Looking for nicer ways to work this out than having to check all permutations.
If we have the polynomial:
$p=x_1^2+x_2^2+x_3^2+x_4$
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
$x_1x_2^2x_3^3+x_3x_4^2x_1^3+x_2x_3^2x_4^3+x_4x_1^2x_2^3$
When I say similar I just mean what's the smart way to solve this :-)