# 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: $\langle x,y,z,a \mid x^2 = y^2 = z^2 = a^4 = e, xy = yx, xz = zx, yz = zy, ax = xa, aya^{-1} = xy, aza^{-1} = yz \rangle$

Explicitly, the acting element $a$ is acting as the following matrix if we use $\{ x,y,z \}$ as the basis for elementary abelian group:E8 viewed as a three-dimensional vector space over field:F2: $\begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \\\end{pmatrix}$

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 $G$:

gap> G := SmallGroup(32,6);

Conversely, to check whether a given group $G$ 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.