Abstract: A skew morphism φ of a finite group G is a permutation of G fixing the identity of G and satisfying the property φ(gh)=φ(g)φ^π(g)(h) for any g,h∈G, where π is a function from G to {1,2,…,d-1} for the order d of φ. If for any g∈G, π(g)=1, then φ is an automorphism of G. Hence a skew morphism is a generalization of an automorphism. When π(φ(g))=π(g)for any g∈G, the skew morphism φ is called a smooth skew morphism. In this paper, we classify all smooth skew morphisms of a kind of maximal class 3-groups which have abelian maximal subgroups.

Key words: maximal class 3-group, smooth skew morphism, regular Cayley map, skew morphism, maximal subgroup