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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	2	Monoclinic/inclined	3
p1	p
p112	p11m	p11a	p112/m	p112/a
p211	p2111	c211	pm11	pb11
cm11	p2/m11	p21/m11	p2/b11	p21/b11
c2/m11
p222	p2122	p21212	c222	pmm2
pma2	pba2	cmm2	pm2m	pm21b
pb21m	pb2b	pm2a	pm21n	pb21a
pb2n	cm2m	cm2e	pmmm	pmaa
pban	pmam	pmma	pman	pbaa
pbam	pbma	pmmn	cmmm	cmme
p4	p	p4/m	p4/n	p422
p4212	p4mm	p4bm	p2m	p21m
pm2	pb2	p4/mmm	p4/nbm	p4/mbm
p4/nmm
p3	p	p312	p321	p3m1
p31m	p1m	pm1
p6	p	p6/m	p622	p6mm
pm2	p2m	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).

#		Layer Group		#		Plane Group
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31
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41
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49
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References

(2002). "International Tables for Crystallography, Volume E: Subperiodic Groups". [[Springer-Verlag]].
(17–19 Aug 2008). "Representation of crystallographic subperiodic groups by geometric algebra". Electronic Proc. of AGACSE.
Sze, W.H.R.. (2025). "Key difference of input data organization to the predictions of symmetry information and layer number for quasi-2D films from band structure". Computational Condensed Matter.

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euclidean-symmetries discrete-groups

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