Consider the group $G = \mathbb{Z}^2$ of translations on a 2D square lattice. The subgroup $H = 3 \mathbb{Z} \times 3 \mathbb{Z}$ consists of all translations with coordinates divisible by $3$. A coset of this subgroup, as seen on the lattice, is the set of points $3$ vertices apart in either direction. Any such coset can be represented by any of its elements. So we can represent all cosets by, in this case, $9$ points on the grid, any $9$ as long as they're in different cosets.
A natural set of coset representatives would be a $3$ by $3$ square, and an even more natural one a $3$ by $3$ square around the origin.
My question is: is there a way to make this rigorous? Not just for this group but for any, or at least some class (concretely, I am interested in wallpaper and space groups)?
If we consider $G$ as being generated by the translations $\hat{x} = (1, 0)$ and $\hat{y} = (0, 1)$, then the square around the origin are the coset representatives requiring the minimal amount of generators(/inverse generators) to construct. That is one attempt, but then the question becomes (apart from how that generalizes), why these generators? $\hat{x}$ and $\hat{y} + 10 \hat{x}$ also generate $G$, and now the point e.g. $(1, 1)$ is no longer minimal in this sense.
(Also, if instead we consider the subgroup $H^\prime = 2\mathbb{Z} \times 2 \mathbb{Z}$, we no longer have one intuitively natural set of coset generators, because there's no square with the origin in the middle, but rather four $2\times 2$ squares with the origin in one corner. But still, any of these $4$ sets seem more natural than any other.)