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

Correspondence Between Layer Groups and Plane Groups

The surjective mapping from a layer group to a wallpaper 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.
#Layer Group#Plane Group
1p11p1
2p2p2
3p1122p2
4p11m1p1
5p11a1p1
6p112/m2p2
7p112/a2p2
8p2113pm
9p2114pg
10c2115cm
11pm113pm
12pb114pg
13cm115cm
14p2/m116p2mm
15p2/m117p2mg
16p2/b117p2mg
17p2/b118p2gg
18c2/m119c2mm
19p2226p2mm
20p2227p2mg
21p2228p2gg
22c2229c2mm
23pmm26p2mm
24pma27p2mg
25pba28p2gg
26cmm29c2mm
27pm2m3pm
28pm2b3pm
29pb2m4pg
30pb2b3pm
31pm2a3pm
32pm2n4pg
33pb2a4pg
34pb2n5cm
35cm2m5cm
36cm2e3pm
37pmmm6p2mm
38pmaa6p2mm
39pban10p4
40pmam7p2mg
41pmma6p2mm
42pman9c2mm
43pbaa7p2mg
44pbam8p2gg
45pbma7p2mg
46pmmn10p4
47cmmm9c2mm
48cmme6p2mm
49p410p4
50p10p4
51p4/m10p4
52p4/n12p4gm
53p42211p4mm
54p42212p4gm
55p4mm11p4mm
56p4bm12p4gm
57p2m11p4mm
58p2m12p4gm
59pm211p4mm
60pb212p4gm
61p4/mmm11p4mm
62p4/nbm11p4mm
63p4/mbm12p4gm
64p4/nmm11p4mm
65p313p3
66p16p6
67p31214p3m1
68p32115p31m
69p3m114p3m1
70p31m15p31m
71p1m17p6mm
72pm117p6mm
73p616p6
74p13p3
75p6/m16p6
76p62217p6mm
77p6mm17p6mm
78pm214p3m1
79p2m15p31m
80p6/mmm17p6mm