Consider the action of $S_3$ on itself by conjugation. Describe the orbits of (1, 2) and (1, 2, 3) under this action.

Question

Find the stabilizer subgroup of each of these elements. How many orbits are there in total?

For the cycles (1 2) and (1 2 3). I am getting the orbit of these cycles as follows:
Orb(1 2) = {(1 3),(1 2),(2 3)}
Orb(1 2 3) = { (1 3 2),(1), (1 2 3)}
The stabilizer group is as follows:
Stab(1 2) = {(1),(1 2)}
Stab(1 2 3) = {(1), (1 2 3)}

I have calculated the Orbit and Stabilizer subgroups of these elements by putting elements of $ S_3$. Is there any easier way to compute them. Are my answers correct?