# Generalized dihedral group for E9

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## Definition

This group is defined as the generalized dihedral group corresponding to the elementary abelian group of order nine. In other words, it is the semidirect product of the elementary abelian group of order nine with a cyclic group of order two acting via the inverse map.

## Arithmetic functions

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

order | 18 | |

exponent | 6 | |

derived length | 2 | |

Frattini length | 1 | |

Fitting length | 2 |

## GAP implementation

### Group ID

This finite group has order 18 and has ID 4 among the groups of order 18 in GAP's SmallGroup library. For context, there are groups of order 18. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(18,4)`

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

`gap> G := SmallGroup(18,4);`

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

`IdGroup(G) = [18,4]`

or just do:

`IdGroup(G)`

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