Group Actions · Superchallenge problem

$SL(2,\mathbb Z_3)$, the $2\times2$ matrices mod $3$ with determinant exactly $1$, has order $24$. It acts transitively on the $8$ nonzero vectors of $\mathbb Z_3^2$ by matrix-vector multiplication. Find $|\operatorname{Stab}((1,0))|$.