Please help me to prove the following result:
I recall that the holomorph $Hol(\Gamma)$ of a group $\Gamma$ is the semidirect product $Hol(\Gamma)=\Gamma\ltimes_{\phi} Aut(\Gamma)$ where $\phi: Aut(\Gamma)\to Aut(\Gamma)$ is the identity map.
Let $G=Hol(C_8)$. In addition let $f(x)\in \mathbb{Q}[x]$ be an irreducible polynomial of degree $8$, $L$ a splitting field of $f(x)$ over $\mathbb{Q}$ and suppose that $Gal(L/\mathbb{Q})\cong G$
Show that there is a subgroup of $S_8$ isomorphic to $G$
Thanks in advance.