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The paper narrows down the possible automorphisms of extremal even unimodularlattices of dimension 72. With extensive computations in {\sc Magma} using thevery sophisticated algorithm for computing class groups of algebraic numberfields written by Steve Donnelly it is shown that the extremal even unimodularlattice $\Gamma _{72}$ from \cite{dim72} is the unique extremal even unimodularlattice 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 ofdegree $> d/2$.

Automorphisms of extremal unimodular lattices in dimension 72