Using mixed dihedral groups to construct normal Cayley graphs and a new bipartite 2-arc-transitive graph which is not a Cayley graph

A mixed dihedral group is a group H with two disjoint subgroups X and Y, each elementary abelian of order 2ⁿ, such that H is generated by X∪Y, and H/H^′≅X×Y. In this paper, we give a sufficient condition such that the automorphism group of the Cayley graph Cay(H,(X∪Y)\{1}) is equal to H⋊A(H,X,Y), where A(H, X, Y) is the setwise stabiliser in Aut(H) of X∪Y. We use this criterion to resolve a question of Li et al. (J Aust Math Soc 86:111-122, 2009), by constructing a 2-arc-transitive normal cover of order 2⁵³ of the complete bipartite graph K_16,16 and prove that it is not a Cayley graph.