In mathematics, a rod group is a three-dimensional line group

Line group

A line group is a mathematical way of describing symmetries associated with moving along a line. These symmetries include repeating along that line, making that line a one-dimensional lattice...

 whose point group

Point group

In geometry, a point group is a group of geometric symmetries that keep at least one point fixed. Point groups can exist in a Euclidean space with any dimension, and every point group in dimension d is a subgroup of the orthogonal group O...

 is one of the axial crystallographic point groups

Crystallographic point group

In crystallography, a crystallographic point group is a set of symmetry operations, like rotations or reflections, that leave a central point fixed while moving other directions and faces of the crystal to the positions of features of the same kind...

. 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

Crystal system

In crystallography, the terms crystal system, crystal family, and lattice system each refer to one of several classes of space groups, lattices, point groups, or crystals...

 or lattice type, and by their point groups:

Triclinic
1	p1	2	p
Monoclinic/inclined
3	p211	4	pm11	5	pc11	6	p2/m11	7	p2/c11
Monoclinic/orthogonal
8	p112	9	p1121	10	p11m	11	p112/m	12	p1121/m
Orthorhombic
13	p222	14	p2221	15	pmm2	16	pcc2	17	pmc21
18	p2mm	19	p2cm	20	pmmm	21	pccm	22	pmcm
Tetragonal
23	p4	24	p41	25	p42	26	p43	27	p
28	p4/m	29	p42/m	30	p422	31	p4122	32	p4222
33	p4322	34	p4mm	35	p42cm, p42mc	36	p4cc	37	p2m, pm2
38	p2c, pc2	39	p4/mmm	40	p4/mcc	41	p42/mmc, p42/mcm
Trigonal
42	p3	43	p31	44	p32	45	p	46	p312, p321
47	p3112, p3121	48	p3212, p3221	49	p3m1, p31m	50	p3c1, p31c	51	pm1, p1m
52	pc1, p1c
Hexagonal
53	p6	54	p61	55	p62	56	p63	57	p64
58	p65	59	p	60	p6/m	61	p63/m	62	p622
63	p6122	64	p6222	65	p6322	66	p6422	67	p6522
68	p6mm	69	p6cc	70	p63mc, p63cm	71	pm2, p2m	72	pc2, p2c
73	p6/mmm	74	p6/mcc	75	p6/mmc, p6/mcm

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

See also

• Point group
  Point group
  In geometry, a point group is a group of geometric symmetries that keep at least one point fixed. Point groups can exist in a Euclidean space with any dimension, and every point group in dimension d is a subgroup of the orthogonal group O...
  
  
• Crystallographic point group
  Crystallographic point group
  In crystallography, a crystallographic point group is a set of symmetry operations, like rotations or reflections, that leave a central point fixed while moving other directions and faces of the crystal to the positions of features of the same kind...
  
  
• Space group
  Space group
  In mathematics and geometry, a space group is a symmetry group, usually for three dimensions, that divides space into discrete repeatable domains.In three dimensions, there are 219 unique types, or counted as 230 if chiral copies are considered distinct...
  
  
• Line group
  Line group
  A line group is a mathematical way of describing symmetries associated with moving along a line. These symmetries include repeating along that line, making that line a one-dimensional lattice...
  
  
• Frieze group
  Frieze group
  A frieze group is a mathematical concept to classify designs on two-dimensional surfaces which are repetitive in one direction, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art...
  
  
• Layer group
  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...
  
  

External links

• Bilbao Crystallographic Server, under "Subperiodic Groups: Layer, Rod and Frieze Groups"
• Nomenclature, Symbols and Classification of the Subperiodic Groups, V. Kopsky and D. B. Litvin