A family of 2-groups and an associated family of semisymmetric, locally 2-arc-transitive graphs

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, for each n ≥ 2,
we construct a mixed dihedral 2-group H of nilpotency class 3 and order
2a where a = (n3 +n2 +4n)/2, and a corresponding graph Σ, which is the
clique graph of a Cayley graph of H. We prove that Σ is semisymmetric,
that is, Aut(Σ) acts transitively on the edges but intransitively on the
vertices of Σ. These graphs are the first known semisymmetric graphs
constructed from groups that are not 2-generated (indeed H requires 2n
generators). Additionally, we prove that Σ is locally 2-arc-transitive, and is
a normal cover of the ‘basic’ locally 2-arc-transitive graph K2n,2n. As such,
the construction of this family of graphs contributes to the investigation of
normal covers of prime-power order of basic locally 2-arc-transitive graphs
– the ‘local’ analogue of a question posed by C. H. Li.