Layer group explained

In mathematics, a layer group is a three-dimensional extension of a wallpaper group, with reflections in the third dimension. It is a space group with a two-dimensional lattice, meaning that it is symmetric over repeats in the two lattice directions. The symmetry group at each lattice point is an axial crystallographic point group with the main axis being perpendicular to the lattice plane.

Table of the 80 layer groups, organized by crystal system or lattice type, and by their point groups:

Triclinic
1	p1	2	p
Monoclinic/inclined
3	p112	4	p11m	5	p11a	6	p112/m	7	p112/a
Monoclinic/orthogonal
8	p211	9	p2111	10	c211	11	pm11	12	pb11
13	cm11	14	p2/m11	15	p21/m11	16	p2/b11	17	p21/b11
18	c2/m11
Orthorhombic
19	p222	20	p2122	21	p21212	22	c222	23	pmm2
24	pma2	25	pba2	26	cmm2	27	pm2m	28	pm21b
29	pb21m	30	pb2b	31	pm2a	32	pm21n	33	pb21a
34	pb2n	35	cm2m	36	cm2e	37	pmmm	38	pmaa
39	pban	40	pmam	41	pmma	42	pman	43	pbaa
44	pbam	45	pbma	46	pmmn	47	cmmm	48	cmme
Tetragonal
49	p4	50	p	51	p4/m	52	p4/n	53	p422
54	p4212	55	p4mm	56	p4bm	57	p2m	58	p21m
59	pm2	60	pb2	61	p4/mmm	62	p4/nbm	63	p4/mbm
64	p4/nmm
Trigonal
65	p3	66	p	67	p312	68	p321	69	p3m1
70	p31m	71	p1m	72	pm1
Hexagonal
73	p6	74	p	75	p6/m	76	p622	77	p6mm
78	pm2	79	p2m	80	p6/mmm

See also

• Point group
• Crystallographic point group
• Space group
• Rod group
• Frieze group
• Wallpaper group

External links

• Bilbao Crystallographic Server, under "Subperiodic Groups: Layer, Rod and Frieze Groups"
• Nomenclature, Symbols and Classification of the Subperiodic Groups, V. Kopsky and D. B. Litvin
• CVM 1.1: Vibrating Wallpaper by Frank Farris. He constructs layer groups from wallpaper groups using negating isometries.