  It has been proven in a series of works that the order of the automorphism
group of a binary [72,36,16] code does not exceed five. We obtain a
parametrization of all self-dual binary codes of length 72 with automorphism of
order 4 which can be extremal. We use extensive computations in MAGMA and on a
supercomputer to show that an extremal binary code of length 72 does not have
an element of order 4.
