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

The automorphism group of a self-dual [72,36,16] code does not contain S_3, A_4, or D_8