I would like to know whether there exists a group with the following structure:
$G$ is a non-split extension of a cyclic group of order 3 ($C_3$) by the Janko group $J_2$ such that $G/C_G(C_3)$ has order $2$ . ($C_3$ is normal in $G$)
Moreover, if such a group exists how I can introduce it to GAP?
$J_2$ is a simple group, so $G/C_G(C_3)$ could not possibly have order $2$. In fact the Schur Multiplier of $J_2$ has order $2$, so there can be no non-split extension $G$ with normal $C_3$ and $G/C_3 \cong J_2$.