# Holomorph of a group

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## Definition

### Symbol-free definition

The **holomorph** of a group is defined as its external semidirect product with its automorphism group.

## Particular cases

Here are some examples for groups of small orders.

Note that for any complete group, i.e., a centerless group in which every automorphism is inner, the holomorph is isomorphic to the square of the group, i.e., its external direct product with itself. This is true even though the action given by the external semidirect product is nontrivial -- what's happening is that if we view the direct product as a semidirect product taking the *diagonal* subgroup as the complement, we get the holomorph. The smallest nontrivial example group is symmetric group:S3.

Group | Order of group | Automorphism group | Order of automorphism group | Holomorph | Order of holomorph |
---|---|---|---|---|---|

trivial group | 1 | trivial group | 1 | trivial group | 1 |

cyclic group:Z2 | 2 | trivial group | 1 | cyclic group:Z2 | 2 |

cyclic group:Z3 | 3 | cyclic group:Z2 | 2 | symmetric group:S3 | 6 |

cyclic group:Z4 | 4 | cyclic group:Z2 | 2 | dihedral group:D8 | 8 |

Klein four-group | 4 | symmetric group:S3 | 6 | symmetric group:S4 | 24 |

cyclic group:Z5 | 5 | cyclic group:Z4 | 4 | general affine group:GA(1,5) | 20 |

symmetric group:S3 | 6 | symmetric group:S3 | 6 | direct product of S3 and S3 | 36 |

cyclic group:Z6 | 6 | cyclic group:Z2 | 2 | dihedral group:D12 | 12 |

cyclic group:Z7 | 7 | cyclic group:Z6 | 6 | general affine group:GA(1,7) | 42 |

cyclic group:Z8 | 8 | Klein four-group | 4 | holomorph of Z8 | 32 |