  A computer calculation with Magma shows that there is no extremal self-dual
binary code C of length 72, whose automorphism group contains the symmetric
group of degree 3, the alternating group of degree 4 or the dihedral group of
order 8. Combining this with the known results in the literature one obtains
that Aut(C) has order at most 5 or isomorphic to the elementary abelian group
of order 8.
