(International Tables) Space group 15

C2/c	C_2h⁶	2/m	Monoclinic
No. 15	C12/c1	Patterson symmetry C12/m1
UNIQUE AXIS b, CELL CHOICE 1

symmetry group diagram

Origin at -1 on glide plane c

Asymmetric unit

0 ≤ x ≤ 1/2; 0 ≤ y ≤ 1/2; 0 ≤ z ≤ 1/2

Symmetry operations

For (0, 0, 0)+ set

(1) 1

(2) 2 0, y, 1/4

(3) -1 0, 0, 0

(4) c x, 0, z

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

(1) t(1/2, 1/2, 0)

(2) 2(0, 1/2, 0) 1/4, y, 1/4

(3) -1 1/4, 1/4, 0

(4) n(1/2, 0, 1/2) x, 1/4, z

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

Positions

Multiplicity, Wyckoff letter,
Site symmetry

Coordinates

Reflection conditions

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

General:

(1) x, y, z

(2) -x, y, -z + 1/2

(3) -x, -y, -z

(4) x, -y, z + 1/2

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

Special: as above, plus

0, y, 1/4

0, -y, 3/4

no extra conditions

-1

1/4, 1/4, 1/2

3/4, 1/4, 0

hkl : k + l = 2n

-1

1/4, 1/4, 0

3/4, 1/4, 1/2

hkl : k + l = 2n

-1

0, 1/2, 0

0, 1/2, 1/2

hkl : l = 2n

-1

0, 0, 0

0, 0, 1/2

hkl : l = 2n

Symmetry of special projections

Along [001] c2mm
a' = a_p b' = b
Origin at 0, 0, z

Along [100] p2gm
a' = 1/2b b' = c_p
Origin at x, 0, 0

Along [010] p2
a' = 1/2c b' = 1/2a
Origin at 0, y, 0

Maximal non-isomorphic subgroups

I	[2] C1c1 (Cc, 9)	(1; 4)+
	[2] C121 (C2, 5)	(1; 2)+
	[2] C-1 (P-1, 2)	(1; 3)+

IIa	[2] P12₁/n1 (P2₁/c, 14)	1; 3; (2; 4) + (1/2, 1/2, 0)
	[2] P12₁/c1 (P2₁/c, 14)	1; 4; (2; 3) + (1/2, 1/2, 0)
	[2] P12/c1 (P2/c, 13)	1; 2; 3; 4
	[2] P12/n1 (P2/c, 13)	1; 2; (3; 4) + (1/2, 1/2, 0)

IIb

none

Maximal isomorphic subgroups of lowest index

IIc

[3] C12/c1 (b' = 3b) (C2/c, 15); [3] C12/c1 (c' = 3c) (C2/c, 15); [3] C12/c1 (a' = 3a or a' = 3a, c' = -a + c or a' = 3a, c' = a + c) (C2/c, 15)

Minimal non-isomorphic supergroups

I	[2] Cmcm (63); [2] Cmce (64); [2] Cccm (66); [2] Ccce (68); [2] Fddd (70); [2] Ibam (72); [2] Ibca (73); [2] Imma (74); [2] I4₁/a (88); [3] P-31c (163); [3] P-3c1 (165); [3] R-3c (167)

II	[2] F12/m1 (C2/m, 12); [2] C12/m1 (c' = 1/2c) (C2/m, 12); [2] P12/c1 (a' = 1/2a, b' = 1/2b) (P2/c, 13)

UNIQUE AXIS b, DIFFERENT CELL CHOICES

symmetry group diagram

C12/c1

UNIQUE AXIS b, CELL CHOICE 1

cell choice

Origin at -1 on glide plane c

Asymmetric unit

0 ≤ x ≤ 1/2; 0 ≤ y ≤ 1/2; 0 ≤ z ≤ 1/2

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

Positions

Multiplicity, Wyckoff letter,
Site symmetry

Coordinates

Reflection conditions

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

General:

(1) x, y, z

(2) -x, y, -z + 1/2

(3) -x, -y, -z

(4) x, -y, z + 1/2

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

Special: as above, plus

0, y, 1/4

0, -y, 3/4

no extra conditions

-1

1/4, 1/4, 1/2

3/4, 1/4, 0

hkl : k + l = 2n

-1

1/4, 1/4, 0

3/4, 1/4, 1/2

-1

0, 1/2, 0

0, 1/2, 1/2

hkl : l = 2n

-1

0, 0, 0

0, 0, 1/2

A12/n1

UNIQUE AXIS b, CELL CHOICE 2

cell choice

Origin at -1 on glide plane n

Asymmetric unit

0 ≤ x ≤ 1/2; 0 ≤ y ≤ 1; 0 ≤ z ≤ 1/4

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

Positions

Multiplicity, Wyckoff letter,
Site symmetry

Coordinates

Reflection conditions

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

General:

(1) x, y, z

(2) -x + 1/2, y, -z + 1/2

(3) -x, -y, -z

(4) x + 1/2, -y, z + 1/2

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

Special: as above, plus

3/4, y, 3/4

1/4, -y, 1/4

no extra conditions

-1

1/2, 1/4, 3/4

0, 1/4, 3/4

hkl : h = 2n

-1

0, 1/4, 1/4

1/2, 1/4, 1/4

-1

0, 1/2, 0

1/2, 1/2, 1/2

hkl : h + k = 2n

-1

0, 0, 0

1/2, 0, 1/2

I12/a1

UNIQUE AXIS b, CELL CHOICE 3

cell choice

Origin at -1 on glide plane a

Asymmetric unit

0 ≤ x ≤ 1; 0 ≤ y ≤ 1/2; 0 ≤ z ≤ 1/4

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

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

(3) -x, -y, -z

(4) x + 1/2, -y, z

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

Special: as above, plus

1/4, y, 0

3/4, -y, 0

no extra conditions

-1

1/4, 1/4, 3/4

1/4, 1/4, 1/4

hkl : l = 2n

-1

3/4, 1/4, 3/4

3/4, 1/4, 1/4

-1

0, 1/2, 0

1/2, 1/2, 0

hkl : h = 2n

-1

0, 0, 0

1/2, 0, 0

C2/c	C_2h⁶	2/m	Monoclinic
No. 15	A112/a	Patterson symmetry A112/m
UNIQUE AXIS c, CELL CHOICE 1

symmetry group diagram

Origin at -1 on glide plane a

Asymmetric unit

0 ≤ x ≤ 1/2; 0 ≤ y ≤ 1/2; 0 ≤ z ≤ 1/2

Symmetry operations

For (0, 0, 0)+ set

(1) 1

(2) 2 1/4, 0, z

(3) -1 0, 0, 0

(4) a x, y, 0

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

(1) t(0, 1/2, 1/2)

(2) 2(0, 0, 1/2) 1/4, 1/4, z

(3) -1 0, 1/4, 1/4

(4) n(1/2, 1/2, 0) x, y, 1/4

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

Positions

Multiplicity, Wyckoff letter,
Site symmetry

Coordinates

Reflection conditions

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

General:

(1) x, y, z

(2) -x + 1/2, -y, z

(3) -x, -y, -z

(4) x + 1/2, y, -z

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

Special: as above, plus

1/4, 0, z

3/4, 0, -z

no extra conditions

-1

1/2, 1/4, 1/4

0, 3/4, 1/4

hkl : h + k = 2n

-1

0, 1/4, 1/4

1/2, 3/4, 1/4

hkl : h + k = 2n

-1

0, 0, 1/2

1/2, 0, 1/2

hkl : h = 2n

-1

0, 0, 0

1/2, 0, 0

hkl : h = 2n

Symmetry of special projections

Along [001] p2
a' = 1/2a b' = 1/2b
Origin at 0, 0, z

Along [100] c2mm
a' = b_p b' = c
Origin at x, 0, 0

Along [010] p2gm
a' = 1/2c b' = a_p
Origin at 0, y, 0

Maximal non-isomorphic subgroups

I	[2] A11a (Cc, 9)	(1; 4)+
	[2] A112 (C2, 5)	(1; 2)+
	[2] A-1 (P-1, 2)	(1; 3)+

IIa	[2] P112₁/n (P2₁/c, 14)	1; 3; (2; 4) + (0, 1/2, 1/2)
	[2] P112₁/a (P2₁/c, 14)	1; 4; (2; 3) + (0, 1/2, 1/2)
	[2] P112/a (P2/c, 13)	1; 2; 3; 4
	[2] P112/n (P2/c, 13)	1; 2; (3; 4) + (0, 1/2, 1/2)

IIb

none

Maximal isomorphic subgroups of lowest index

IIc

[3] A112/a (c' = 3c) (C2/c, 15); [3] A112/a (a' = 3a) (C2/c, 15); [3] A112/a (b' = 3b or a' = a - b, b' = 3b or a' = a + b, b' = 3b) (C2/c, 15)

Minimal non-isomorphic supergroups

I	[2] Cmcm (63); [2] Cmce (64); [2] Cccm (66); [2] Ccce (68); [2] Fddd (70); [2] Ibam (72); [2] Ibca (73); [2] Imma (74); [2] I4₁/a (88); [3] P-31c (163); [3] P-3c1 (165); [3] R-3c (167)

II	[2] F112/m (C2/m, 12); [2] A112/m (a' = 1/2a) (C2/m, 12); [2] P112/a (b' = 1/2b, c' = 1/2c) (P2/c, 13)

UNIQUE AXIS c, DIFFERENT CELL CHOICES

symmetry group diagram

A112/a

UNIQUE AXIS c, CELL CHOICE 1

cell choice

Origin at -1 on glide plane a

Asymmetric unit

0 ≤ x ≤ 1/2; 0 ≤ y ≤ 1/2; 0 ≤ z ≤ 1/2

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

Positions

Multiplicity, Wyckoff letter,
Site symmetry

Coordinates

Reflection conditions

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

General:

(1) x, y, z

(2) -x + 1/2, -y, z

(3) -x, -y, -z

(4) x + 1/2, y, -z

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

Special: as above, plus

1/4, 0, z

3/4, 0, -z

no extra conditions

-1

1/2, 1/4, 1/4

0, 3/4, 1/4

hkl : h + k = 2n

-1

0, 1/4, 1/4

1/2, 3/4, 1/4

-1

0, 0, 1/2

1/2, 0, 1/2

hkl : h = 2n

-1

0, 0, 0

1/2, 0, 0

B112/n

UNIQUE AXIS c, CELL CHOICE 2

cell choice

Origin at -1 on glide plane n

Asymmetric unit

0 ≤ x ≤ 1/4; 0 ≤ y ≤ 1/2; 0 ≤ z ≤ 1

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

Positions

Multiplicity, Wyckoff letter,
Site symmetry

Coordinates

Reflection conditions

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

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

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

Special: as above, plus

3/4, 3/4, z

1/4, 1/4, -z

no extra conditions

-1

3/4, 1/2, 1/4

3/4, 0, 1/4

hkl : k = 2n

-1

1/4, 0, 1/4

1/4, 1/2, 1/4

-1

0, 0, 1/2

1/2, 1/2, 1/2

hkl : h + k = 2n

-1

0, 0, 0

1/2, 1/2, 0

I112/b

UNIQUE AXIS c, CELL CHOICE 3

cell choice

Origin at -1 on glide plane b

Asymmetric unit

0 ≤ x ≤ 1/4; 0 ≤ y ≤ 1; 0 ≤ z ≤ 1/2

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

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, -y + 1/2, z

(3) -x, -y, -z

(4) x, y + 1/2, -z

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

Special: as above, plus

0, 1/4, z

0, 3/4, -z

no extra conditions

-1

3/4, 1/4, 1/4

1/4, 1/4, 1/4

hkl : h = 2n

-1

3/4, 3/4, 1/4

1/4, 3/4, 1/4

-1

0, 0, 1/2

0, 1/2, 1/2

hkl : k = 2n

-1

0, 0, 0

0, 1/2, 0

International Tables for Crystallography (2006). Vol. A, ch. 7.1, Space group 15.