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 | p1 | ||||||
| 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 | p4 | 51 | p4/m | 52 | p4/n | 53 | p422 |
| 54 | p4212 | 55 | p4mm | 56 | p4bm | 57 | p42m | 58 | p421m |
| 59 | p4m2 | 60 | p4b2 | 61 | p4/mmm | 62 | p4/nbm | 63 | p4/mbm |
| 64 | p4/nmm | ||||||||
| Trigonal | |||||||||
| 65 | p3 | 66 | p3 | 67 | p312 | 68 | p321 | 69 | p3m1 |
| 70 | p31m | 71 | p31m | 72 | p3m1 | ||||
| Hexagonal | |||||||||
| 73 | p6 | 74 | p6 | 75 | p6/m | 76 | p622 | 77 | p6mm |
| 78 | p6m2 | 79 | p62m | 80 | p6/mmm | ||||