Mixed degree number field computations

We present a method for computing complete lists of number fields in cases where the Galois group, as an abstract group, appears as a Galois group in smaller degree. We apply this method to find the twenty-five octic fields with Galois group $\textrm{PSL}_2(7)$ and smallest absolute discriminant. We carry out a number of related computations, including determining the octic field with Galois group $2^3{:}\textrm{GL}_3(2)$ of smallest absolute discriminant.