We know that every finitely generated metabelian group $\Gamma$ has a faithful representation over a finite product of fields. Denoting $C_n$ the cyclic group of order $n$, the restricted wreath product $C_6\wr C_\infty$ is linear over a product of two fields ( a field of char. 2 and a field of char. 3). May I ask how to construct this injective homomorphism from $C_6\wr C_\infty$ to the product of general linear group?
Let $t$ be the generator for $C_\infty$ and $s$ be the generator for $C_6$, I know that $C_6\wr C_\infty = \langle s, t \rangle$, but I am not sure where the generator should be mapped to. Any hint/reference would be really appreciated.