Looking for nicer ways to work this out than having to check all permutations.
If we have the polynomial:
Then In order for $S_4$ to stabilize it it must leave $x_4$ uuntouched and we can move around the other variables so the stabilizer is just the number of permutations which leave $x_4$ which is quite straightforward. Is there a similar approach with the polynomial
When I say similar I just mean what's the smart way to solve this :-)