  We give a complete description of the normal subgroups of arboreal Galois
groups of Belyi maps. The normal groups form a unique chief series. We also
carefully compute the discriminate of the iterate of a polynomial minus an
algebraic number, which allows us to predict when a such discriminate is a
perfect square in the base field or intermediate field for a postcritically
finite polynomials (PCF). As a consequence we are able to find another PCF
cubic polynomial that has the same arboreal Galois group as the one of Belyi
maps.
