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clickbait: '$A_5$ has easily-characterised conjugacy classes, based on a rather surprising theorem about when conjugacy classes in the symmetric group split.',
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text: 'This page lists the [4bj conjugacy classes] of the [-4hf] $A_5$ on five elements.\nSee a [4l0 different lens] for a derivation of this result using less theory.\n\n$A_5$ has size $5!/2 = 60$, where the exclamation mark denotes the [-factorial] function.\nWe will assume access to [4bk the conjugacy class table of $S_5$] the [-497] on five elements; $A_5$ is a [quotient_group quotient] of $S_5$ by the [4hk sign homomorphism].\n\nWe have that a conjugacy class splits if and only if its [49f cycle type] is all odd, all distinct. ([4kv Proof.])\nThis makes the classification of conjugacy classes very easy.\n\n# The table\n\nWe 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 [4cn transpositions]). Indeed, those lines are classes which are not even in $A_5$.\n\nWe are left with cycle types $(5)$, $(3, 1, 1)$, $(2, 2, 1)$, $(1,1,1,1,1)$.\nOnly the $(5)$ cycle type can split into two, by the splitting condition.\nIt splits into the class containing $(12345)$ and the class which is $(12345)$ conjugated by odd permutations in $S_5$.\nA representative for that latter class is $(12)(12345)(12)^{-1} = (21345)$.\n\n$$\\begin{array}{|c|c|c|c|}\n\\hline\n\\text{Representative}& \\text{Size of class} & \\text{Cycle type} & \\text{Order of element} \\\\ \\hline\n(12345) & 12 & 5 & 5 \\\\ \\hline\n(21345) & 12 & 5 & 5 \\\\ \\hline\n(123) & 20 & 3,1,1 & 3 \\\\ \\hline\n(12)(34) & 15 & 2,2,1 & 2 \\\\ \\hline\ne & 1 & 1,1,1,1,1 & 1 \\\\ \\hline\n\\end{array}$$',
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