# 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](../../references.md#cite-autoisom)], 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](../../references.md#cite-plesken-thesis)]) 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.