Suppose $S_6$ is symmetricthe symmetric group on six lettersletters and consider subsetlet $X$ is conjugacydenote the conjugacy class containing $(12)(34)$. Define a graph $\Gamma$ such that its vertices iswith vertex set $X$ and between two vertices has an edge if theyedges precisely the 2-element subsets of $X$ which commute as two elements of $S_6$. I would like to know the automorphism group structure of the graph $\Gamma$.
P.S. These graphs hasare known in scientific texts as commuting graph'commuting graphs'.