm
a
=
1
2
(−a + b) b
=
1
2
c
Origin at x,x, 0
Maximal non-isomorphic subgroups
I
[2] P
¯
4c2 (116) 1; 2; 7; 8; 11; 12; 13; 14
[2] P
¯
42
1
c (114) 1; 2; 5; 6; 11; 12; 15; 16
[2] P4cc (103) 1; 2; 3; 4; 13; 14; 15; 16
[2] P42
1
2 (90) 1; 2; 3; 4; 5; 6; 7; 8
[2] P4/n11(P4/n, 85) 1; 2; 3; 4; 9; 10; 11; 12
[2] P2/n12/c (Ccce, 68) 1; 2; 7; 8; 9; 10; 15; 16
[2] P2/n2
1
/c1(Pccn, 56) 1; 2; 5; 6; 9; 10; 13; 14
IIa none
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] P4/ncc(c
= 3c) (130); [9] P4/ncc(a
= 3a,b
= 3b) (130)
Minimal non-isomorphic supergroups
I
none
II [2] C 4/mcc (P4/mcc, 124); [2] I 4/mcm(140); [2] P4 /nmm(c
=
1
2
c) (129)
451
P4/ncc D
8
4h
4/mmm Tetragonal
No. 130 P 4/n 2
1
/c 2/c Patterson symmetry P4/mmm
ORIGIN CHOICE 2
Origin at
¯
1atn1(c,n),at
1
4
,−
1
4
,0 from
¯
4
Asymmetric unit −
1
4
≤ x ≤
1
4
; −
1
4
≤ y ≤
1
4
;0≤ z ≤
1
4
Symmetry operations
(1) 1 (2) 2
1
4
,
1
4
,z (3) 4
+
1
4
,
1
4
,z (4) 4
−
1
4
,
1
4
,z
(5) 2(0,
1
2
,0) 0,y,
1
4
(6) 2(
1
2
,0, 0) x,0,
1
4
(7) 2(
1
2
,
1
2
,0) x, x,
1
4
(8) 2 x, ¯x,
1
4
(9)
¯
10, 0,0 (10) n(
1
2
,
1
2
,0) x,y,0 (11)
¯
4
+
1
4
,−
1
4
,z;
1
4
,−
1
4
,0 (12)
¯
4
−
−
1
4
,
1
4
,z; −
1
4
,
1
4
,0
(13) cx,
1
4
,z (14) c
1
4
,y,z (15) cx+
1
2
, ¯x, z (16) cx,x,z
452
CONTINUED No. 130 P4/ncc
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3); (5); (9)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
General:
16 g 1(1)x, y,z (2) ¯x +
1
2
, ¯y +
1
2
,z (3) ¯y +
1
2
,x,z (4) y, ¯x +
1
2
,z
(5) ¯x,y +
1
2
, ¯z +
1
2
(6) x +
1
2
, ¯y, ¯z +
1
2
(7) y +
1
2
,x +
1
2
, ¯z +
1
2
(8) ¯y, ¯x, ¯z +
1
2
(9) ¯x, ¯y, ¯z (10) x +
1
2
,y +
1
2
, ¯z (11) y +
1
2
, ¯x, ¯z (12) ¯y,x +
1
2
, ¯z
(13) x, ¯y +
1
2
,z +
1
2
(14) ¯x +
1
2
,y,z +
1
2
(15) ¯y +
1
2
, ¯x +
1
2
,z +
1
2
(16) y,x , z +
1
2
hk0: h + k = 2n
0kl : l = 2n
hhl : l = 2n
00l : l = 2n
h00 : h = 2n
Special: as above, plus
8 f ..2 x, ¯x,
1
4
¯x+
1
2
,x +
1
2
,
1
4
x +
1
2
,x,
1
4
¯x, ¯x +
1
2
,
1
4
¯x,x,
3
4
x +
1
2
, ¯x +
1
2
,
3
4
¯x +
1
2
, ¯x,
3
4
x,x +
1
2
,
3
4
hkl : h + k + l = 2n
8 e 2 ..
3
4
,
1
4
,z
1
4
,
3
4
,z
1
4
,
3
4
, ¯z +
1
2
3
4
,
1
4
, ¯z +
1
2
1
4
,
3
4
, ¯z
3
4
,
1
4
, ¯z
3
4
,
1
4
,z +
1
2
1
4
,
3
4
,z +
1
2
hkl : h + k,l = 2n
8 d
¯
10, 0,0
1
2
,
1
2
,0
1
2
,0, 00,
1
2
,00,
1
2
,
1
2
1
2
,0,
1
2
1
2
,
1
2
,
1
2
0,0,
1
2
hkl : h,k,l = 2n
4 c 4 ..
1
4
,
1
4
,z
3
4
,
3
4
, ¯z +
1
2
3
4
,
3
4
, ¯z
1
4
,
1
4
,z +
1
2
hkl : l = 2n
4 b
¯
4 ..
3
4
,
1
4
,0
1
4
,
3
4
,0
1
4
,
3
4
,
1
2
3
4
,
1
4
,
1
2
hkl : h + k,l = 2n
4 a 2 . 22
3
4
,
1
4
,
1
4
1
4
,
3
4
,
1
4
1
4
,
3
4
,
3
4
3
4
,
1
4
,
3
4
hkl : h + k,l = 2n
Symmetry of special projections
Along [001] p4mm
a
=
1
2
(a − b) b
=
1
2
(a + b)
Origin at
1
4
,
1
4
,z
Along [100] p2mg
a
= bb
=
1
2
c
Origin at x, 0,0
Along [110] p2mm
a
=
1
2
(−a + b) b
=
1
2
c
Origin at x,x, 0
Maximal non-isomorphic subgroups
I
[2] P
¯
4c2 (116) 1; 2; 7; 8; 11; 12; 13; 14
[2] P
¯
42
1
c (114) 1; 2; 5; 6; 11; 12; 15; 16
[2] P4cc (103) 1; 2; 3; 4; 13; 14; 15; 16
[2] P42
1
2 (90) 1; 2; 3; 4; 5; 6; 7; 8
[2] P4/n11(P4/n, 85) 1; 2; 3; 4; 9; 10; 11; 12
[2] P2/n12/c (Ccce, 68) 1; 2; 7; 8; 9; 10; 15; 16
[2] P2/n2
1
/c1(Pccn, 56) 1; 2; 5; 6; 9; 10; 13; 14
IIa none
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] P4/ncc(c
= 3c) (130); [9] P4/ncc(a
= 3a,b
= 3b) (130)
Minimal non-isomorphic supergroups
I
none
II [2] C 4/mcc (P4/mcc, 124); [2] I 4/mcm(140); [2] P4 /nmm(c
=
1
2
c) (129)
453