This is a follow up on [another MO question](https://mathoverflow.net/questions/265438/n-divides-a-product).



> 
> **Question.** For $n\geq2$, the following is always an integer. Is it not?
>  $$\frac{(2^n-2)(2^{n-1}-2)\cdots(2^3-2)(2^2-2)}{n!}.$$
> 
> 
> 



n! divides a product: Part II
As $S\_n$ also embeds in $GL\_{n-1}({\mathbb F}\_2)$, you only need to check that $n!$ is not divisible by $2^n$. 


