Let $f:\mathbb{ P}^1\rightarrow\mathbb{ P}^1$ be a quadratic rational map defined over the rational field $\mathbb{ Q}$ with nonabelian automorphism group. We prove that no such map has a $\mathbb{ Q}$-rational periodic point with exact period $N\ge 4$. We also give an explicit parametrization of such maps that have $\mathbb{ Q}$-rational periodic points of period $1$, $2$, and $3$. Consequently, we show that the number of $\mathbb{ Q}$-rational preperiodic points of such a map is at most $6$; establishing Morton-Silverman Uniform Boundedness Conjecture for this family of quadratic rational maps. As a result, we completely classify all portraits of $\mathbb{ Q}$-rational preperiodic points for such $f$ showing that there are exactly $5$ such portraits.
Categories: math.NT math.DS