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Let C be a binary extremal self-dual code of length n Â¿ 48. We prove that for each Â¿ Â¿ Aut(C) of prime order p Â¿ 5 the number of fixed points in the permutation action on the coordinate positions is bounded by the number of p-cycles. It turns out that large primes p, i.e., n-p small, seem to occur in |Aut(C)| very rarely. Examples are the extended quadratic residue codes. We further prove that doubly even extended quadratic residue codes of length n = p + 1 are extremal only in the cases n =8, 24, 32, 48, 80, and 104.