$2S_4 * Q_8$-extensions in characteristic 3

Abstract

We present an explicit construction of the complete family of Galois extensions of a field $K$ of characteristic 3 with Galois group the central product $2S_4 * Q_8$ of a double cover $2S_4$ of the symmetric group $S_4$ and the quaternion group $Q_8$, containing a given $S_4$-extension of the field $K$.