# Layer group

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 | p2$_1$11 | 10 | c211 | 11 | pm11 | 12 | pb11 |
| 13 | cm11 | 14 | p2/m11 | 15 | p2$_1$/m11 | 16 | p2/b11 | 17 | p2$_1$/b11 |
| 18 | c2/m11 |
| Orthorhombic ||||||||||
| 19 | p222 | 20 | p2$_1$22 | 21 | p2$_1$2$_1$2 | 22 | c222 | 23 | pmm2 |
| 24 | pma2 | 25 | pba2 | 26 | cmm2 | 27 | pm2m | 28 | pm2$_1$b |
| 29 | pb2$_1$m | 30 | pb2b | 31 | pm2a | 32 | pm2$_1$n | 33 | pb2$_1$a |
| 34 | pb2n | 35 | cm2m | 36 | cm2e | 37 | pmmm | 38 | pmaa |
| 39 | pban | 40 | pmam | 41 | pmma | 42 | pman | 43 | pbaa |
| 44 | pbam | 45 | pbma | 46 | pmmn | 47 | cmmm | 48 | cmme |
| Tetragonal ||||||||||
| 49 | p4 | 50 | p4 | 51 | p4/m | 52 | p4/n | 53 | p422 |
| 54 | p42$_1$2 | 55 | p4mm | 56 | p4bm | 57 | p42m | 58 | p42$_1$m |
| 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 |

## Correspondence Between Layer Groups and Plane Groups

The surjective mapping from a layer group to a wallpaper group (plane group) can be obtained by disregarding symmetry elements along the stacking direction, typically denoted as the z-axis, and aligning the remaining elements with those of the plane groups. The resulting surjective mapping provides a direct correspondence between layer groups and plane groups (wallpaper groups).
Surjective mapping from Layer Groups to Plane Groups :
| # | Layer Group | # | Plane Group |
| 1 | p1 | 1 | p1 |
| 2 | p1 | 2 | p2 |
| 3 | p112 | 2 | p2 |
| 4 | p11m | 1 | p1 |
| 5 | p11a | 1 | p1 |
| 6 | p112/m | 2 | p2 |
| 7 | p112/a | 2 | p2 |
| 8 | p211 | 3 | pm |
| 9 | p2$_1$11 | 4 | pg |
| 10 | c211 | 5 | cm |
| 11 | pm11 | 3 | pm |
| 12 | pb11 | 4 | pg |
| 13 | cm11 | 5 | cm |
| 14 | p2/m11 | 6 | p2mm |
| 15 | p2$_1$/m11 | 7 | p2mg |
| 16 | p2/b11 | 7 | p2mg |
| 17 | p2$_1$/b11 | 8 | p2gg |
| 18 | c2/m11 | 9 | c2mm |
| 19 | p222 | 6 | p2mm |
| 20 | p2$_1$22 | 7 | p2mg |
| 21 | p2$_1$2$_1$2 | 8 | p2gg |
| 22 | c222 | 9 | c2mm |
| 23 | pmm2 | 6 | p2mm |
| 24 | pma2 | 7 | p2mg |
| 25 | pba2 | 8 | p2gg |
| 26 | cmm2 | 9 | c2mm |
| 27 | pm2m | 3 | pm |
| 28 | pm2$_1$b | 3 | pm |
| 29 | pb2$_1$m | 4 | pg |
| 30 | pb2b | 3 | pm |
| 31 | pm2a | 3 | pm |
| 32 | pm2$_1$n | 4 | pg |
| 33 | pb2$_1$a | 4 | pg |
| 34 | pb2n | 5 | cm |
| 35 | cm2m | 5 | cm |
| 36 | cm2e | 3 | pm |
| 37 | pmmm | 6 | p2mm |
| 38 | pmaa | 6 | p2mm |
| 39 | pban | 10 | p4 |
| 40 | pmam | 7 | p2mg |
| 41 | pmma | 6 | p2mm |
| 42 | pman | 9 | c2mm |
| 43 | pbaa | 7 | p2mg |
| 44 | pbam | 8 | p2gg |
| 45 | pbma | 7 | p2mg |
| 46 | pmmn | 10 | p4 |
| 47 | cmmm | 9 | c2mm |
| 48 | cmme | 6 | p2mm |
| 49 | p4 | 10 | p4 |
| 50 | p4 | 10 | p4 |
| 51 | p4/m | 10 | p4 |
| 52 | p4/n | 12 | p4gm |
| 53 | p422 | 11 | p4mm |
| 54 | p42$_1$2 | 12 | p4gm |
| 55 | p4mm | 11 | p4mm |
| 56 | p4bm | 12 | p4gm |
| 57 | p42m | 11 | p4mm |
| 58 | p42$_1$m | 12 | p4gm |
| 59 | p4m2 | 11 | p4mm |
| 60 | p4b2 | 12 | p4gm |
| 61 | p4/mmm | 11 | p4mm |
| 62 | p4/nbm | 11 | p4mm |
| 63 | p4/mbm | 12 | p4gm |
| 64 | p4/nmm | 11 | p4mm |
| 65 | p3 | 13 | p3 |
| 66 | p3 | 16 | p6 |
| 67 | p312 | 14 | p3m1 |
| 68 | p321 | 15 | p31m |
| 69 | p3m1 | 14 | p3m1 |
| 70 | p31m | 15 | p31m |
| 71 | p31m | 17 | p6mm |
| 72 | p3m1 | 17 | p6mm |
| 73 | p6 | 16 | p6 |
| 74 | p6 | 13 | p3 |
| 75 | p6/m | 16 | p6 |
| 76 | p622 | 17 | p6mm |
| 77 | p6mm | 17 | p6mm |
| 78 | p6m2 | 14 | p3m1 |
| 79 | p62m | 15 | p31m |
| 80 | p6/mmm | 17 | p6mm |