  We construct three new families of fibrations $\pi : S \to B$ where $S$ is an
algebraic complex surface and $B$ a curve that violate Xiao's conjecture
relating the relative irregularity and the genus of the general fiber. The
fibers of $\pi$ are certain \'etale cyclic covers of hyperelliptic curves that
give coverings of $P^1$ with dihedral monodromy.
  As an application, we also show the existence of big and nef effective
divisors in the Brill-Noether range.
