Let $H$ be the subgroup of $\mathrm{GL}_4(\mathbb{Z})$ generated by the $4!$ permutation matrices together with $$ \begin{pmatrix} 1 & 0 & 0 & 0 \\ -1 & 0 & 1 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1\end{pmatrix} $$
What is the index of $H$ in $\mathrm{GL}_4(\mathbb{Z})$? (I'm actually hoping the index is $1$.)
This seems like the sort of thing that GAP or some other software might be able to do directly, but I'm not familiar with the area. My motivation is that I've found certain symmetries on objects indexed by $\mathbb{Z}_{\geq 0}^4$ and I'm probing whether or not I've "found all of them."
The group $H$ is reducible: it fixes the column vector $(1\ 1\ 1\ 1)^{\mathsf T}$.
So it has infinite index in ${\rm GL}(4,{\mathbb Z})$.