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A computer calculation with Magma shows that there is no extremal self-dualbinary code C of length 72 that has an automorphism group containing D10,Z3xZ3, or Z7. Combining this with the known results in the literature oneobtains that Aut(C) is either Z5 or has order dividing 24.

The automorphism group of a self-dual binary [72,36,16] code does not contain Z7, Z3xZ3, or D10