$S_4$ as a subgroup of $GL_3(\mathbb{F}_2)$

Question

I am trying to find a subgroup of $GL_3(\mathbb{F}_2)$, which is isomorphic to $S_4$. Our teacher gave us a hint: we should look at matrices with first column $(1, 0, 0)^T$.

But what is the next step?

My guess would be that there is some relation between this fact and the fact that $S_4$ is isomorphic to $GA(2,2)$ (general affine group), but I don't know how to proceed futher.

Answer 1

The affine group $GA_n(K)$ (for any field $K$) is embedded into the linear group $GL_{n+1}(K)$ by: $$[x\mapsto Ax+b]\mapsto\begin{pmatrix}1&0\\b&A\end{pmatrix}.$$