Another 80-dimensional extremal lattice

We show that the unimodular lattice associated to the rank 20 quaternionic matrix group SL2(F41)⊗ S̃3 ⊂ GL80(Z) is a fourth example of an 80-dimensional extremal lattice. Our method is to use the positivity of the Θ-series in conjunction with an enumeration of all the norm 10 vectors. The use of Aschbacher’s theorem on subgroups of finite classical groups (reliant on the classification of finite simple groups) provides one proof that this lattice is distinct from the previous three, while computing the inner product distribution of the minimal vectors is an alternative method. We give details of the latter, and this method also enables us to find the full automorphism group for each of the four lattices. As already noted by Nebe, this fourth lattice has an additional 2-extension in its automorphism group.