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 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

See also

References
  • Hitzer, E.S.M.; Ichikawa, D. (2008), "Representation of crystallographic subperiodic groups by geometric algebra", Electronic Proc. of AGACSE, Leipzig, Germany (3, 17-19 Aug. 2008), arXiv:1306.1280, Bibcode:2013arXiv1306.1280H
  • Kopsky, V.; Litvin, D.B., eds. (2002), International Tables for Crystallography, Volume E: Subperiodic groups, E (5th ed.), Berlin, New York: Springer-Verlag, doi:10.1107/97809553602060000105, ISBN 978-1-4020-0715-6
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