# Dihedral group:D10

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## Contents

## Definition

This group is defined as the dihedral group of order ten. In other words, it is the semidirect product of the cyclic group of order five and a cyclic group of order two.

## Arithmetic functions

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

order | 10 | |

exponent | 10 | |

Fitting length | 2 | |

Frattini length | 1 | |

derived length | 2 |

## Group properties

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

cyclic group | No | |

abelian group | No | |

nilpotent group | No | |

metacyclic group | Yes | |

supersolvable group | Yes | |

solvable group | Yes |

## GAP implementation

### Group ID

This finite group has order 10 and has ID 1 among the groups of order 10 in GAP's SmallGroup library. For context, there are groups of order 10. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(10,1)`

For instance, we can use the following assignment in GAP to create the group and name it :

`gap> G := SmallGroup(10,1);`

Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:

`IdGroup(G) = [10,1]`

or just do:

`IdGroup(G)`

to have GAP output the group ID, that we can then compare to what we want.

### Other descriptions

This group can be defined using GAP's DihedralGroup function:

`DihedralGroup(10)`