# Semidihedral group:SD32

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## Contents

## Definition

The group is the semidihedral group of order . In other words, it is defined by the following 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 | |

metacyclic group | Yes | |

metabelian group | Yes | |

nilpotent group | Yes | |

maximal class group | Yes | |

directly indecomposable group | Yes | |

centrally indecomposable group | Yes | |

splitting-simple group | No |

## GAP implementation

### Group ID

This finite group has order 32 and has ID 19 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(32,19)`

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

`gap> G := SmallGroup(32,19);`

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

`IdGroup(G) = [32,19]`

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 defined as:

gap> F := FreeGroup(2); <free group on the generators [ f1, f2 ]> gap> G := F/[F.1^(16), F.2^2, F.2 * F.1 * F.2 * F.1^9]; <fp group on the generators [ f1, f2 ]>

is the semidihedral group of order .