Symmetric groups isomorph to dihedral groups. I've noticed, that $S_2 \cong D_1$ and $S_3 \cong D_3$. Is every symmetric group $S_n$ (no including $S_1$) isomorph to the dihedral group $D_{n!/2}$?

No. The symmetric groups $S_n$ for $n\geq 3$ have trivial centre, but the dihedral groups $D_{n!/2}$ all have a centre of order $2$, since $n!/2$ is even.