In Magma, a G-lattice L is a lattice upon which a finite integral matrix group G acts by right multiplication. Magma allows various computations with lattices associated with finite integral matrix groups by use of G-lattices.
The computation of the automorphism group of a lattice (i.e. the largest matrix group that acts on the lattice) and the testing of lattices for isometry is performed within Magma by a search designed by Bill Unger, which is based on the Plesken-Souvignier backtrack algorithm [PS97], together with ordered partition methods. Optionally, this may be combined with orthogonal decomposition code of Gabi Nebe.
If G is a finite integral matrix group, then Magma uses Plesken's centering algorithm ([Ple74]) to construct all G-invariant sublattices of a given G-lattice L. The lattice of G-invariant sublattices of L can be explored much like the lattice of submodules over finite fields.