How many distinct orbits there are of $S_4$ when $S_4$ acts on itself by conjugation?

Question

The question is, how many distinct orbits there are of $S_4$ when $S_4$ acts on itself by conjugation?

I was thinking the number would be the same as the cycle types of $S_4$. However, the orbits must be distinct, so here is where I get slightly confused.

Answer 1

As discussed in the comments, two permutations are conjugates iff they have the same cycle type. Thus, it comes down to the number of cycle types in $S_4$. This is the same as the number of integer partitions of $4$. We have:

1. $1 + 1 + 1 + 1$
2. $2 + 1 + 1$
3. $2 + 2$
4. $3 + 1$
5. $4$

How do we know we've got all? Well, count the number of permutations in each of the cycle type listed above. It is not too tough to see that they are (in the same order) given as:

1. $1$
2. $6$
3. $3$
4. $8$
5. $6$

Since $1 + 6 + 3 + 8 + 6 = 24 = 4! = |S_4|$, we know that we've covered all.

Answer 2

The class equation will give this information: the number of terms is the number of classes.

The class equation for $S_4$ is $1+6+3+8+6$.
Thus we have $5$ orbits.