Let $S=S_g$ be the closed orientable surface of genus $g$ and let $\Gamma_3(S)$ be the subgroup of the mapping class group, $Mod(S)$, which acts trivially on $H_1(S;\mathbb{Z/3\mathbb{Z}})$. Define $\Theta(g):=[Mod(S):\Gamma_3(S)]$.

Question: Is it known if $\Theta(g)$ is exponentially large in $g$?