# 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) |

Somewhat important (importance rank 3):

## Pages in category "Particular groups"

The following 200 pages are in this category, out of 492 total. The count *includes* redirect pages that have been included in the category. Redirect pages are shown in italics.

### D

- Direct product of Z4 and Z4 and Z4
- Direct product of Z6 and Z2
- Direct product of Z6 and Z3
- Direct product of Z6 and Z4
- Direct product of Z8 and D8
- Direct product of Z8 and E8
- Direct product of Z8 and V4
- Direct product of Z8 and Z2
- Direct product of Z8 and Z4
- Direct product of Z8 and Z4 and V4
- Direct product of Z8 and Z4 and Z2
- Direct product of Z8 and Z8
- Direct product of Z81 and Z3
- Direct product of Z9 and E27
- Direct product of Z9 and E9
- Direct product of Z9 and Z3
- Direct product of Z9 and Z9
- Direct product of Z9 and Z9 and Z3
- Double cover of alternating group:A7
- Double cover of alternating group:A8
- Double cover of projective special linear group:PSL(3,4)
- Double cover of symmetric group:S5 of minus type
- Double cover of symmetric group:S5 of plus type

### E

### F

- Finitary alternating group of countable degree
- Finitary symmetric group of countable degree
- Fischer group:Fi22
- Fischer group:Fi23
- Free abelian group of countable rank
- Free abelian group of rank three
- Free abelian group of rank two
- Free group of countable rank
- Free group:F2
- Free group:F3
- Free product of class two of two Klein four-groups
- Fundamental group of Klein bottle

### G

- GAPlus(1,R)
- General affine group:GA(1,5)
- General affine group:GA(1,7)
- General affine group:GA(1,8)
- General affine group:GA(1,Q)
- General affine group:GA(2,3)
- General linear group:GL(2,3)
- General linear group:GL(2,4)
- General linear group:GL(2,5)
- General linear group:GL(2,H)
- General linear group:GL(2,Q)
- General linear group:GL(2,Z)
- General linear group:GL(2,Z4)
- General linear group:GL(2,Z9)
- General linear group:GL(3,3)
- General semiaffine group:GammaA(1,9)
- General semilinear group:GammaL(1,8)
- Generalized dihedral group for 2-quasicyclic group
- Generalized dihedral group for additive group of 2-adic integers
- Generalized dihedral group for direct product of Z4 and Z4
- Generalized dihedral group for E9
- Generalized quaternion group:Q16
- Generalized quaternion group:Q32
- Generalized quaternion group:Q64
- Grigorchuk group
- Group of integers
- Group of rational numbers
- Group of rational numbers modulo integers
- Group of rational numbers with square-free denominators
- Group of recursive permutations
- Group of unit quaternions

### I

### M

### N

### P

- Panferov Lie group for 5
- Panferov Lie group for 7
- Profinite completion of the integers
- Projective general linear group:PGL(2,11)
- Projective general linear group:PGL(2,7)
- Projective general linear group:PGL(2,9)
- Projective general linear group:PGL(2,Z9)
- Projective special linear group:PSL(2,11)
- Projective special linear group:PSL(2,13)
- Projective special linear group:PSL(2,17)
- Projective special linear group:PSL(2,19)
- Projective special linear group:PSL(2,23)
- Projective special linear group:PSL(2,25)
- Projective special linear group:PSL(2,27)
- Projective special linear group:PSL(2,8)
- Projective special linear group:PSL(2,C)
- Projective special linear group:PSL(2,R)
- Projective special linear group:PSL(2,Z)
- Projective special linear group:PSL(2,Z9)
- Projective special linear group:PSL(3,2)
- Projective special linear group:PSL(3,3)
- Projective special linear group:PSL(3,4)
- Projective special linear group:PSL(3,5)
- Projective special linear group:PSL(3,9)
- Projective special unitary group:PSU(3,3)
- Projective symplectic group:PSp(4,3)
- Projective symplectic group:PSp(4,4)
- Projective symplectic group:PSp(6,2)
- Projective symplectic group:PSp(6,3)
- Pure braid group:P3

### Q

### S

- Schur cover of alternating group:A6
- Schur cover of alternating group:A7
- Semidihedral group:SD128
- Semidihedral group:SD16
- Semidihedral group:SD256
- Semidihedral group:SD32
- Semidihedral group:SD64
- Semidirect product of Z16 and Z4 of dihedral type
- Semidirect product of Z16 and Z4 of M-type
- Semidirect product of Z16 and Z4 of semidihedral type
- Semidirect product of Z16 and Z4 via cube map
- Semidirect product of Z16 and Z4 via fifth power map
- Semidirect product of Z3 and D8 with action kernel V4
- Semidirect product of Z5 and Z8 via inverse map
- Semidirect product of Z5 and Z8 via square map
- Semidirect product of Z8 and Z4 of dihedral type
- Semidirect product of Z8 and Z4 of M-type
- Semidirect product of Z8 and Z4 of semidihedral type
- Semidirect product of Z8 and Z8 of dihedral type
- Semidirect product of Z8 and Z8 of M-type
- Semidirect product of Z8 and Z8 of semidihedral type
- SmallGroup(128,1015)
- SmallGroup(16,3)
- SmallGroup(243,16)
- SmallGroup(243,19)
- SmallGroup(243,20)
- SmallGroup(256,27799)
- SmallGroup(256,29626)
- SmallGroup(256,6745)
- SmallGroup(32,10)
- SmallGroup(32,15)
- SmallGroup(32,2)
- SmallGroup(32,24)
- SmallGroup(32,27)
- SmallGroup(32,28)
- SmallGroup(32,29)
- SmallGroup(32,30)
- SmallGroup(32,31)
- SmallGroup(32,32)
- SmallGroup(32,33)
- SmallGroup(32,35)
- SmallGroup(32,44)
- SmallGroup(32,5)
- SmallGroup(32,7)
- SmallGroup(32,8)
- SmallGroup(32,9)
- SmallGroup(36,1)
- SmallGroup(36,12)