Extensions with Galois group \(2^+S_4*D_8\) in characteristic 3 (Q931797)

Let \(2^+S_n\) be the double cover of \(S_n\) such that transpositions lift to involutions and products of two disjoint transpositions lift to elements of order \(4\), and let \(D_8\) be the dihedral group of order 8. For groups \(G_1\) and \(G_2\) with isomorphic centers, denote by \(G_1 * G_2\) their central product, defined to be the subgroup of \(G_1 \times G_2\) obtained by identifying the centers of \(G_1\) and \(G_2\). Suppose that \(K\) is a field of characteristic \(3\). This note describes explicitly all extensions \(L/K\) with \(\text{Gal}(L/K) \cong 2^+S_4 * D_8\), thereby improving on previous results of the authors. The calculation is intricate, and relies in part on work of Abhyankar. The case where \(K\) is the fraction field of the formal power series ring \(k[x_1, x_2, x_3]\) for some \(k\) of characteristic \(3\) is relevant to Abhyankar's normal crossings local conjecture.