The Automorphism Group of an Extremal [72,36,16] Code does not contain elements of order 6

The existence of an extremal code of length 72 is a long-standing open problem. Let C be a putative extremal code of length 72 and suppose that C has an automorphism g of order 6. We show that C, as an F_2<g>-module, is the direct sum of two modules, one easily determinable and the other one which has a very restrictive structure. We use this fact to do an exhaustive search and we do not find any code. This proves that the automorphism group of an extremal code of length 72 does not contain elements of order 6.