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.