What is the name of the permutation group corresponding to all the translation operations in the $3$ directions $x$, $y$ and $z$ (with periodic boundary conditions) of a general rectangular discrete lattice of size $m \times n \times k$, where $m$, $n$ and $k$ are the number of lattice sites in the each direction?
What algorithms/libraries are there to obtain the basis for irreducible representations of this group? What I have in mind as way to construct this basis is assigning a "bit" at each site that can take one of two values either $1$ or $0$.
Tarek