An extremal [72, 36, 16] binary code has no automorphism group containing Z2 x Z4, Q8, or Z10

Let $C$ be an extremal self-dual binary code of length 72 and $g\in \Aut(C) $ be an automorphism of order 2. We show that $C$ is a free $\F_2 $ module and use this to exclude certain subgroups of order 8 of $\Aut (C)$. We also show that $\Aut(C)$ does not contain an element of order 10.

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