# 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):

## 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.

### A

- Alternating group:A10
- Alternating group:A11
- Alternating group:A12
- Alternating group:A13
- Alternating group:A14
- Alternating group:A4
- Alternating group:A5
- Alternating group:A6
- Alternating group:A7
- Alternating group:A8
- Alternating group:A9
- Amalgamated free product of two copies of group of rational numbers over group of integers
- Amalgamated free product of Z and Z over 2Z
- Automorphism group of alternating group:A6
- Automorphism group of free group:F2

### B

### C

- Central product of D16 and Z4
- Central product of D8 and Q8
- Central product of D8 and Z12
- Central product of D8 and Z16
- Central product of D8 and Z4
- Central product of D8 and Z8
- Central product of M16 and Z8 over common Z2
- Central product of SL(2,3) and Z4
- Central product of SL(2,5) and SL(2,7)
- Central product of SL(2,5) and Z4
- Central product of UT(3,3) and Z9
- Central product of UT(3,Z) and Q
- Central product of UT(3,Z) and Z identifying center with 2Z
- Central product of Z9 and wreath product of Z3 and Z3
- Chevalley group of type B:B3(3)
- Circle group
- Conway group:Co0
- Conway group:Co1
- Conway group:Co2
- Conway group:Co3
- Cyclic group:Z10
- Cyclic group:Z12
- Cyclic group:Z128
- Cyclic group:Z16
- Cyclic group:Z18
- Cyclic group:Z2
- Cyclic group:Z20
- Cyclic group:Z24
- Cyclic group:Z243
- Cyclic group:Z27
- Cyclic group:Z3
- Cyclic group:Z32
- Cyclic group:Z36
- Cyclic group:Z4
- Cyclic group:Z40
- Cyclic group:Z5
- Cyclic group:Z6
- Cyclic group:Z64
- Cyclic group:Z7
- Cyclic group:Z8
- Cyclic group:Z81
- Cyclic group:Z9

### D

- Dicyclic group:Dic12
- Dicyclic group:Dic20
- Dicyclic group:Dic24
- Dihedral group:D10
- Dihedral group:D12
- Dihedral group:D128
- Dihedral group:D14
- Dihedral group:D16
- Dihedral group:D18
- Dihedral group:D20
- Dihedral group:D24
- Dihedral group:D256
- Dihedral group:D32
- Dihedral group:D64
- Dihedral group:D8
- Direct product of A4 and A4
- Direct product of A4 and D8
- Direct product of A4 and E8
- Direct product of A4 and Q8
- Direct product of A4 and S3
- Direct product of A4 and V4
- Direct product of A4 and Z2
- Direct product of A4 and Z3
- Direct product of A4 and Z4
- Direct product of A4 and Z4 and Z2
- Direct product of A4 and Z5
- Direct product of A4 and Z8
- Direct product of A5 and S3
- Direct product of A5 and SL(2,7)
- Direct product of A5 and V4
- Direct product of A5 and Z2
- Direct product of A5 and Z4
- Direct product of A5 and Z7
- Direct product of A6 and Z2
- Direct product of D12 and Z2
- Direct product of D12 and Z3
- Direct product of D16 and V4
- Direct product of D16 and Z2
- Direct product of D16 and Z3
- Direct product of D16 and Z4
- Direct product of D32 and Z2
- Direct product of D8 and D8
- Direct product of D8 and E8
- Direct product of D8 and Q8
- Direct product of D8 and S3
- Direct product of D8 and V4
- Direct product of D8 and Z2
- Direct product of D8 and Z3
- Direct product of D8 and Z4
- Direct product of D8 and Z4 and Z2
- Direct product of D8 and Z5
- Direct product of D8 and Z6
- Direct product of D8 and Z7
- Direct product of Dic12 and Z2
- Direct product of E16 and Z4
- Direct product of E8 and Z3
- Direct product of E8 and Z4
- Direct product of group of rational numbers and group of rational numbers modulo integers
- Direct product of holomorph of Z8 and Z2
- Direct product of M16 and V4
- Direct product of M16 and Z2
- Direct product of M16 and Z3
- Direct product of M16 and Z4
- Direct product of M27 and Z3
- Direct product of M32 and Z2
- Direct product of prime-cube order group:U(3,3) and Z3
- Direct product of Q16 and V4
- Direct product of Q16 and Z2
- Direct product of Q16 and Z3
- Direct product of Q16 and Z4
- Direct product of Q32 and Z2
- Direct product of Q8 and Q8
- Direct product of Q8 and S3
- Direct product of Q8 and V4
- Direct product of Q8 and Z3
- Direct product of Q8 and Z4
- Direct product of Q8 and Z5
- Direct product of S3 and Z3
- Direct product of S3 and Z4
- Direct product of S4 and V4
- Direct product of S4 and Z2
- Direct product of S4 and Z3
- Direct product of S4 and Z4
- Direct product of S4 and Z5
- Direct product of S5 and V4
- Direct product of S5 and Z2
- Direct product of SD16 and V4
- Direct product of SD16 and Z2
- Direct product of SD16 and Z3
- Direct product of SD16 and Z4
- Direct product of SD32 and Z2
- Direct product of SL(2,3) and V4
- Direct product of SL(2,3) and Z2
- Direct product of SL(2,3) and Z3
- Direct product of SL(2,3) and Z4
- Direct product of SL(2,5) and PSL(3,2)
- Direct product of SL(2,5) and SL(2,7)
- Direct product of SL(2,5) and Z2
- Direct product of SmallGroup(16,13) and V4
- Direct product of SmallGroup(16,13) and Z2
- Direct product of SmallGroup(16,13) and Z4
- Direct product of SmallGroup(16,3) and V4
- Direct product of SmallGroup(16,3) and Z3
- Direct product of SmallGroup(16,3) and Z4
- Direct product of SmallGroup(16,4) and V4
- Direct product of SmallGroup(16,4) and Z4
- Direct product of SmallGroup(32,12) and Z2
- Direct product of SmallGroup(32,13) and Z2
- Direct product of SmallGroup(32,14) and Z2
- Direct product of SmallGroup(32,2) and Z2
- Direct product of SmallGroup(32,24) and Z2
- Direct product of SmallGroup(32,27) and Z2
- Direct product of SmallGroup(32,33) and Z2
- Direct product of SmallGroup(32,4) and Z2
- Direct product of SmallGroup(32,49) and Z2
- Direct product of SmallGroup(32,50) and Z2
- Direct product of Z and Z2
- Direct product of Z10 and Z2
- Direct product of Z16 and V4
- Direct product of Z16 and Z2
- Direct product of Z16 and Z4
- Direct product of Z27 and E9
- Direct product of Z27 and Z3
- Direct product of Z27 and Z9
- Direct product of Z32 and Z2
- Direct product of Z4 and V4
- Direct product of Z4 and Z2
- Direct product of Z4 and Z4
- Direct product of Z4 and Z4 and V4
- Direct product of Z4 and Z4 and Z2
- Direct product of Z4 and Z4 and Z4
- Direct product of Z6 and Z2
- Direct product of Z6 and Z3