# Rod group

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In mathematics, a '''rod group''' is a three-dimensional [line group](/source/line_group) whose [point group](/source/point_group) is one of the axial [crystallographic point group](/source/crystallographic_point_group)s. 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](/source/crystal_system) or lattice type, and by their point groups:

{| class="wikitable"
! colspan=10 | [Triclinic](/source/Triclinic)
|-
!1
| p1
!2
| p{{overline|1}}
|-
! colspan=10 | [Monoclinic](/source/Monoclinic)/inclined
|-
!3
| p211
!4
| pm11
!5
| pc11
!6
| p2/m11
!7
| p2/c11
|-
! colspan=10 | [Monoclinic](/source/Monoclinic)/orthogonal
|-
!8
| p112
!9
| p112<sub>1</sub>
!10
| p11m
!11
| p112/m
!12
| p112<sub>1</sub>/m
|-
! colspan=10 | [Orthorhombic](/source/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](/source/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](/source/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](/source/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](/source/Point_group)
* [Crystallographic point group](/source/Crystallographic_point_group)
* [Space group](/source/Space_group)
* [Line group](/source/Line_group)
* [Frieze group](/source/Frieze_group)
* [Layer group](/source/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](/source/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](/source/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

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Adapted from the Wikipedia article [Rod group](https://en.wikipedia.org/wiki/Rod_group) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Rod_group?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.