Computing character degrees via a Galois connection

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In a previous paper, the second author established that, given finite fields $F < E$ and certain subgroups $C leq E^times$, there is a Galois connection between the intermediate field lattice ${L mid F leq L leq E}$ and $C$ʹs subgroup lattice. Based on the Galois connection, the paper then calculated the irreducible, complex character degrees of the semi-direct product $C rtimes {rm Gal} (E/F)$. However, the analysis when $|F|$ is a Mersenne prime is more complicated, so certain cases were omitted from that paper.The present exposition, which is a reworking of the previous article, provides a uniform analysis over all the families, including the previously undetermined ones. In the group $Crtimes{rm Gal}(E/F)$, we use the Galois connection to calculate stabilizers of linear characters, and these stabilizers determine the full character degree set. This is shown for each subgroup $Cleq E^times$ which satisfies the condition that every prime dividing $|E^times :C|$ divides $|F^times|$.