In mathematics, a '''rod group''' is a three-dimensional line group whose point group is one of the axial crystallographic point groups. This constraint means that the point group must be the symmetry of some three-dimensional lattice.

Table of the 75 rod groups, organized by crystal system or lattice type, and by their point groups:

{| class="wikitable" ! colspan=10 | Triclinic |- !1 | p1 !2 | p{{overline|1}} |- ! colspan=10 | Monoclinic/inclined |- !3 | p211 !4 | pm11 !5 | pc11 !6 | p2/m11 !7 | p2/c11 |- ! colspan=10 | Monoclinic/orthogonal |- !8 | p112 !9 | p112<sub>1</sub> !10 | p11m !11 | p112/m !12 | p112<sub>1</sub>/m |- ! colspan=10 | Orthorhombic |- !13 | p222 !14 | p222<sub>1</sub> !15 | pmm2 !16 | pcc2 !17 | pmc2<sub>1</sub> |- !18 | p2mm !19 | p2cm !20 | pmmm !21 | pccm !22 | pmcm |- ! colspan=10 | Tetragonal |- !23 | p4 !24 | p4<sub>1</sub> !25 | p4<sub>2</sub> !26 | p4<sub>3</sub> !27 | p{{overline|4}} |- !28 | p4/m !29 | p4<sub>2</sub>/m !30 | p422 !31 | p4<sub>1</sub>22 !32 | p4<sub>2</sub>22 |- !33 | p4<sub>3</sub>22 !34 | p4mm !35 | p4<sub>2</sub>cm, p4<sub>2</sub>mc !36 | p4cc !37 | p{{overline|4}}2m, p{{overline|4}}m2 |- !38 | p{{overline|4}}2c, p{{overline|4}}c2 !39 | p4/mmm !40 | p4/mcc !41 | p4<sub>2</sub>/mmc, p4<sub>2</sub>/mcm |- ! colspan=10 | Trigonal |- !42 | p3 !43 | p3<sub>1</sub> !44 | p3<sub>2</sub> !45 | p{{overline|3}} !46 | p312, p321 |- !47 | p3<sub>1</sub>12, p3<sub>1</sub>21 !48 | p3<sub>2</sub>12, p3<sub>2</sub>21 !49 | p3m1, p31m !50 | p3c1, p31c !51 | p{{overline|3}}m1, p{{overline|3}}1m |- !52 | p{{overline|3}}c1, p{{overline|3}}1c |- ! colspan=10 | Hexagonal |- !53 | p6 !54 | p6<sub>1</sub> !55 | p6<sub>2</sub> !56 | p6<sub>3</sub> !57 | p6<sub>4</sub> |- !58 | p6<sub>5</sub> !59 | p{{overline|6}} !60 | p6/m !61 | p6<sub>3</sub>/m !62 | p622 |- !63 | p6<sub>1</sub>22 !64 | p6<sub>2</sub>22 !65 | p6<sub>3</sub>22 !66 | p6<sub>4</sub>22 !67 | p6<sub>5</sub>22 |- !68 | p6mm !69 | p6cc !70 | p6<sub>3</sub>mc, p6<sub>3</sub>cm !71 | p{{overline|6}}m2, p{{overline|6}}2m !72 | p{{overline|6}}c2, p{{overline|6}}2c |- !73 | p6/mmm !74 | p6/mcc !75 | p6{{sub|3}}/mmc, p6{{sub|3}}/mcm |}

The double entries are for orientation variants of a group relative to the perpendicular-directions lattice.

Among these groups, there are 8 enantiomorphic pairs.

== See also ==

* Point group * Crystallographic point group * Space group * Line group * Frieze group * Layer group

== References ==

* {{Citation | last1=Hitzer | first1=E.S.M. | last2=Ichikawa | first2=D. | title=Representation of crystallographic subperiodic groups by geometric algebra | url=http://sinai.apphy.u-fukui.ac.jp/gcj/publications/RCSGGA/RCSGGA.pdf | journal=Electronic Proc. Of AGACSE | issue=3, 17–19 Aug. 2008 | location=Leipzig, Germany | year=2008 | url-status=dead | archiveurl=https://web.archive.org/web/20120314155923/http://sinai.apphy.u-fukui.ac.jp/gcj/publications/RCSGGA/RCSGGA.pdf | archivedate=2012-03-14 }} * {{Citation | editor1-last=Kopsky | editor1-first=V. | editor2-last=Litvin | editor2-first=D.B. | title=International Tables for Crystallography, Volume E: Subperiodic groups | url=http://it.iucr.org/E/ | publisher=Springer-Verlag | location=Berlin, New York | edition=5th | isbn=978-1-4020-0715-6 |doi= 10.1107/97809553602060000105 | year=2002 | volume=E| url-access=subscription }}

== External links == * [http://www.cryst.ehu.es/ "Subperiodic Groups: Layer, Rod and Frieze Groups"] on Bilbao Crystallographic Server * [http://www.bk.psu.edu/faculty/litvin/Download/D_IUCr_Report.pdf Nomenclature, Symbols and Classification of the Subperiodic Groups, V. Kopsky and D. B. Litvin]

Category:Euclidean symmetries Category:Discrete groups