I have learned that lattices defined on the complex plane can be defined by $\mathbb Z\tau_1\oplus\mathbb Z\tau_2$, where $\tau_1,\tau_2\in\mathbb C$ are linearly independent over $\mathbb R$. I have also learned that $(\tau_1,\tau_2)$ and $(\tau_1',\tau_2')$ define the same lattice when: $$(\tau_1',\tau_2')=(a\tau_1+b\tau_2,c\tau_1+d\tau_2)$$ For $\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\mathrm{SL}_2(\mathbb Z)$.
I'm wondering what the more general form of this is. I don't know what the technical term is, but let's call $\mathrm{SL}_2(\mathbb Z)$ the "Lattice Transition Group" of $\mathbb C$, in that it defines all isomorphisms between lattices in $\mathbb C$.
If we fix the lattice transition group of some group $G$ as $\mathrm{SL}_n(\mathbb Z)$, what can be said of $G$? Is there even a group for every $n$ (or even any $n>2$?) that would satisfy this property? If so, what are the properties of this group?