The automorphism group of a self-dual binary [72,36,16] code does not   contain Z7, Z3xZ3, or D10
  A computer calculation with Magma shows that there is no extremal self-dual binary code C of length 72 that has an automorphism group containing D10, Z3xZ3, or Z7. Combining this with the known results in the literature one obtains that Aut(C) is either Z5 or has order dividing 24. 
Thomas Feulner and Gabriele Nebe
cs.IT math.IT