(International Tables) Space group 142

I4₁/acd	D_4h²⁰	4/mmm	Tetragonal
No. 142	I4₁/a2/c2/d	Patterson symmetry I4/mmm
ORIGIN CHOICE 1

symmetry group diagram

Origin at -4c 2₁, at 0, 1/4, -1/8 from -1

Asymmetric unit

0 ≤ x ≤ 1/2; 0 ≤ y ≤ 1/2; 0 ≤ z ≤ 1/8

Symmetry operations

For (0, 0, 0)+ set

(1) 1	(2) 2(0, 0, 1/2) 1/4, 1/4, z	(3) 4⁺(0, 0, 1/4) -1/4, 1/4, z	(4) 4^-(0, 0, 3/4) 1/4, -1/4, z
(5) 2 1/4, y, 1/8	(6) 2 x, 1/4, 3/8	(7) 2(1/2, 1/2, 0) x, x, 0	(8) 2 x, -x, 1/4
(9) -1 0, 1/4, 1/8	(10) a x, y, 3/8	(11) -4⁺ 0, 0, z; 0, 0, 0	(12) -4^- 0, 1/2, z; 0, 1/2, 1/4
(13) a x, 1/4, z	(14) c 0, y, z	(15) d(1/4, -1/4, 1/4) x + 1/4, -x, z	(16) d(1/4, 1/4, 3/4) x - 1/4, x, z

For (1/2, 1/2, 1/2)+ set

(1) t(1/2, 1/2, 1/2)	(2) 2 0, 0, z	(3) 4⁺(0, 0, 3/4) 1/4, 1/4, z	(4) 4^-(0, 0, 1/4) 1/4, 1/4, z
(5) 2(0, 1/2, 0) 0, y, 3/8	(6) 2(1/2, 0, 0) x, 0, 1/8	(7) 2 x, x, 1/4	(8) 2 x, -x + 1/2, 0
(9) -1 1/4, 0, 3/8	(10) b x, y, 1/8	(11) -4⁺ 1/2, 0, z; 1/2, 0, 1/4	(12) -4^- 0, 0, z; 0, 0, 0
(13) c x, 0, z	(14) b 1/4, y, z	(15) d(-1/4, 1/4, 3/4) x + 1/4, -x, z	(16) d(1/4, 1/4, 1/4) x + 1/4, x, z

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t(1/2, 1/2, 1/2); (2); (3); (5); (9)

Positions

Multiplicity, Wyckoff letter,
Site symmetry

Coordinates

Reflection conditions

(0, 0, 0)+ (1/2, 1/2, 1/2)+

General:

(1) x, y, z	(2) -x + 1/2, -y + 1/2, z + 1/2	(3) -y, x + 1/2, z + 1/4	(4) y + 1/2, -x, z + 3/4
(5) -x + 1/2, y, -z + 1/4	(6) x, -y + 1/2, -z + 3/4	(7) y + 1/2, x + 1/2, -z	(8) -y, -x, -z + 1/2
(9) -x, -y + 1/2, -z + 1/4	(10) x + 1/2, y, -z + 3/4	(11) y, -x, -z	(12) -y + 1/2, x + 1/2, -z + 1/2
(13) x + 1/2, -y + 1/2, z	(14) -x, y, z + 1/2	(15) -y + 1/2, -x, z + 1/4	(16) y, x + 1/2, z + 3/4

hkl : h + k + l = 2n
hk0 : h, k = 2n
0kl : k, l = 2n
hhl : 2h + l = 4n
00l : l = 4n
h00 : h = 2n
h-h0 : h = 2n

Special: as above, plus

. . 2

x, x, 1/4	-x + 1/2, -x + 1/2, 3/4	-x, x + 1/2, 1/2	x + 1/2, -x, 0
-x, -x + 1/2, 0	x + 1/2, x, 1/2	x, -x, 3/4	-x + 1/2, x + 1/2, 1/4

hkl : l = 2n + 1
or 2h + l = 4n

. 2 .

1/4, y, 1/8	1/4, -y + 1/2, 5/8	-y, 3/4, 3/8	y + 1/2, 3/4, 7/8
3/4, -y + 1/2, 1/8	3/4, y, 5/8	y, 3/4, 7/8	-y + 1/2, 3/4, 3/8

hkl : l = 2n + 1
or h = 2n

2 . .

0, 0, z	0, 1/2, z + 1/4	1/2, 0, -z + 1/4	1/2, 1/2, -z
0, 1/2, -z + 1/4	0, 0, -z	1/2, 1/2, z	1/2, 0, z + 1/4

hkl : 2h + l = 4n

-1

0, 1/4, 1/8

1/2, 1/4, 5/8

3/4, 1/2, 3/8

3/4, 0, 7/8

1/2, 1/4, 1/8

0, 1/4, 5/8

3/4, 1/2, 7/8

3/4, 0, 3/8

hkl : h, k = 2n, h + k + l = 4n

2 . 2 2

0, 0, 1/4

0, 1/2, 1/2

0, 1/2, 0

0, 0, 3/4

hkl : 2h + l = 4n

-4 . .

0, 0, 0

0, 1/2, 1/4

1/2, 0, 1/4

1/2, 1/2, 0

hkl : 2h + l = 4n

Symmetry of special projections

Along [001] p4mm
a' = 1/2 a b' = 1/2b
Origin at 0, 0, z

Along [100] p2mm
a' = 1/2b b' = 1/2c
Origin at x, 0, 1/8

Along [110] c2mm
a' = 1/2(-a + b) b' = 1/2c
Origin at x, x, 0

Maximal non-isomorphic subgroups

I	[2] I-42d (122)	(1; 2; 5; 6; 11; 12; 15; 16)+
	[2] I-4c2 (120)	(1; 2; 7; 8; 11; 12; 13; 14)+
	[2] I4₁cd (110)	(1; 2; 3; 4; 13; 14; 15; 16)+
	[2] I4₁22 (98)	(1; 2; 3; 4; 5; 6; 7; 8)+
	[2] I4₁/a11 (I4₁/a, 88)	(1; 2; 3; 4; 9; 10; 11; 12)+
	[2] I2/a2/c1 (Ibca, 73)	(1; 2; 5; 6; 9; 10; 13; 14)+
	[2] I2/a12/d (Fddd, 70)	(1; 2; 7; 8; 9; 10; 15; 16)+

IIa

none

IIb

none

Maximal isomorphic subgroups of lowest index

IIc

[3] I4₁/acd (c' = 3c) (142); [9] I4₁/acd (a' = 3a, b' = 3b) (142)

Minimal non-isomorphic supergroups

I	[3] Fd-3c (228); [3] Ia-3d (230)

II	[2] C4₂/amd (c' = 1/2c) (P4₂/nnm, 134)

I4₁/acd	D_4h²⁰	4/mmm	Tetragonal
No. 142	I4₁/a2/c2/d	Patterson symmetry I4/mmm
ORIGIN CHOICE 2

symmetry group diagram

Origin at -1 at b (c, a)d, at 0, -1/4, 1/8 from -4

Asymmetric unit

0 ≤ x ≤ 1/2; -1/4 ≤ y ≤ 1/4; 0 ≤ z ≤ 1/8

Symmetry operations

For (0, 0, 0)+ set

(1) 1	(2) 2(0, 0, 1/2) 1/4, 0, z	(3) 4⁺(0, 0, 1/4) -1/4, 1/2, z	(4) 4^-(0, 0, 3/4) 1/4, 0, z
(5) 2 1/4, y, 0	(6) 2 x, 0, 1/4	(7) 2(1/2, 1/2, 0) x, x + 1/4, 3/8	(8) 2 x, -x + 1/4, 1/8
(9) -1 0, 0, 0	(10) a x, y, 1/4	(11) -4⁺ 1/2, -1/4, z; 1/2, -1/4, 3/8	(12) -4^- 0, 3/4, z; 0, 3/4, 1/8
(13) a x, 0, z	(14) c 0, y, z	(15) d(1/4, -1/4, 1/4) x + 1/2, -x, z	(16) d(3/4, 3/4, 3/4) x, x, z

For (1/2, 1/2, 1/2)+ set

(1) t(1/2, 1/2, 1/2)	(2) 2 0, 1/4, z	(3) 4⁺(0, 0, 3/4) 1/4, 1/2, z	(4) 4^-(0, 0, 1/4) 3/4, 0, z
(5) 2(0, 1/2, 0) 0, y, 1/4	(6) 2(1/2, 0, 0) x, 1/4, 0	(7) 2(1/2, 1/2, 0) x, x - 1/4, 1/8	(8) 2 x, -x + 3/4, 3/8
(9) -1 1/4, 1/4, 1/4	(10) b x, y, 0	(11) -4⁺ 1/2, 1/4, z; 1/2, 1/4, 1/8	(12) -4^- 0, 1/4, z; 0, 1/4, 3/8
(13) c x, 1/4, z	(14) b 1/4, y, z	(15) d(-1/4, 1/4, 3/4) x + 1/2, -x, z	(16) d(1/4, 1/4, 1/4) x, x, z

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t(1/2, 1/2, 1/2); (2); (3); (5); (9)

Positions

Multiplicity, Wyckoff letter,
Site symmetry

Coordinates

Reflection conditions

(0, 0, 0)+ (1/2, 1/2, 1/2)+

General:

(1) x, y, z	(2) -x + 1/2, -y, z + 1/2	(3) -y + 1/4, x + 3/4, z + 1/4	(4) y + 1/4, -x + 1/4, z + 3/4
(5) -x + 1/2, y, -z	(6) x, -y, -z + 1/2	(7) y + 1/4, x + 3/4, -z + 3/4	(8) -y + 1/4, -x + 1/4, -z + 1/4
(9) -x, -y, -z	(10) x + 1/2, y, -z + 1/2	(11) y + 3/4, -x + 1/4, -z + 3/4	(12) -y + 3/4, x + 3/4, -z + 1/4
(13) x + 1/2, -y, z	(14) -x, y, z + 1/2	(15) -y + 3/4, -x + 1/4, z + 1/4	(16) y + 3/4, x + 3/4, z + 3/4

hkl : h + k + l = 2n
hk0 : h, k = 2n
0kl : k, l = 2n
hhl : 2h + l = 4n
00l : l = 4n
h00 : h = 2n
h-h0 : h = 2n

Special: as above, plus

. . 2

x, x + 1/4, 1/8	-x + 1/2, -x + 3/4, 5/8	-x, x + 3/4, 3/8	x + 1/2, -x + 1/4, 7/8
-x, -x + 3/4, 7/8	x + 1/2, x + 1/4, 3/8	x, -x + 1/4, 5/8	-x + 1/2, x + 3/4, 1/8

hkl : l = 2n + 1
or 2h + l = 4n

. 2 .

x, 0, 1/4	-x + 1/2, 0, 3/4	1/4, x + 3/4, 1/2	1/4, -x + 1/4, 0
-x, 0, 3/4	x + 1/2, 0, 1/4	3/4, -x + 1/4, 1/2	3/4, x + 3/4, 0

hkl : l = 2n + 1
or h = 2n

2 . .

0, 1/4, z	0, 3/4, z + 1/4	1/2, 1/4, -z	1/2, 3/4, -z + 3/4
0, 3/4, -z	0, 1/4, -z + 3/4	1/2, 3/4, z	1/2, 1/4, z + 1/4

hkl : 2h + l = 4n

-1

0, 0, 0

1/2, 0, 1/2

1/4, 3/4, 1/4

1/4, 1/4, 3/4

1/2, 0, 0

0, 0, 1/2

1/4, 3/4, 3/4

1/4, 1/4, 1/4

hkl : h, k = 2n, h + k + l = 4n

2 . 2 2

0, 1/4, 1/8

0, 3/4, 3/8

0, 3/4, 7/8

0, 1/4, 5/8

hkl : 2h + l = 4n

-4 . .

0, 1/4, 3/8

0, 3/4, 5/8

1/2, 1/4, 5/8

1/2, 3/4, 3/8

hkl : 2h + l = 4n

Symmetry of special projections

Along [001] p4mm
a' = 1/2 a b' = 1/2b
Origin at 1/4, 0, z

Along [100] p2mm
a' = 1/2b b' = 1/2c
Origin at x, 0, 0

Along [110] c2mm
a' = 1/2(-a + b) b' = 1/2c
Origin at x, x + 1/4, 1/8

Maximal non-isomorphic subgroups

I	[2] I-42d (122)	(1; 2; 5; 6; 11; 12; 15; 16)+
	[2] I-4c2 (120)	(1; 2; 7; 8; 11; 12; 13; 14)+
	[2] I4₁cd (110)	(1; 2; 3; 4; 13; 14; 15; 16)+
	[2] I4₁22 (98)	(1; 2; 3; 4; 5; 6; 7; 8)+
	[2] I4₁/a11 (I4₁/a, 88)	(1; 2; 3; 4; 9; 10; 11; 12)+
	[2] I2/a2/c1 (Ibca, 73)	(1; 2; 5; 6; 9; 10; 13; 14)+
	[2] I2/a12/d (Fddd, 70)	(1; 2; 7; 8; 9; 10; 15; 16)+

IIa

none

IIb

none

Maximal isomorphic subgroups of lowest index

IIc

[3] I4₁/acd (c' = 3c) (142); [9] I4₁/acd (a' = 3a, b' = 3b) (142)

Minimal non-isomorphic supergroups

I	[3] Fd-3c (228); [3] Ia-3d (230)

II	[2] C4₂/amd (c' = 1/2c) (P4₂/nnm, 134)

International Tables for Crystallography (2006). Vol. A, ch. 7.1, Space group 142.