On colourings of cubic lattices

Given the integral lattice ${\mathrm{\Lambda }}^d$ in $d$-dimensional Euclidean space, partitions of the lattice nodes into orbits of finite-index subgroups of $Aut({\mathrm{\Lambda }}^d)$ have been computed for $d\le 4$. These partitions can be interpreted as colourings of orbits defined up to permutation of colours. Complete results are obtained for $d=2$ up to 64 orbits, for $d=3$ up to 8 orbits, and for 2 orbits in dimension 4. The automorphism groups of the partitions are also determined. Our results for two orbits in dimension 3 correct the old result of H. Heesch [Z. Kristallogr., (1933), 85, 335–344] who overlooked one partition.