I know how to find structure of Rational group algebras of finite cyclic groups such as for cyclic group $C_6=\langle a \rangle$ we have $\Bbb{Q}C_6 \cong \Bbb{Q} \oplus \Bbb{Q}\ \oplus\ \Bbb{Q}(\omega)\ \oplus \Bbb{Q}(\omega^2)$ where $a \to (1,1,\omega, \omega^2)$.
But when we are finding structure over finite abelian groups which are not cyclic, like $C_2 \times C_4$, then by perlis walker theorem, I can find that $\Bbb{Q}(C_2\times C_2) \cong \Bbb{Q} \oplus\Bbb{Q} \oplus\Bbb{Q} \oplus\Bbb{Q} $ but how do I determine what is the isomorphism and where do I send which elements. The proof is by induction and does not reveal anything specific in case of non cyclic groups.
Thanks.
Similarly how to find what does $\Bbb{Q}(C_2\times C_4)$ is mapped into as isomorphism.