Shifts-induced group of a toroidal cube

Consider $ [n]^d$ — a $ d$ -dimensional toroidal cube with side length $ n$ divided into $ n^d$ unit cubes. Define $ k$ -shift as a following permutation type on unit cubes: choose $ S \subset [n]$ with $ |S| = k$ , $ i \in S$ , and assign a number $ y_j \in [n]$ to every $ j \not \in S$ . For a unit cube $ (x_1, \ldots, x_d)$ , we change $ x_i$ to $ x_i + 1$ with looping over the edge if $ x_j = y_j$ for every $ j \not \in S$ , otherwise the cube stays in place. For instance, a $ 1$ -shift in a toroidal square would be cyclic shift of a row or a column, a $ 2$ -shift in a toroidal 3D cube is a shift of a planar layer, and so on.

Let $ G_d(n, k)$ be the group induced by all possible $ k$ -shifts in $ [n]^d$ . This question is concerned with identifying $ G_d(n, k)$ , or at least some of its properties.

  1. In which cases can one identify $ G_d(n, k)$ ? What is its size, simple subgroups, composition series?
  2. Clearly, for any $ k \in [d – 1]$ we have that $ G_d(n, k + 1)$ is a subgroup $ G_d(n, k)$ . Can we answer some of the questions above about $ G_d(n, k) / G_d(n, k + 1)$ ? Are there $ n, k, d$ such that the quotient above is trivial?
  3. Given a permutation $ \sigma \in \mathrm{Sym}([n]^d)$ , can one efficiently determine the highest $ k$ such that $ \sigma \in G_d(n, k)$ ?

Choice of $ n$ is not particularly important, for simplicity we may take $ n$ to be a prime number.