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A toric hyperk\"{a}hler variety is determined by combinatorial data A and$\alpha$. Here A is an integer valued matrix and $\alpha$ is a character of analgebraic torus $T^d$. $Y(A, \alpha)$ is a crepant partial resolution of anaffine toric hyperk\"{a}hler variety $Y(A,0)$. However, $Y(A, \alpha)$ is notgenerally a Q-factorial terminalization of $Y(A,0)$ even if $\alpha$ isgeneric.  In this article, we realize $Y(A,0)$ as another toric hyperk\"{a}hler variety$Y(A^{\sharp}, 0)$ so that $Y(A^{\sharp}, \alpha^{\sharp})$ is aQ-factorialization of $Y(A^{\sharp}, 0)$ for a generic $\alpha^{\sharp}$. As anapplication, we give a necessary and sufficient condition for $Y(A,0)$ to havea crepant resolution. Moreover, we construct explicitly the universal Poissondeformation of $Y(A,0)$ in terms of $A^{\sharp}$.

Toric hyperk\"{a}hler varieties and Q-factorial terminalizations