# Element structure of groups of order 24

From Groupprops

This article gives specific information, namely, element structure, about a family of groups, namely: groups of order 24.

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## Conjugacy class structure

FACTS TO CHECK AGAINST FOR CONJUGACY CLASS SIZES AND STRUCTURE:Divisibility facts: size of conjugacy class divides order of group | size of conjugacy class divides index of center | size of conjugacy class equals index of centralizerBounding facts: size of conjugacy class is bounded by order of derived subgroupCounting facts: number of conjugacy classes equals number of irreducible representations | class equation of a group

### Full listing

Group | Second part of GAP ID (ID is (24,second part)) | Nilpotency class | Derived length | Number of conjugacy classes of size 1 (= order of center) | Number of conjugacy classes of size 2 | Number of conjugacy classes of size 3 | Number of conjugacy classes of size 4 | Number of conjugacy classes of size 6 | Number of conjugacy classes of size 8 | Total number of conjugacy classes |
---|---|---|---|---|---|---|---|---|---|---|

nontrivial semidirect product of Z3 and Z8 | 1 | not nilpotent | 2 | 4 | 4 | 4 | 0 | 0 | 0 | 12 |

cyclic group:Z24 | 2 | 1 | 1 | 24 | 0 | 0 | 0 | 0 | 0 | 24 |

special linear group:SL(2,3) | 3 | not nilpotent | 3 | 2 | 0 | 0 | 4 | 1 | 0 | 7 |

dicyclic group:Dic24 | 4 | not nilpotent | 2 | 2 | 5 | 0 | 0 | 2 | 0 | 9 |

direct product of S3 and Z4 | 5 | not nilpotent | 2 | 4 | 4 | 4 | 0 | 0 | 0 | 12 |

dihedral group:D24 | 6 | not nilpotent | 2 | 2 | 5 | 0 | 0 | 2 | 0 | 9 |

direct product of Dic12 and Z2 | 7 | not nilpotent | 2 | 4 | 4 | 4 | 0 | 0 | 0 | 12 |

SmallGroup(24,8) | 8 | not nilpotent | 2 | 2 | 5 | 0 | 0 | 2 | 0 | 9 |

direct product of Z6 and Z4 (also, direct product of Z12 and Z2) | 9 | 1 | 1 | 24 | 0 | 0 | 0 | 0 | 0 | 24 |

direct product of D8 and Z3 | 10 | 2 | 2 | 6 | 9 | 0 | 0 | 0 | 0 | 15 |

direct product of Q8 and Z3 | 11 | 2 | 2 | 6 | 9 | 0 | 0 | 0 | 0 | 15 |

symmetric group:S4 | 12 | not nilpotent | 3 | 1 | 0 | 1 | 0 | 2 | 1 | 5 |

direct product of A4 and Z2 | 13 | not nilpotent | 2 | 2 | 0 | 2 | 4 | 0 | 0 | 8 |

direct product of D12 and Z2 (also direct product of S3 and V4) | 14 | not nilpotent | 2 | 4 | 4 | 4 | 0 | 0 | 0 | 12 |

direct product of E8 and Z3 | 15 | 1 | 1 | 24 | 0 | 0 | 0 | 0 | 0 | 24 |

### Grouping by conjugacy class sizes

Number of conjugacy classes of size 1 (= order of center) | Number of conjugacy classes of size 2 | Number of conjugacy classes of size 3 | Number of conjugacy classes of size 4 | Number of conjugacy classes of size 6 | Number of conjugacy classes of size 8 | Total number of conjugacy classes | Number of groups with these conjugacy class size statistics | Nilpotency class(es) attained | Derived lengths attained | Description of groups | List of groups | List of GAP IDs (second part) |
---|---|---|---|---|---|---|---|---|---|---|---|---|

24 | 0 | 0 | 0 | 0 | 0 | 24 | 3 | 1 | 1 | all the abelian groups of order 24 | cyclic group:Z24, direct product of Z6 and Z4, direct product of E8 and Z3 | 2, 9, 15 |

6 | 9 | 0 | 0 | 0 | 0 | 15 | 2 | 2 | 2 | nilpotent non-abelian groups of order 24 | direct product of D8 and Z3, direct product of Q8 and Z3 | 10, 11 |

4 | 4 | 4 | 0 | 0 | 0 | 12 | 4 | non-nilpotent | 2 | nontrivial semidirect product of Z3 and Z8, direct product of S3 and Z4, direct product of Dic12 and Z2, direct product of D12 and Z2 (also described as direct product of S3 and V4) | 1, 5, 7, 14 | |

2 | 0 | 2 | 4 | 0 | 0 | 8 | 1 | non-nilpotent | 2 | direct product of A4 and Z2 | 13 | |

2 | 5 | 0 | 0 | 2 | 0 | 9 | 3 | non-nilpotent | 2 | dicyclic group:Dic24, dihedral group:D24, SmallGroup(24,8) | 4, 6, 8 | |

2 | 0 | 0 | 4 | 1 | 0 | 7 | 1 | non-nilpotent | 3 | special linear group:SL(2,3) | 3 | |

1 | 0 | 1 | 0 | 2 | 1 | 5 | 1 | non-nilpotent | 3 | symmetric group:S4 | 12 |

### Correspondence between degrees of irreducible representations and conjugacy class sizes

See also linear representation theory of groups of order 24#Degrees of irreducible representations.

For groups of order 24, it is true that the list of conjugacy class sizes completely determines the list of degrees of irreducible representations, and vice versa. The details are given below. The middle column, which is the total number of each, separates the description of the list of conjugacy class sizes and the list of degrees of irreducible representations:

Number of conjugacy classes of size 1 | Number of conjugacy classes of size 2 | Number of conjugacy classes of size 3 | Number of conjugacy classes of size 4 | Number of conjugacy classes of size 6 | Number of conjugacy classes of size 8 | Total number of conjugacy classes = number of irreducible representations | Number of irreps of degree 1 | Number of irreps of degree 2 | Number of irreps of degree 3 |
---|---|---|---|---|---|---|---|---|---|

24 | 0 | 0 | 0 | 0 | 0 | 24 | 24 | 0 | 0 |

6 | 9 | 0 | 0 | 0 | 0 | 15 | 12 | 3 | 0 |

4 | 4 | 4 | 0 | 0 | 0 | 12 | 8 | 4 | 0 |

2 | 0 | 2 | 4 | 0 | 0 | 8 | 6 | 0 | 2 |

2 | 5 | 0 | 0 | 2 | 0 | 9 | 4 | 5 | 0 |

2 | 0 | 0 | 4 | 1 | 0 | 7 | 3 | 3 | 1 |

1 | 0 | 1 | 0 | 2 | 1 | 5 | 2 | 1 | 2 |

Note that the phenomenon of the conjugacy class size statistics and degrees of irreducible representations determining one another is not true for all orders: