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Abstract: In this paper we study the Selmer groups of elliptic curves over Galois extensions of number fields whose Galois group G is isomorphic to the semidirect product of two couples of the p-adic numbers. In particular, we give examples where its Pontryagin dual is a faithful torsion module under the Iwasawa algebra of G. Then we calculate its Euler characteristic and give a criterion for the Selmer group being trivial. Furthermore, we describe a new asymptotic bound of the rank of the Mordell-Weil group in these towers of number fields.