The automorphism group of a self-dual [72,36,16][72,36,16] code is not an elementary abelian group of order 8

The existence of an extremal self-dual binary linear code CC of length 72 is a long-standing open problem. We continue the investigation of its automorphism group: looking at the combination of the subcodes fixed by different involutions and doing a computer calculation with Magma, we prove that Aut(C)Aut(C) is not isomorphic to the elementary abelian group of order 8. Combining this with the known results in the literature one obtains that Aut(C)Aut(C) has order at most 5.