List of planar symmetry groups
# List of planar symmetry groups

This article summarizes the classes of discrete symmetry groups of the Euclidean plane. The symmetry groups are named here by three naming schemes: International notation, orbifold notation, and Coxeter notation. There are three kinds of symmetry groups of the plane:

* 2 families of rosette groups – 2D point groups
* 7 frieze groups – 2D line groups
* 17 wallpaper groups – 2D space groups.

## Rosette groups

There are two families of discrete two-dimensional point groups, and they are specified with parameter n, which is the order of the group of the rotations in the group.

| Family | Intl (orbifold) | Schön. | Geo Coxeter | Order | Examples | Examples | Examples | Examples | Examples | Examples |
| - | - | - | - | - | - | - | - | - | - | - |
| Cyclic symmetry | n (n•) | $C_n$ | n $[n]^+$ | n | $C_1,$ [ $]^+$ (•) | $C_2,$ $[2]^+$ (2•) | $C_3,$ $[3]^+$ (3•) | $C_4,$ $[4]^+$ (4•) | $C_5,$ $[5]^+$ (5•) | $C_6,$ $[6]^+$ (6•) |
| Dihedral symmetry | nm (*n•) | $D_n$ | n [n] | 2n | $D_1,$ [ ] (*•) | $D_2,$ [2] (*2•) | $D_3,$ [3] (*3•) | $D_4,$ [4] (*4•) | $D_5,$ [5] (*5•) | $D_6,$ [6] (*6•) |

## Frieze groups

The 7 frieze groups, the two-dimensional line groups, with a direction of periodicity are given with five notational names. The Schönflies notation is given as infinite limits of 7 dihedral groups. The yellow regions represent the infinite fundamental domain in each.

[1,∞], :

| IUC (orbifold) | Geo | Schönflies | Coxeter | Fundamental domain | Example |
| - | - | - | - | - | - |
| p1m1 (*∞•) | p1 | $C_{∞v}$ | [1,∞] | | sidle |
| p1 (∞•) | p1 | $C_∞$ | $[1,∞]^+$ | | hop |

$[2,∞^+],$ :

| IUC (orbifold) | Geo | Schönflies | Coxeter | Fundamental domain | Example |
| - | - | - | - | - | - |
| p11m (∞*) | p. 1 | $C_{∞h}$ | $[2,∞^+]$ | | jump |
| p11g (∞×) | $p._g1$ | $S_{2∞}$ | $[2^+,∞^+]$ | | step |

[2,∞], :

| IUC (orbifold) | Geo | Schönflies | Coxeter | Example |
| - | - | - | - | - |
| p2mm (*22∞) | p2 | $D_{∞h}$ | [2,∞] | spinning jump |
| p2mg (2*∞) | $p2_g$ | $D_{∞d}$ | $[2^+,∞]$ | spinning sidle |
| p2 (22∞) | p2 | $D_∞$ | $[2,∞]^+$ | spinning hop |

## Wallpaper groups

The 17 wallpaper groups, with finite fundamental domains, are given by International notation, orbifold notation, and Coxeter notation, classified by the 5 Bravais lattices in the plane: square, oblique (parallelogrammatic), hexagonal (equilateral triangular), rectangular (centered rhombic), and rhombic (centered rectangular).

The p1 and p2 groups, with no reflectional symmetry, are repeated in all classes. The related pure reflectional Coxeter group are given with all classes except oblique.

Square [4,4], :

| IUC (Orb.) Geo | Coxeter | Domain Conway name |
| - | - | - |
| p1 (°) p1 | | Monotropic |
| p2 (2222) p2 | $[4,1^+,4]^+$ $[1^+,4,4,1^+]^+$ | Ditropic |
| pgg (22×) $p_g2_g$ | $[4^+,4^+]$ | Diglide |
| pmm (*2222) p2 | $[4,1^+,4]$ $[1^+,4,4,1^+]$ | Discopic |
| cmm (2*22) c2 | $[(4,4,2^+)]$ | Dirhombic |
| p4 (442) p4 | $[4,4]^+$ | Tetratropic |
| p4g (4*2) $p_g4$ | $[4^+,4]$ | Tetragyro |
| p4m (*442) p4 | [4,4] | Tetrascopic |

Rectangular $[∞_h,2,∞_v],$ :

| IUC (Orb.) Geo | Coxeter | Domain Conway name |
| - | - | - |
| p1 (°) p1 | $[∞^+,2,∞^+]$ | Monotropic |
| p2 (2222) p2 | $[∞,2,∞]^+$ | Ditropic |
| pg(h) (××) $p_g1$ | h: $[∞^+,(2,∞)^+]$ | Monoglide |
| pg(v) (××) $p_g1$ | v: $[(∞,2)^+,∞^+]$ | Monoglide |
| pgm (22*) $p_g2$ | h: $[(∞,2)^+,∞]$ | Digyro |
| pmg (22*) $p_g2$ | v: $[∞,(2,∞)^+]$ | Digyro |
| pm(h) (**) p1 | h: $[∞^+,2,∞]$ | Monoscopic |
| pm(v) (**) p1 | v: $[∞,2,∞^+]$ | Monoscopic |
| pmm (*2222) p2 | [∞,2,∞] | Discopic |

Rhombic $[∞_h,2^+,∞_v],$ :

| IUC (Orb.) Geo | Coxeter | Domain Conway name |
| - | - | - |
| p1 (°) p1 | $[∞^+,2^+,∞^+]$ | Monotropic |
| p2 (2222) p2 | $[∞,2^+,∞]^+$ | Ditropic |
| cm(h) (*×) c1 | h: $[∞^+,2^+,∞]$ | Monorhombic |
| cm(v) (*×) c1 | v: $[∞,2^+,∞^+]$ | Monorhombic |
| pgg (22×) $p_g2_g$ | $[((∞,2)^+)^{[2]}]$ | Diglide |
| cmm (2*22) c2 | $[∞,2^+,∞]$ | Dirhombic |

Parallelogrammatic (oblique) :

| p1 (°) p1 | Monotropic |
| - | - |
| p2 (2222) p2 | Ditropic |

Hexagonal/Triangular [6,3], / $[3^{[3]}],$ :

| IUC (Orb.) Geo | Coxeter | Domain Conway name |
| - | - | - |
| p1 (°) p1 | | Monotropic |
| p2 (2222) p2 | $[6,3]^Δ$ | Ditropic |
| cmm (2*22) c2 | $[6,3]^⅄$ | Dirhombic |
| p3 (333) p3 | $[1^+,6,3^+]$ $[3^{[3]}]^+$ | Tritropic |
| p3m1 (*333) p3 | $[1^+,6,3]$ $[3^{[3]}]$ | Triscopic |
| p31m (3*3) h3 | $[6,3^+]$ | Trigyro |
| p6 (632) p6 | $[6,3]^+$ | Hexatropic |
| p6m (*632) p6 | [6,3] | Hexascopic |

## Wallpaper subgroup relationships

Subgroup relationships among the 17 wallpaper group :

| | | o | 2222 | ×× | ** | *× | 22× | 22* | *2222 | 2*22 | 442 | 4*2 | *442 | 333 | *333 | 3*3 | 632 | *632 |
| - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - |
| | | p1 | p2 | pg | pm | cm | pgg | pmg | pmm | cmm | p4 | p4g | p4m | p3 | p3m1 | p31m | p6 | p6m |
| o | p1 | 2 | | | | | | | | | | | | | | | | |
| 2222 | p2 | 2 | 2 | 2 | | | | | | | | | | | | | | |
| ×× | pg | 2 | | 2 | | | | | | | | | | | | | | |
| ** | pm | 2 | | 2 | 2 | 2 | | | | | | | | | | | | |
| *× | cm | 2 | | 2 | 2 | 3 | | | | | | | | | | | | |
| 22× | pgg | 4 | 2 | 2 | | | 3 | | | | | | | | | | | |
| 22* | pmg | 4 | 2 | 2 | 2 | 4 | 2 | 3 | | | | | | | | | | |
| *2222 | pmm | 4 | 2 | 4 | 2 | 4 | 4 | 2 | 2 | 2 | | | | | | | | |
| 2*22 | cmm | 4 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | | | | | | | | |
| 442 | p4 | 4 | 2 | | | | | | | | 2 | | | | | | | |
| 4*2 | p4g | 8 | 4 | 4 | 8 | 4 | 2 | 4 | 4 | 2 | 2 | 9 | | | | | | |
| *442 | p4m | 8 | 4 | 8 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | | | | | |
| 333 | p3 | 3 | | | | | | | | | | | | 3 | | | | |
| *333 | p3m1 | 6 | | 6 | 6 | 3 | | | | | | | | 2 | 4 | 3 | | |
| 3*3 | p31m | 6 | | 6 | 6 | 3 | | | | | | | | 2 | 3 | 4 | | |
| 632 | p6 | 6 | 3 | | | | | | | | | | | 2 | | | 4 | |
| *632 | p6m | 12 | 6 | 12 | 12 | 6 | 6 | 6 | 6 | 3 | | | | 4 | 2 | 2 | 2 | 3 |
