Classifying permutations in terms of their cycle notation Is there are a standard way of referring to permutations in terms of their cycle notation? For example: Does the set of all permutations in $S_4$ that can be expressed as the composition of two two-cycle permutations $\left\{ (12)(34), (13)(43), (14)(23)\right\}$ has a name?  

The sets you're looking for are the conjugacy classes of $S_n$.
It's a good exercise to convince oneself that "$a$ and $b$ are conjugates in $S_n$" is equivalent to "$a$ and $b$ have the same cycle type".
