# Faithful semidirect product of E8 and Z4

## Definition

This group can be defined as the unique (up to isomorphism) group obtained as the semidirect product of elementary abelian group:E8 and cyclic group:Z4 acting on the elementary abelian group faithfully. An explicit presentation is given by:

Explicitly, the acting element is acting as the following matrix if we use as the basis for elementary abelian group:E8 viewed as a three-dimensional vector space over field:F2:

Note that this matrix has order four, explaining the group structure.

## 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

Want to compare and contrast group properties with other groups of the same order? Check out groups of order 32#Group properties

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

group of prime power order | Yes | ||

nilpotent group | Yes | prime power order implies nilpotent | |

supersolvable group | Yes | via nilpotent: finite nilpotent implies supersolvable | |

solvable group | Yes | via nilpotent: nilpotent implies solvable | |

abelian group | No |

## GAP implementation

### Group ID

This finite group has order 32 and has ID 6 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(32,6)`

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

`gap> G := SmallGroup(32,6);`

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

`IdGroup(G) = [32,6]`

or just do:

`IdGroup(G)`

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