# Sylow subgroup of holomorph of cyclic group of odd prime-cube order

## Contents

## Definition

Let be an *odd* prime number. This group is defined as the -Sylow subgroup of the holomorph of the cyclic group of order . Equivalently, it is the semidirect product of the cyclic group of order and the cyclic group of order , where the generator of the latter acts on the former by the -power map.

It can be given by the explicit presentation:

Note that the case is different, because the automorphism structure of the cyclic group of order is different, with a non-cyclic 2-Sylow subgroup of automorphisms. See holomorph of Z8 for details on that group.

## Arithmetic functions

Function | Value | Explanation |
---|---|---|

order | semidirect product of groups of order and | |

exponent | element has order , no element of order | |

derived length | 2 | commutator subgroup cyclic of order |

nilpotency class | 3 | |

Frattini length | 3 | Frattini subgroup is , isomorphic to semidirect product of cyclic group of prime-square order and cyclic group of prime order |

Fitting length | 1 | nilpotent. |

minimum size of generating set | 2 | |

subgroup rank | 2 | |

rank as p-group | 2 | abelian subgroup |

normal rank | 2 | abelian normal subgroup |

characteristic rank | 2 | first omega subgroup is |

## Group properties

Property | Satisfied | Explanation |
---|---|---|

abelian group | No | |

group of prime power order | Yes | |

metabelian group | Yes | |

group of nilpotency class two | No | |

metacyclic group | Yes | |

finite p-group that is not characteristic in any finite p-group properly containing it | Yes | |

regular p-group | Yes for | |

Lazard Lie group | Yes for |

## Subgroup-defining functions

## Facts

- For odd, this is a finite p-group that is not characteristic in any finite p-group properly containing it.

## GAP implementation

Here, `p</math> is the previously assigned value of the prime number, that we assume here to be odd. You can replace <tt>p` by an actual numerical value of a prime or precede these commands by an assignment `p := ` the value.

gap> C := CyclicGroup(p^3); <pc group of size 343 with 3 generators> gap> A := AutomorphismGroup(C); <group of size 294 with 6 generators> gap> S := SylowSubgroup(A,p); <group> gap> G := SemidirectProduct(S,C); <pc group with 5 generators>