A permutation $\pi$ of $\{1,\dots ,n\}$ can be represented in one-line notation as $\pi (1),\dots ,\pi (n)$. If all $n!$ permutations are written in lexicographic order then $\text{rank}(\pi )$ is the position of $\pi$ in this 1-based list.

For example, $\text{rank}(2,1,3)=3$ because the six permutations of $\{1,2,3\}$ in lexicographic order are: $$1,2,31,3,22,1,32,3,13,1,23,2,1$$

Let $Q(n)$ be the sum $\sum _{\pi }\sum _{i=1}^{n!}\text{rank}({\pi }^i)$, where $\pi$ ranges over all permutations of $\{1,\dots ,n\}$, and ${\pi }^i$ is the permutation arising from applying $\pi$ $i$ times.

For example, $Q(2)=5$, $Q(3)=88$, $Q(6)=133103808$ and $Q(10)\equiv 468421536(\mathrm{mod}{10}^9+7)$.

Find $Q({10}^6)$. Give your answer modulo $({10}^9+7)$.