Suppose a group $G$ acts faithfully on a set of five elements, inducing two orbits of size $3$ and $2$ respectively. What group may $G$ be?

There is clearly a homomorphism $G \mapsto S\_3$ and another $G \mapsto S\_2$.

$|G|$ cannot be greater than $|S\_2 \times S\_3| = 12$, the total number of simultaneous permutations, or its action would not be faithful.

It seems to me that $D\_3=S\_3$, $C\_6$, and $S\_3 \times C\_2$ are possibilities. Is this list correct and exhaustive, and how do I prove it?