# Groups of order 24

This article gives information about, and links to more details on, groups of order 24

See pages on algebraic structures of order 24| See pages on groups of a particular order

This article gives basic information comparing and contrasting groups of order 24. See also more detailed information on specific subtopics through the links:

Information type | Page summarizing information for groups of order 24 |
---|---|

element structure (element orders, conjugacy classes, etc.) | element structure of groups of order 24 |

subgroup structure | subgroup structure of groups of order 24 |

linear representation theory | linear representation theory of groups of order 24 projective representation theory of groups of order 24 modular representation theory of groups of order 24 |

endomorphism structure, automorphism structure | endomorphism structure of groups of order 24 |

group cohomology | group cohomology of groups of order 24 |

## Contents

## Statistics at a glance

To understand these in a broader context, see groups of order 3.2^n

### Factorization and useful forms

The number 24 has prime factors 2 and 3 and prime factorization:

Other expressions for this number are:

### Group counts

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 up to isomorphism | 15 | |

Number of abelian groups (i.e., finite abelian groups) up to isomorphism | 3 | (number of abelian groups of order ) times (number of abelian groups of order ) = (number of unordered integer partitions of 3) 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 (i.e., finite nilpotent groups) up to isomorphism | 5 | (number of groups of order 8) 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 (i.e., finite solvable groups) up to isomorphism | 15 | 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 possible multisets of composition factors, i.e., number of equivalence classes under composition factor-equivalence | 1 | Follows from all groups of this order being solvable. The only possibility is: cyclic group:Z2 (3 times), cyclic group:Z3 (1 time). See order of group is product of orders of composition factors and classification of possible multisets of composition factors for groups of a given order. |

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

## The list

There are 15 groups of order 24.

Group | Second part of GAP ID (ID is (24,second part)) | Nilpotency class | Derived length |
---|---|---|---|

nontrivial semidirect product of Z3 and Z8 | 1 | not nilpotent | 2 |

cyclic group:Z24 | 2 | 1 | 1 |

special linear group:SL(2,3) | 3 | not nilpotent | 3 |

dicyclic group:Dic24 | 4 | not nilpotent | 2 |

direct product of S3 and Z4 | 5 | not nilpotent | 2 |

dihedral group:D24 | 6 | not nilpotent | 2 |

direct product of Dic12 and Z2 | 7 | not nilpotent | 2 |

semidirect product of Z3 and D8 with action kernel V4 | 8 | not nilpotent | 2 |

direct product of Z6 and Z4 (also, direct product of Z12 and Z2) | 9 | 1 | 1 |

direct product of D8 and Z3 | 10 | 2 | 2 |

direct product of Q8 and Z3 | 11 | 2 | 2 |

symmetric group:S4 | 12 | not nilpotent | 3 |

direct product of A4 and Z2 | 13 | not nilpotent | 2 |

direct product of D12 and Z2 (also direct product of S3 and V4) | 14 | not nilpotent | 2 |

direct product of E8 and Z3 | 15 | 1 | 1 |

## Relation with other orders

### Divisors of the order

More in-depth information can be found under subgroup structure of groups of order 24.

Divisor | Quotient value | Number of groups of the order | Information on groups of the order | Relationship (subgroup perspective) | Relationship (quotient value) |
---|---|---|---|---|---|

2 | 12 | 1 | cyclic group:Z2 | ||

3 | 8 | 1 | cyclic group:Z3 | ||

4 | 6 | 2 | groups of order 4 | ||

6 | 4 | 2 | groups of order 6 | ||

8 | 3 | 5 | groups of order 8 | ||

12 | 2 | 5 | groups of order 12 |

### Multiples of the order

More in-depth information can be found under supergroups of groups of order 24.

Multiplier (other factor) | Multiple | Number of groups | Information on groups of the order | Relationship (subgroup perspective) | Relationship (quotient perspective) |
---|---|---|---|---|---|

2 | 48 | 52 | groups of order 48 | ||

3 | 72 | 50 | groups of order 72 | ||

4 | 96 | 231 | groups of order 96 | ||

5 | 120 | 47 | groups of order 120 | ||

6 | 144 | 197 | groups of order 144 | ||

7 | 168 | 57 | groups of order 168 | ||

8 | 192 | 1543 | groups of order 192 | ||

9 | 216 | 177 | groups of order 216 | ||

10 | 240 | 208 | groups of order 240 |

## GAP implementation

The order 24 is part of GAP's SmallGroup library. Hence, any group of order 24 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 24 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(24); There are 15 groups of order 24. They are sorted by their Frattini factors. 1 has Frattini factor [ 6, 1 ]. 2 has Frattini factor [ 6, 2 ]. 3 has Frattini factor [ 12, 3 ]. 4 - 8 have Frattini factor [ 12, 4 ]. 9 - 11 have Frattini factor [ 12, 5 ]. 12 - 15 have trivial Frattini subgroup. For the selection functions the values of the following attributes are precomputed and stored: IsAbelian, IsNilpotentGroup, IsSupersolvableGroup, IsSolvableGroup, LGLength, FrattinifactorSize and FrattinifactorId. This size belongs to layer 2 of the SmallGroups library. IdSmallGroup is available for this size.