Group Actions · Hard problem

Let $V=\{e,(1\,2)(3\,4),(1\,3)(2\,4),(1\,4)(2\,3)\}\le S_4$ (the Klein four-group). $V$ acts on $\{1,2,3,4\}$ in the natural way, and this action is transitive. Find $|\operatorname{Stab}(1)|$.