Permutation polynomials and orthomorphism polynomials of degree six

A classic paper of Dickson gives a complete list of permutation polynomials of degree less than 6 over arbitrary finite fields, and degree 6 over finite fields of odd characteristic. However, some published statements have hinted that Dicksonʼs classification might be incomplete in the degree 6 case. We uncover the reason for this confusion, and confirm the list of degree 6 permutation polynomials over all finite fields. Using this classification, we determine the complete list of degree 6 orthomorphism polynomials. Additionally, we note that a family of permutation polynomials from Dicksonʼs list provides counterexamples to a published conjecture of Mullen.