Find the semidirect products of $C_2 \times C_2$ by $C_3$, that is: $(C_2 \times C_2) \rtimes C_3$
My approach: I let $C_3:=\langle y\rangle$ and let $\phi : C_3 \to \mathrm{Aut}(C_2 \times C_2)$ be a homomorphism, observe that $\mathrm{Aut}(C_2 \times C_2) \cong \mathrm{GL}(2,2)$ and $\phi(y)$ has order 1 or 3.
Case 1: If $\phi(y)$ has order 1, then $\phi$ is the trivial homomorphism thus $(C_2 \times C_2) \rtimes C_3 \cong C_2 \times C_2 \times C_3 $
Case2: If $\phi(y)$ has order 3, then $\phi(y)= \begin{pmatrix}0 & 1\\ 1 & 1\end{pmatrix} $ which is the only matrix of order 3 inside $\mathrm{GL}(2,2)$, in the next step one needs to find the isomorphism type of $(C_2 \times C_2) \rtimes C_3$ relised by $\phi$ in this case, this is where my problem is: I know its presentation $\langle a,b,y\rangle$ contains $ \{ a^2=1,\ b^2=1,\ [a,b]=1,\ y^3=1 \} $ as a part of the relations, but I do not know how to get all the relations, and also how to get its isomorphism type.