Abstract
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 25 octic fields with Galois group $${{\mathrm{PSL}}}_2(7)$$ and smallest absolute discriminant. We carry out a number of related computations, including determining the octic field with Galois group $$2^3{:}{{\mathrm{GL}}}_3(2)$$ of smallest absolute discriminant.
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