# Groups of order 12

This article gives information about, and links to more details on, groups of order 12

See pages on algebraic structures of order 12| See pages on groups of a particular order

This article gives basic information comparing and contrasting groups of order 12. See also more detailed information on specific subtopics through the links:

Information type | Page summarizing information for groups of order 12 |
---|---|

element structure (element orders, conjugacy classes, etc.) | element structure of groups of order 12 |

subgroup structure | subgroup structure of groups of order 12 |

linear representation theory | linear representation theory of groups of order 12 projective representation theory of groups of order 12 modular representation theory of groups of order 12 |

endomorphism structure, automorphism structure | endomorphism structure of groups of order 12 |

group cohomology | group cohomology of groups of order 12 |

## Contents

## Statistics at a glance

The number 12 has prime factorization . There are only two prime factors of this number. Order has only two prime factors implies solvable (by Burnside's -theorem) and hence all groups of this order are solvable groups (specifically, finite solvable groups). Another way of putting this is that the order is a solvability-forcing number. In particular, there is no simple non-abelian group of this order.

Quantity | Value | Explanation |
---|---|---|

Total number of groups | 5 | |

Number of abelian groups | 2 | (number of abelian groups of order ) times (number of abelian groups of order ) = (number of unordered integer partitions of 2) times (number of unordered integer partitions of 1) = . See classification of finite abelian groups and structure theorem for finitely generated abelian groups. |

Number of nilpotent groups | 2 | (number of groups of order 4) times (number of groups of order 3) = See number of nilpotent groups equals product of number of groups of order each maximal prime power divisor, which in turn follows from equivalence of definitions of finite nilpotent group. |

Number of solvable groups | 5 | There are only two prime factors of this number. Order has only two prime factors implies solvable (by Burnside's -theorem) and hence all groups of this order are solvable groups (specifically, finite solvable groups). Another way of putting this is that the order is a solvability-forcing number. In particular, there is no simple non-abelian group of this order. |

Number of simple groups | 0 | Follows from all groups of this order being solvable. |

CLASSIFICATION: For an explanation of how toarriveat this list of groups, and prove that it is comprehensive, see classification of groups of order 12.

## The list

Group | Second part of GAP ID (GAP ID is (12,second part)) | Abelian? | 2-Sylow subgroup | Is the 2-Sylow subgroup normal? | Is the 3-Sylow subgroup normal? |
---|---|---|---|---|---|

dicyclic group:Dic12 | 1 | No | cyclic group:Z4 | No | Yes |

cyclic group:Z12 | 2 | Yes | cyclic group:Z4 | Yes | Yes |

alternating group:A4 | 3 | No | Klein four-group | Yes | No |

dihedral group:D12 | 4 | No | Klein four-group | No | Yes |

direct product of Z6 and Z2 | 5 | Yes | Klein four-group | Yes | Yes |

## Arithmetic functions

Function | dicyclic group:Dic12 | cyclic group:Z12 | alternating group:A4 | dihedral group:D12 | direct product of Z6 and Z2 |
---|---|---|---|---|---|

exponent | 12 | 12 | 6 | 6 | 6 |

derived length | 2 | 1 | 2 | 2 | 1 |

Fitting length | 2 | 1 | 2 | 2 | 1 |

nilpotency class | -- | 1 | -- | -- | 1 |

minimum size of generating set | 2 | 1 | 2 | 2 | 2 |

subgroup rank | 2 | 1 | 2 | 2 | 2 |

number of conjugacy classes | 6 | 12 | 4 | 6 | 12 |

### Numerical invariants

Group | Conjugacy class sizes | degrees of irreducible representations |
---|---|---|

dicyclic group:Dic12 | 1,1,2,2,3,3 | 1,1,1,1,2,2 |

cyclic group:Z12 | 1 (12 times) | 1 (12 times) |

alternating group:A4 | 1,3,4,4 | 1,1,1,3 |

dihedral group:D12 | 1,1,2,2,3,3 | 1,1,1,1,2,2 |

direct product of Z6 and Z2 | 1 (12 times) | 1 (12 times) |

## Group properties

Function | dicyclic group:Dic12 | cyclic group:Z12 | alternating group:A4 | dihedral group:D12 | direct product of Z6 and Z2 |
---|---|---|---|---|---|

cyclic group | No | Yes | No | No | No |

abelian group | No | Yes | No | No | Yes |

metacyclic group | Yes | Yes | No | Yes | Yes |

nilpotent group | No | Yes | No | No | Yes |

supersolvable group | Yes | Yes | No | Yes | Yes |

solvable group | Yes | Yes | Yes | Yes | Yes |

T-group | Yes | Yes | No | Yes | Yes |

## Sylow subgroups

### 2-Sylow subgroups

Here is the occurrence summary:

Group of order 4 | GAP ID (second part) | Number of groups of order 12 in which it is a 2-Sylow subgroup | List of these groups | Second part of GAP IDs of these groups |
---|---|---|---|---|

cyclic group:Z4 | 1 | 2 | dicyclic group:Dic12, cyclic group:Z12 | 1,2 |

Klein four-group | 2 | 3 | alternating group:A4, dihedral group:D12, direct product of Z6 and Z2 | 3,4,5 |

Note that the number of 2-Sylow subgroups is either 1 or 3. The former happens if and only if we have a normal Sylow subgroup for the prime 2. The latter happens if and only if we have a self-normalizing Sylow subgroup for the prime 2.

Group | Second part of GAP ID (ID is (12,second part)) | 2-Sylow subgroup | Second part of GAP ID | Number of 2-Sylow subgroups | Subgroup structure page |
---|---|---|---|---|---|

dicyclic group:Dic12 | 1 | cyclic group:Z4 | 1 | 3 | subgroup structure of dicyclic group:Dic12, subgroup structure of dicyclic groups |

cyclic group:Z12 | 2 | cyclic group:Z4 | 1 | 1 | subgroup structure of cyclic group:Z12, subgroup structure of cyclic groups |

alternating group:A4 | 3 | Klein four-group | 2 | 1 | subgroup structure of alternating group:A4, subgroup structure of alternating groups |

dihedral group:D12 | 4 | Klein four-group | 2 | 3 | subgroup structure of dihedral group:D12, subgroup structure of dihedral groups |

direct product of Z6 and Z2 | 5 | Klein four-group | 2 | 1 | subgroup structure of direct product of Z6 and Z2, subgroup structure of finite abelian groups |

The only *non-abelian* (and also the only *non-nilpotent*) example where there is a normal Sylow subgroup is the case of alternating group:A4.

## GAP implementation

The order 12 is part of GAP's SmallGroup library. Hence, any group of order 12 can be constructed using the SmallGroup function by specifying its group ID. Also, IdGroup is available, so the group ID of any group of this order can be queried.

Further, the collection of all groups of order 12 can be accessed as a list using GAP's AllSmallGroups function.

Here is GAP's summary information about how it stores groups of this order, accessed using GAP's SmallGroupsInformation function:

gap> SmallGroupsInformation(12); There are 5 groups of order 12. 1 is of type 6.2. 2 is of type c12. 3 is of type A4. 4 is of type D12. 5 is of type 2^2x3. The groups whose order factorises in at most 3 primes have been classified by O. Hoelder. This classification is used in the SmallGroups library. This size belongs to layer 1 of the SmallGroups library. IdSmallGroup is available for this size.