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

Let C be an extremal self-dual binary code of length 72 and g∈Aut(C) be an automorphism of order 2. We show that C is a free F2〈g〉 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. Combining these results with the ones obtained in earlier papers we find that the order of Aut(C) is either 5 or divides 24. If 8 divides the order of Aut(C) then its Sylow 2-subgroup is either D8 or Z2×Z2×Z2.