On finite self-complementary metacirculants

Abstract

We prove that the automorphism group of a self-complementary metacirculant is either soluble or has $$\mathrm{A}_5$$ A 5 as the only insoluble composition factor, extending a result of Li and Praeger which says the automorphism group of a self-complementary circulant is soluble. The proof involves a construction of self-complementary metacirculants which are Cayley graphs and have insoluble automorphism groups. To the best of our knowledge, these are the first examples of self-complementary graphs with this property.