Nonuniqueness of semidirect decompositions for semidirect products with directly decomposable factors and applications for dihedral groups

Nonuniqueness of semidirect decompositions of groups is an insufficiently studied question in contrast to direct decompositions. We obtain some results about semidirect decompositions for semidirect products with factors which are nontrivial direct products. We deal with a special case of semidirect product when the twisting homomorphism acts diagonally on a direct product, as well as with the case when the extending group is a direct product. We give applications of these results in the case of generalized dihedral groups and classic dihedral groups $D_{2n}$. For $D_{2n}$ we give a complete description of semidirect decompositions and values of minimal permutation degrees.