The degree of the field $\mathbb Q[a,b,c]$ over $\mathbb Q$ where $a^5 = 1, b^7 = 12, c^5 = 58$

Question

I want to calculate the degree of the field $\mathbb Q[a,b,c]$ over $\mathbb Q$ where $a^5 = 1, b^7 = 12, c^5 = 58$. $\mathbb Q(a,c)$ is of degree $20$ because it is the splitting field of $x^5 - 58$ and $c \notin \mathbb Q(a)$. Now I want know whether $b$ is in $\mathbb Q(a,c)$ and is $x^7-12$ irreducible over it in order to calculate the degree.