Determine the symmetric group for $(x - y)(y - b)(b - a)(a - x)$
I've been trying to find this for a while now, but I cannot get the polynomial to a form where I could see this. Fully expanding I have $$b^2x^2-b^2ax+ba^2x-bax^2+bxy^2-axy^2+2baxy-bx^2y-a^2xy+ax^2y-b^2xy-bay^2+a^2y^2+b^2ay-ba^2y$$
and factoring I get $$\left(xy-bx-y^2+by\right)\left(b-a\right)\left(a-x\right) = \left(xy-bx-y^2+by\right)(ba-bx-a^2+ax)$$
but neither one of these doesn't seem to show any symmetry to me. The term $\left(xy-bx-y^2+by\right)$ it seems that $x$ and $b$ would be symmetric and similarly $y$ and $b$ so for this polynomial I would have $\{e, (xb)(yb)\}$ as the symmetric group. $(ba-bx-a^2+ax)$ would seem to be symmetric with $\{e, (xb)\}$?