# Dihedral group:D32

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]

## Contents

## Definition

This group is the dihedral group of degree sixteen and order thirty-two. It is given by the presentation:

.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 32#Arithmetic functions

## Group properties

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

abelian group | No | |

group of prime power order | Yes | |

nilpotent group | Yes | |

maximal class group | Yes | |

metacyclic group | Yes | |

metabelian group | Yes | |

maximal class group | Yes | |

directly indecomposable group | Yes | |

splitting-simple group | No | |

centrally indecomposable group | Yes |

## GAP implementation

### Group ID

This finite group has order 32 and has ID 18 among the groups of order 32 in GAP's SmallGroup library. For context, there are 51 groups of order 32. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(32,18)`

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

`gap> G := SmallGroup(32,18);`

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

`IdGroup(G) = [32,18]`

or just do:

`IdGroup(G)`

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

### Other descriptions

The group can be described using GAP's DihedralGroup function:

`DihedralGroup(32)`