# Dicyclic group:Dic20

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## Contents

## Definition

This group is defined as the dicyclic group of order , and hence degree . In other words, it has the presentation:

Alternatively, it has the presentation:

.

## Arithmetic functions

Function | Value | Similar groups | Explanation |
---|---|---|---|

order (number of elements, equivalently, cardinality or size of underlying set) | 20 | groups with same order | As dicyclic group of degree : |

exponent of a group | 10 | groups with same order and exponent of a group | groups with same exponent of a group | As dicyclic group of degree : . |

Frattini length | 2 | groups with same order and Frattini length | groups with same Frattini length | The Frattini subgroup is isomorphic to cyclic group:Z2 -- specifically, it is the center. |

derived length | 2 | groups with same order and derived length | groups with same derived length | The group is in fact a metacyclic group, hence metabelian, but it is not abelian. |

nilpotency class | -- | -- | Not a nilpotent group. |

minimum size of generating set | 2 | groups with same order and minimum size of generating set | groups with same minimum size of generating set | |

subgroup rank of a group | 2 | groups with same order and subgroup rank of a group | groups with same subgroup rank of a group |

## Group properties

### Important properties

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

cyclic group | No | |

abelian group | No | |

nilpotent group | No | |

metacyclic group | Yes | |

supersolvable group | Yes | |

solvable group | Yes |

### Other properties

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

ambivalent group | No | |

Schur-trivial group | Yes | This is true for all dicyclic groups. |

finite group with periodic cohomology | Yes | This is true for all dicyclic groups. |

## GAP implementation

### Group ID

This finite group has order 20 and has ID 1 among the groups of order 20 in GAP's SmallGroup library. For context, there are 5 groups of order 20. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(20,1)`

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

`gap> G := SmallGroup(20,1);`

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

`IdGroup(G) = [20,1]`

or just do:

`IdGroup(G)`

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