Compute the Galois group of L over K

Question

$\textbf{Problem}$ $K=\mathbb{Z}/p\mathbb{Z}$; L=splitting field of $\prod_{i=1}^{p-1} (t^2-i)$, where $p$ is an odd prime

$\textbf{Attempt}$ Since $(-k)^2=(p-k)^2=k^2$, we know that there are $\frac{p-1}{2}$ many irreducible polynomials among $t^2-1,t^2-2,\cdots,t^2-(p-1)$. This implies that among $1,2,\cdots,p-1$, $\frac{p-1}{2}$ are squares and other $\frac{p-1}{2}$ are not.

Then, the splitting field of the polynomial be $K(\sqrt{k_1},\cdots,\sqrt{k_{\frac{p-1}{2}}})$...

I don't know how to compute the Galois group...

Any help is appreciated..

Thank you!