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

Abstract

A computer calculation with $M$AGMA shows that there is noextremal self-dual binary code $\mathcal{C}$ of length $72$ whoseautomorphism group contains the symmetric group of degree $3$, thealternating group of degree $4$ or the dihedral group of order $8$.Combining this with the known results in the literature one obtainsthat $Aut(\mathcal{C})$ has order at most $5$ or isisomorphic to the elementary abelian group of order $8$.