= Conjugacy classes of the alternating group on six letters
{title2=$|A_6|=360$}
= Conjugacy classes of A6
{synonym}
The <alternating conjugacy class splitting criterion> gives seven <conjugacy classes> in $A_6$. Their cycle types and sizes are $1^6:1$, $2^2 1^2:45$, $3 1^3:40$, $3^2:40$, $4\,2:90$, and two $5\,1$ classes of size 72. Representatives of the split classes are a 5-cycle and its square. Their sizes sum to 360. A <normal subgroup> must be a union of these classes containing the identity, so these explicit sizes can prove simplicity using <Lagrange's theorem>.
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