The Automorphism Group of an Extremal $[{72,36,16}]$ Code Does Not Contain $Z_{7}$, $Z_{3}\times Z_{3}$, or $D_{10}$

A computer calculation with Magma shows that there is no extremal self-dual binary code
$C$
of length 72 that has an automorphism group containing either the dihedral group of order 10, the elementary abelian group of order 9, or the cyclic group of order 7. Combining this with the known results in the literature, one obtains that the order of
${\rm Aut}(C)$
is either 5 or divides 24.