we study galois covers of the projective line branched at three points with galois group $g$. when such a cover is defined over a $p$-adic field, it is known to have potentially good reduction to characteristic $p$ if $p$ does not divide the order of $g$. we give a sufficient criterion for good reduction, even when $p$ does divide the order of $g$, so long as the $p$-sylow subgroup of $g$ is cyclic and the absolute ramification index of a field of definition of the cover is small enough. this extends work of (and answers a question of) raynaud. our proof depends on working very explicitly with kummer extensions of complete discrete valuation rings with imperfect residue fields.