Splitting field of $x^6-6x^4-10x^3+12x^2-60x+17$

Question

The one I want to know exactly is the Galois group of minimal polynomial of $a=\sqrt 2+\sqrt[3]5$. If the splitting field $E$ of $m_a$ is the subfield of $Q(\sqrt 2,\sqrt[3]5,\eta)$, where $\eta$ is primitive 3rd root of 1, then by using Galois theory, I show that the size of orbit of a is 6, so $x^6-6x^4-10x^3+12x^2-60x+17$ is irreducible and minimal polynomial of $a$, and hence $m_a$ is irreducible, hence separable. Since $Q(\sqrt 2+\sqrt[3]5)$ is included in $E$, not same, and $[Q(\sqrt 2+\sqrt[3]5):Q]=6$, $E=Q(\sqrt 2,\sqrt[3]5,\eta)$. The problem is the first part: is $E$ the subfield of $Q(\sqrt 2,\sqrt[3]5,\eta)$? I think so, but I can't prove this without direct calculation. Am I wrong, or am I missing some simple facts?