Introduction

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 [Plesken and Souvignier, 1997], 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 ([Plesken, 1974]) 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.