Let $n\in\mathbb{N}$, $n>1$. Let $(\mathbb{Z}/n\mathbb{Z})^\times$ be the Multiplicative group of primitive residue classes modulo $n$. By Wilson's theorem, it's clear that $$\prod_{k\in (\mathbb{Z}/n\mathbb{Z})^\times}k=-1$$ if $n$ is prime. Also, for all $n$, we can say that $$\prod_{k\in (\mathbb{Z}/n\mathbb{Z})^\times}k^2=1$$ since $\prod_{k\in (\mathbb{Z}/n\mathbb{Z})^\times}k=\prod_{k\in (\mathbb{Z}/n\mathbb{Z})^\times}k^{-1}$. How can I find the product $\prod_{k\in (\mathbb{Z}/n\mathbb{Z})^\times}k$ if $n$ is not a prime ? Is there a generalization for the Wilson's theorem ?
It's clear that $\text{ord}(\prod_{k\in (\mathbb{Z}/n\mathbb{Z})^\times}k)$ is $1$ or $2$ since $(\prod_{k\in (\mathbb{Z}/n\mathbb{Z})^\times}k)^2=1$.
