Crystal Lab › The 80 layer groups

The 80 layer groups

3D objects periodic in two directions. Open a group to see its symmetry elements drawn in the cell, its operations, and its Wyckoff positions with a model atom you can move.

The 230 space groups · The 17 plane groups · The 7 frieze groups · The 75 rod groups · The 80 layer groups

Triclinic (2)

No.	Symbol	Operations	Wyckoff positions	Settings
1	p 1	1	1	1
2	p -1	2	5	1

Monoclinic (5)

No.	Symbol	Operations	Wyckoff positions	Settings
3	p 1 1 2	2	5	1
4	p 1 1 m	2	2	1
5	p 1 1 a	2	1	3
6	p 1 1 2/m	4	10	1
7	p 1 1 2/a	4	5	3

Orthorhombic (41)

No.	Symbol	Operations	Wyckoff positions	Settings
8	p 2 1 1	2	3	2
9	p 21 1 1	2	1	2
10	c 2 1 1	4	2	2
11	p m 1 1	2	3	2
12	p b 1 1	2	1	2
13	c m 1 1	4	2	2
14	p 2/m 1 1	4	9	2
15	p 21/m 1 1	4	4	2
16	p 2/b 1 1	4	4	2
17	p 21/b 1 1	4	3	2
18	c 2/m 1 1	8	6	2
19	p 2 2 2	4	13	1
20	p 21 2 2	4	4	2
21	p 21 21 2	4	3	1
22	c 2 2 2	8	8	1
23	p m m 2	4	9	1
24	p m a 2	4	4	2
25	p b a 2	4	3	1
26	c m m 2	8	6	1
27	p m 2 m	4	6	2
28	p m 21 b	4	3	2
29	p b 21 m	4	2	2
30	p b 2 b	4	3	2
31	p m 2 a	4	3	2
32	p m 21 n	4	2	2
33	p b 21 a	4	1	2
34	p b 2 n	4	2	2
35	c m 2 m	8	4	2
36	c m 2 e	8	3	2
37	p m m m	8	18	1
38	p m a a	8	11	2
39	p b a n	8	8	1
40	p m a m	8	8	2
41	p m m a	8	9	2
42	p m a n	8	6	2
43	p b a a	8	4	2
44	p b a m	8	6	1
45	p b m a	8	4	2
46	p m m n	8	6	1
47	c m m m	16	12	1
48	c m m e	16	10	1

Tetragonal (16)

No.	Symbol	Operations	Wyckoff positions	Settings
49	p 4	4	4	1
50	p -4	4	6	1
51	p 4/m	8	8	1
52	p 4/n :2	8	5	2
53	p 4 2 2	8	10	1
54	p 4 21 2	8	5	1
55	p 4 m m	8	7	1
56	p 4 b m	8	4	1
57	p -4 2 m	8	10	1
58	p -4 21 m	8	5	1
59	p -4 m 2	8	9	1
60	p -4 b 2	8	6	1
61	p 4/m m m	16	14	1
62	p 4/n b m :2	16	9	2
63	p 4/m b m	16	8	1
64	p 4/n m m :2	16	8	2

Trigonal (8)

No.	Symbol	Operations	Wyckoff positions	Settings
65	p 3	3	4	1
66	p -3	6	5	1
67	p 3 1 2	6	8	1
68	p 3 2 1	6	5	1
69	p 3 m 1	6	5	1
70	p 3 1 m	6	4	1
71	p -3 1 m	12	8	1
72	p -3 m 1	12	7	1

Hexagonal (8)

No.	Symbol	Operations	Wyckoff positions	Settings
73	p 6	6	4	1
74	p -6	6	8	1
75	p 6/m	12	8	1
76	p 6 2 2	12	9	1
77	p 6 m m	12	6	1
78	p -6 m 2	12	10	1
79	p -6 2 m	12	8	1
80	p 6/m m m	24	12	1

Operations are counted per conventional cell (lattice centring included). Symbols follow the International Tables; the first listed setting is the standard one.