# Category:Particular groups

*This is a category of articles about particular objects or specific concrete examples in a certain collection of objects*

This page lists particular groups, viz groups, each unique up to isomorphism.

See also number of groups of given order to get an idea of how many groups there are of particular orders, along with links to pages that compare and contrast groups of a particular order.

If you want to search for a group given its group ID as per the SmallGroup library for GAP or Magma, type in SmallGroup(order,ID) into the search bar at the top right of the page. For instance, if the ID is (32,33), type in SmallGroup(32,33) in the search bar. If the ID is (12,3), type in SmallGroup(12,3) in the search bar.

## Particular groups of importance

Extremely important (importance rank 1):

GAP ID | |
---|---|

Cyclic group:Z2 | 2 (1) |

Cyclic group:Z3 | 3 (1) |

Cyclic group:Z4 | 4 (1) |

Group of integers | |

Klein four-group | 4 (2) |

Symmetric group:S3 | 6 (1) |

Trivial group | 1 (1) |

Very important (importance rank 2):

GAP ID | |
---|---|

Alternating group:A4 | 12 (3) |

Alternating group:A5 | 60 (5) |

Alternating group:A6 | 360 (118) |

Dihedral group:D8 | 8 (3) |

Direct product of Z4 and Z2 | 8 (2) |

Free group:F2 | |

Projective special linear group:PSL(3,2) | 168 (42) |

Quaternion group | 8 (4) |

Special linear group:SL(2,3) | 24 (3) |

Special linear group:SL(2,5) | 120 (5) |

Symmetric group:S4 | 24 (12) |

Symmetric group:S5 | 120 (34) |

Symmetric group:S6 | 720 (763) |

Somewhat important (importance rank 3):

GAP ID | |
---|---|

Alternating group:A7 | |

Alternating group:A8 | |

Binary octahedral group | 48 (28) |

Central product of D8 and Z4 | 16 (13) |

Dihedral group:D16 | 16 (7) |

Direct product of A4 and Z2 | 24 (13) |

Direct product of D8 and Z2 | 16 (11) |

Direct product of S4 and Z2 | 48 (48) |

Double cover of symmetric group:S5 of minus type | 240 (89) |

Double cover of symmetric group:S5 of plus type | 240 (90) |

General linear group:GL(2,3) | 48 (29) |

Generalized quaternion group:Q16 | 16 (9) |

M16 | 16 (6) |

Mathieu group:M10 | 720 (765) |

Mathieu group:M9 | 72 (41) |

Nontrivial semidirect product of Z4 and Z4 | 16 (4) |

Projective general linear group:PGL(2,7) | 336 (208) |

Projective general linear group:PGL(2,9) | 720 (764) |

Projective special linear group:PSL(2,11) | 660 (13) |

Projective special linear group:PSL(2,8) | 504 (156) |

… further results |

## Subcategories

This category has only the following subcategory.

### G

## Pages in category "Particular groups"

The following 200 pages are in this category, out of 466 total. The count *includes* redirect pages that have been included in the category. Redirect pages are shown in italics.