I am trying to classify the groups of order $56=2^3\cdot 7$. I think I have successfully eliminated most of the cases, and I am left with the last: When the group of order 8 is normal and it is isomorphic to $Z_2\times Z_2\times Z_2$. So, we seek homomorphisms $\varphi:Z_7\to \mathrm{Aut}(Z_2\times Z_2\times Z_2)\cong \mathrm{GL}(3,\mathbb{Z}_2)$.
$\varphi$ is completely determined by its action on $1\in Z_7$, and since its order is 7, a prime, its image, $\varphi(1)$, must have order 1 or 7. The former corresponds to the trivial product $Z_2\times Z_2\times Z_2\times Z_7$. The interesting one would arise from the latter, and here is where I am lost. Yes, $|\mathrm{GL}(3,\mathbb{Z}_2)|=168=2^3\cdot 3\cdot 7$, so by Cauchy, the existence of such an element (of order 7) is guaranteed. But how do I find it?
In general, I seek advice on problems like this where the automorphism group is given by the general linear group. From here and there, I saw people using eigenvalues, but I am not familiar with that technique, given my training. If someone could be so kind as to explain that method in detail, it would be very greatly appreciated.