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Various stylistic and grammatical improvements. Style of question was not preserved, yet it seems that the errors were too many, and in the new form this question is more useful to others. (In particular, with the keyword "commuting graph" in the new title.) Content preserved.
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Automorphism group of a special commuting graph

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'.

Automorphism group of a special graph

Suppose $S_6$ is symmetric group on six letters and consider subset $X$ is conjugacy class containing $(12)(34)$. Define graph $\Gamma$ such that its vertices is $X$ and between two vertices has an edge if they commute as two elements of  $S_6$. I would like to know the automorphism group structure of graph $\Gamma$.

P.S. These graphs has known in scientific texts as commuting graph.

Automorphism group of a special commuting graph

Suppose $S_6$ is the symmetric group on six letters and let $X$ denote the conjugacy class containing $(12)(34)$. Define a graph $\Gamma$ with vertex set $X$ and edges precisely the 2-element subsets of $X$ which commute as elements of $S_6$. I would like to know the automorphism group of the graph $\Gamma$.

P.S. These graphs are known in scientific texts as 'commuting graphs'.

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Automorphism group of a special graph

Suppose $S_6$ is symmetric group on six letters and consider subset $X$ is conjugacy class containing $(12)(34)$. Define graph $\Gamma$ such that its vertices is $X$ and between two vertices has an edge if they commute as two elements of $S_6$. I would like to know the automorphism group structure of graph $\Gamma$.

P.S. These graphs has known in scientific texts as commuting graph.