When $(1+\sqrt{7})$ is raised to an integral power, $n$, we always get a number of the form $(a+b\sqrt{7})$.
We write $(1+\sqrt{7}{)}^n=\alpha (n)+\beta (n)\sqrt{7}$.

For a given number $x$ we define $g(x)$ to be the smallest positive integer $n$ such that: $$\begin{array}{rl}\alpha (n)& \equiv 1(\mathrm{mod}x)\text{and }\\ \beta (n)& \equiv 0(\mathrm{mod}x)\end{array}$$ and $g(x)=0$ if there is no such value of $n$. For example, $g(3)=0$, $g(5)=12$.

Further define $$G(N)=\sum _{x=2}^{N}g(x)$$ You are given $G({10}^2)=28891$ and $G({10}^3)=13131583$.

Find $G({10}^6)$.