I have checked that $X^4 - X^3 - 7X + 19$ is coprime to its derivative. So it is separable. So I know that the Galois group of $X^4 - X^3 - 7X + 19$ must be a subgroup of $S_4$.
But I need to prove (or disprove) that is is not the whole of $S_4$. Can anyone give me an idea of how to solve this problem? Thanks.