  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, for each $n\geq 2$, we
construct a mixed dihedral $2$-group $H$ of nilpotency class $3$ and order
$2^a$ where $a=(n^3+n^2+4n)/2$, and a corresponding graph $\Sigma$, which is
the clique graph of a Cayley graph of $H$. We prove that $\Sigma$ is
semisymmetric, that is, ${\rm Aut}(\Sigma)$ acts transitively on the edges, but
intransitively on the vertices, of $\Sigma$. 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 $\Sigma$ is locally
$2$-arc-transitive, and is a normal cover of the `basic' locally
$2$-arc-transitive graph ${\rm K}_{2^n,2^n}$. 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.
