Galois realizability of non-split group extensions of C2 by (C2)r×(C4)s×(D4)t

Abstract

The focus of this paper is Galois embedding problems associated with extensions of C 2 by groups of the form (C 2 ) r × (C 4 ) s × (D 4 ) t over arbitrary fields of characteristic not 2. As an application of our general solution, we obtain obstructions to the realizability of 13 groups of order 32 for which obstructions were not previously known.

Galois realizability of a central C4-extension of D8

“…A compilation of conditions for the realizability of groups of order 2, 4, 8, or 16, as well as descriptions of extensions with those Galois groups, can be found in [5] and [8]. A compilation of conditions for all groups of order 32, based on the work in this paper as well as in [2,6,[10][11][12], can be found in [4].…”