# 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.