  The paper narrows down the possible automorphisms of extremal even unimodular
lattices of dimension 72. With extensive computations in {\sc Magma} using the
very sophisticated algorithm for computing class groups of algebraic number
fields written by Steve Donnelly it is shown that the extremal even unimodular
lattice $\Gamma _{72}$ from \cite{dim72} is the unique extremal even unimodular
lattice of dimension 72 that admits a large automorphism, where a $d\times d$
matrix is called large, if its minimal polynomial has an irreducible factor of
degree $> d/2$.
