I am working through a set of problems and a question states:
(1) Let $G$ be finite. Show that there exists a finite set $X$ and an injective group homomorphism $\rho:G\rightarrow Sym(X)$, where $Sym(X)$ denotes the group of bijections from $X$ to itself.
(2) Let $\Gamma$ be the symmetry group of all isometries of the regular octahedron, and let $X$ be the set of eight faces of the octahedron. Show that $\Gamma$ acts transitively on $X$, describe the stabilizer of an element of $X$, and hence find the order of $Γ$.
(3) The action of $\Gamma$ on $X$ yields an injective homomorphism from $\Gamma$ to $Sym(X)$. By numbering the faces of the octahedron from $1$ to $8$ we can identify $Sym(X)$ with $S_8$, and hence we obtain a homomorphism $\phi: \Gamma\to S_8$. Is the image $\phi(\Gamma)$ of $\Gamma$ a normal subgroup of $S_8$?
I have completed the first 2 parts. For (3) I have an idea but I'm not convinced.
The image is a normal subgroup if and only if given $x\in \Gamma$, $\sigma\in S_8$ we can find $z\in \Gamma$ such that $\sigma \phi(x)\sigma^{-1}=\phi(z)$. Obviously then I could answer the question via listing each type of isometry and seeing if I can find a corresponding $z$, however I don't think the question actually requires me to do this. Any hints would be great