This page lists the conjugacy classes of the Alternating group $~$A_5$~$ on five elements. See a different lens for a derivation of this result using less theory.

$~$A_5$~$ has size $~$5!/2 = 60$~$, where the exclamation mark denotes the Factorial function. We will assume access to [4bk the conjugacy class table of $~$S_5$~$] the Symmetric group on five elements; $~$A_5$~$ is a quotient of $~$S_5$~$ by the sign homomorphism.

We have that a conjugacy class splits if and only if its cycle type is all odd, all distinct. (Proof.) This makes the classification of conjugacy classes very easy.

# The table

We must remove all the lines of [4bk $~$S_5$~$'s table] which correspond to odd permutations (that is, those which are the product of odd-many transpositions). Indeed, those lines are classes which are not even in $~$A_5$~$.

We are left with cycle types $~$(5)$~$, $~$(3, 1, 1)$~$, $~$(2, 2, 1)$~$, $~$(1,1,1,1,1)$~$. Only the $~$(5)$~$ cycle type can split into two, by the splitting condition. It splits into the class containing $~$(12345)$~$ and the class which is $~$(12345)$~$ conjugated by odd permutations in $~$S_5$~$. A representative for that latter class is $~$(12)(12345)(12)^{-1} = (21345)$~$.

$$~$\begin{array}{|c|c|c|c|} \hline \text{Representative}& \text{Size of class} & \text{Cycle type} & \text{Order of element} \\ \hline (12345) & 12 & 5 & 5 \\ \hline (21345) & 12 & 5 & 5 \\ \hline (123) & 20 & 3,1,1 & 3 \\ \hline (12)(34) & 15 & 2,2,1 & 2 \\ \hline e & 1 & 1,1,1,1,1 & 1 \\ \hline \end{array}$~$$