(International Tables) Space group 68

Ccce	D_2h²²	mmm	Orthorhombic
No. 68	C2/c2/c2/e	Patterson symmetry Cmmm
ORIGIN CHOICE 1

symmetry group diagram

Origin at 2 2 2 at 2/n 2/n 2, at 0, 1/4, 1/4 from -1

Asymmetric unit

0 ≤ x ≤ 1/4; 0 ≤ y ≤ 1/2; 0 ≤ z ≤ 1/2

Symmetry operations

For (0, 0, 0)+ set

(1) 1	(2) 2 1/4, 1/4, z	(3) 2 0, y, 0	(4) 2(1/2, 0, 0) x, 1/4, 0
(5) -1 0, 1/4, 1/4	(6) a x, y, 1/4	(7) c x, 1/4, z	(8) c 1/4, y, z

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

(1) t(1/2, 1/2, 0)	(2) 2 0, 0, z	(3) 2(0, 1/2, 0) 1/4, y, 0	(4) 2 x, 0, 0
(5) -1 1/4, 0, 1/4	(6) b x, y, 1/4	(7) n(1/2, 0, 1/2) x, 0, z	(8) n(0, 1/2, 1/2) 0, y, z

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

Positions

Multiplicity, Wyckoff letter,
Site symmetry

Coordinates

Reflection conditions

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

General:

(1) x, y, z	(2) -x + 1/2, -y + 1/2, z	(3) -x, y, -z	(4) x + 1/2, -y + 1/2, -z
(5) -x, -y + 1/2, -z + 1/2	(6) x + 1/2, y, -z + 1/2	(7) x, -y + 1/2, z + 1/2	(8) -x + 1/2, y, z + 1/2

hkl : h + k = 2n
0kl : k, l = 2n
h0l : h, l = 2n
hk0 : h, k = 2n
h00 : h = 2n
0k0 : k = 2n
00l : l = 2n

Special: as above, plus

. . 2

1/4, 1/4, z

3/4, 1/4, -z

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

1/4, 1/4, z + 1/2

hkl : l = 2n

. . 2

0, 0, z

0, 0, -z

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

0, 1/2, z + 1/2

hkl : k + l = 2n

. 2 .

0, y, 0

1/2, -y + 1/2, 0

0, -y + 1/2, 1/2

1/2, y, 1/2

hkl : k + l = 2n

2 . .

x, 0, 0

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

-x, 1/2, 1/2

x + 1/2, 0, 1/2

hkl : k + l = 2n

-1

0, 1/4, 1/4

1/2, 1/4, 1/4

0, 1/4, 3/4

1/2, 1/4, 3/4

hkl : k, l = 2n

-1

1/4, 0, 1/4

1/4, 1/2, 1/4

3/4, 0, 3/4

3/4, 1/2, 3/4

hkl : k, l = 2n

2 2 2

0, 0, 1/2

0, 1/2, 0

hkl : k + l = 2n

2 2 2

0, 0, 0

0, 1/2, 1/2

hkl : k + l = 2n

Symmetry of special projections

Along [001] p2mm
a' = 1/2a b' = 1/2b
Origin at 0, 0, z

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

Along [010] p2mm
a' = 1/2c b' = 1/2a
Origin at 0, y, 0

Maximal non-isomorphic subgroups

I	[2] Cc2e (Aea2, 41)	(1; 3; 6; 8)+
	[2] C2ce (Aea2, 41)	(1; 4; 6; 7)+
	[2] Ccc2 (37)	(1; 2; 7; 8)+
	[2] C222 (21)	(1; 2; 3; 4)+
	[2] C12/c1 (C2/c, 15)	(1; 3; 5; 7)+
	[2] C2/c11 (C2/c, 15)	(1; 4; 5; 8)+
	[2] C112/e (P2/c, 13)	(1; 2; 5; 6)+

IIa	[2] Pcnb (Pbcn, 60)	1; 4; 5; 8; (2; 3; 6; 7) + (1/2, 1/2, 0)
	[2] Pnca (Pbcn, 60)	1; 4; 6; 7; (2; 3; 5; 8) + (1/2, 1/2, 0)
	[2] Pcca (54)	1; 2; 3; 4; 5; 6; 7; 8
	[2] Pccb (Pcca, 54)	1; 2; 7; 8; (3; 4; 5; 6) + (1/2, 1/2, 0)
	[2] Pnnb (Pnna, 52)	1; 2; 3; 4; (5; 6; 7; 8) + (1/2, 1/2, 0)
	[2] Pnna (52)	1; 2; 5; 6; (3; 4; 7; 8) + (1/2, 1/2, 0)
	[2] Pncb (Pban, 50)	1; 3; 5; 7; (2; 4; 6; 8) + (1/2, 1/2, 0)
	[2] Pcna (Pban, 50)	1; 3; 6; 8; (2; 4; 5; 7) + (1/2, 1/2, 0)

IIb

none

Maximal isomorphic subgroups of lowest index

IIc

[3] Ccce (a' = 3a or b' = 3b) (68); [3] Ccce (c' = 3c) (68)

Minimal non-isomorphic supergroups

I	[2] P4/nnc (126); [2] P4/ncc (130); [2] P4₂/nbc (133); [2] P4₂/nmc (137)

II	[2] Fmmm (69); [2] Pccm (a' = 1/2a, b' = 1/2b) (49); [2] Cmme (c' = 1/2c) (67)

Ccce	D_2h²²	mmm	Orthorhombic
No. 68	C2/c2/c2/e	Patterson symmetry Cmmm
ORIGIN CHOICE 2

symmetry group diagram

Origin at -1 at n c e, at 0, -1/4, -1/4 from 222

Asymmetric unit

0 ≤ x ≤ 1/2; 0 ≤ y ≤ 1/4; 0 ≤ z ≤ 1/2

Symmetry operations

For (0, 0, 0)+ set

(1) 1	(2) 2 1/4, 0, z	(3) 2 0, y, 1/4	(4) 2(1/2, 0, 0) x, 0, 1/4
(5) -1 0, 0, 0	(6) a x, y, 0	(7) c x, 0, z	(8) c 1/4, y, z

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

(1) t(1/2, 1/2, 0)	(2) 2 0, 1/4, z	(3) 2(0, 1/2, 0) 1/4, y, 1/4	(4) 2 x, 1/4, 1/4
(5) -1 1/4, 1/4, 0	(6) b x, y, 0	(7) n(1/2, 0, 1/2) x, 1/4, z	(8) n(0, 1/2, 1/2) 0, y, z

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

Positions

Multiplicity, Wyckoff letter,
Site symmetry

Coordinates

Reflection conditions

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

General:

(1) x, y, z	(2) -x + 1/2, -y, z	(3) -x, y, -z + 1/2	(4) x + 1/2, -y, -z + 1/2
(5) -x, -y, -z	(6) x + 1/2, y, -z	(7) x, -y, z + 1/2	(8) -x + 1/2, y, z + 1/2

hkl : h + k = 2n
0kl : k, l = 2n
h0l : h, l = 2n
hk0 : h, k = 2n
h00 : h = 2n
0k0 : k = 2n
00l : l = 2n

Special: as above, plus

. . 2

1/4, 0, z

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

3/4, 0, -z

1/4, 0, z + 1/2

hkl : l = 2n

. . 2

0, 1/4, z

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

0, 3/4, -z

0, 3/4, z + 1/2

hkl : k + l = 2n

. 2 .

0, y, 1/4

1/2, -y, 1/4

0, -y, 3/4

1/2, y, 3/4

hkl : k + l = 2n

2 . .

x, 1/4, 1/4

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

-x, 3/4, 3/4

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

hkl : k + l = 2n

-1

0, 0, 0

1/2, 0, 0

0, 0, 1/2

1/2, 0, 1/2

hkl : k, l = 2n

-1

1/4, 3/4, 0

1/4, 1/4, 0

3/4, 3/4, 1/2

3/4, 1/4, 1/2

hkl : k, l = 2n

2 2 2

0, 1/4, 3/4

0, 3/4, 1/4

hkl : k + l = 2n

2 2 2

0, 1/4, 1/4

0, 3/4, 3/4

hkl : k + l = 2n

Symmetry of special projections

Along [001] p2mm
a' = 1/2a b' = 1/2b
Origin at 0, 0, z

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

Along [010] p2mm
a' = 1/2c b' = 1/2a
Origin at 0, y, 0

Maximal non-isomorphic subgroups

I	[2] Cc2e (Aea2, 41)	(1; 3; 6; 8)+
	[2] C2ce (Aea2, 41)	(1; 4; 6; 7)+
	[2] Ccc2 (37)	(1; 2; 7; 8)+
	[2] C222 (21)	(1; 2; 3; 4)+
	[2] C12/c1 (C2/c, 15)	(1; 3; 5; 7)+
	[2] C2/c11 (C2/c, 15)	(1; 4; 5; 8)+
	[2] C112/e (P2/c, 13)	(1; 2; 5; 6)+

IIa	[2] Pcnb (Pbcn, 60)	1; 4; 5; 8; (2; 3; 6; 7) + (1/2, 1/2, 0)
	[2] Pnca (Pbcn, 60)	1; 4; 6; 7; (2; 3; 5; 8) + (1/2, 1/2, 0)
	[2] Pcca (54)	1; 2; 3; 4; 5; 6; 7; 8
	[2] Pccb (Pcca, 54)	1; 2; 7; 8; (3; 4; 5; 6) + (1/2, 1/2, 0)
	[2] Pnnb (Pnna, 52)	1; 2; 3; 4; (5; 6; 7; 8) + (1/2, 1/2, 0)
	[2] Pnna (52)	1; 2; 5; 6; (3; 4; 7; 8) + (1/2, 1/2, 0)
	[2] Pncb (Pban, 50)	1; 3; 5; 7; (2; 4; 6; 8) + (1/2, 1/2, 0)
	[2] Pcna (Pban, 50)	1; 3; 6; 8; (2; 4; 5; 7) + (1/2, 1/2, 0)

IIb

none

Maximal isomorphic subgroups of lowest index

IIc

[3] Ccce (a' = 3a or b' = 3b) (68); [3] Ccce (c' = 3c) (68)

Minimal non-isomorphic supergroups

I	[2] P4/nnc (126); [2] P4/ncc (130); [2] P4₂/nbc (133); [2] P4₂/nmc (137)

II	[2] Fmmm (69); [2] Pccm (a' = 1/2a, b' = 1/2b) (49); [2] Cmme (c' = 1/2c) (67)

International Tables for Crystallography (2006). Vol. A, ch. 7.1, Space group 68.