The Automorphism Group of a Self-Dual $[72, 36, 16]$ Binary 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 \BBF
₂
〈
g
〉 -module, is the direct sum of two modules; it is easy to determine one of them, while the other one has a very restrictive structure. We use this fact to do an exhaustive search and we do not find an extremal code. This proves that the automorphism group of an extremal code of length 72 does not contain elements of order 6.