Abstract
Let C be a binary extremal self-dual code of length 96. We prove that an automorphism of C of order 3 has 6 or no fixed points and an automorphism of order 5 has 6 fixed points. Moreover, if all automorphisms of order 3 are fixed point free then Aut(C) is solvable and its order divides 253 or 255 or Aut(C) is the alternating group A5 which is the only possible group of order 60. Furthermore, |Aut(C)| = 20 or 40 cannot occur.
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