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 copies of the $p$-adic numbers $\bbfZ_p$. 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.
