Translation Groups for arbitrary Gauge Fields in Synthetic Crystals with real hopping amplitudes

The Cayley-crystals introduced in [F. R. Lux and E. Prodan, Annales Henri Poincar\'e 25(8), 3563 (2024)] are a class of lattices endowed with a Hamiltonian whose translation group $G$ is generic and possibly non-commutative. We show that these systems naturally realize the generalization of the so-called magnetic translation groups to arbitrary discrete gauge groups. A one-body dynamics emulates that of a particle carrying a superposition of charges, each coupled to distinct static gauge-field configuration. The possible types of gauge fields are determined by the irreducible representations of the commutator subgroup $C \subset G$, while the Wilson-loop configurations - which need not be homogeneous - are fixed by the embedding of $C$ in $G$. The role of other subgroups in shaping both the lattice geometry and the dynamics is analyze in depth assuming $C$ finite. We discuss a theorem of direct engineering relevance that, for any cyclic gauge group, yields all compatible translation groups. We then construct two-dimensional examples of Cayley-crystals equivalent to square lattices threaded by inhomogeneous magnetic fluxes. Importantly, Cayley-crystals can be realized with only real hopping amplitudes and in scalable geometries that can fit higher-than-3D dynamics, enabling experimental exploration and eventual exploitation in metamaterials, cQED, and other synthetic platforms.