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 \(2^n\), such that H is generated by \(X\cup Y\), and \(H/H'\cong X\times Y\). In this paper, we give a sufficient condition such that the automorphism group of the Cayley graph \(\textrm{Cay}(H,(X\cup Y){\setminus }\{1\})\) is equal to \(H\rtimes A(H,X,Y)\), where A(H, X, Y) is the setwise stabiliser in \({{\,\textrm{Aut}\,}}(H)\) of \(X\cup Y\). 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 \(2^{53}\) of the complete bipartite graph \({{\textbf {K}}}_{16,16}\) and prove that it is not a Cayley graph.