# Dihedral group:D20

From Groupprops

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this groupView a complete list of particular groups (this is a very huge list!)[SHOW MORE]

## Definition

This group is defined in the following equivalent ways:

- It is the dihedral group of order .
- It is the direct product of the dihedral group of order ten and the cyclic group of order two.

## Arithmetic functions

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

order | 20 | |

exponent | 10 | |

Frattini length | 1 | |

Fitting length | 1 | |

derived length | 2 |

## Group properties

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

cyclic group | No | |

abelian group | No | |

metacyclic group | Yes | |

supersolvable group | Yes | |

solvable group | Yes |

## GAP implementation

### Group ID

This finite group has order 20 and has ID 4 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(20,4)`

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

`gap> G := SmallGroup(20,4);`

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

`IdGroup(G) = [20,4]`

or just do:

`IdGroup(G)`

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