As a follow-up to the construction of a graph that $S_6$ acts on in Constructing a graph based on numbers of vertices, incident edges, and incident triangles, I am now specifically looking at the relationship between the 6 sets of sets of 5 disjoint 2-cycles (ie $\{(ab),(ac),(ad),(ae),(af)\}$ to the 3 sets of 5 pairwise disjoint 2-cycle triangles (ie $\{\{(ab),(cd),(ef)\},\{(ac),(be),(df)\},\{(ad),(bf),(ce)\},\{((ae),(bd),(cf)\},\{(af),(bc),(de)\}\}$. I think it's important that $\frac{6}{3} =2$ but I am still a bit lost on how to construct an outer automorphism from this info. I would appreciate it if someone could steer me in the right direction.
2026-04-05 21:17:23.1775423843
Using the 6 sets of sets of 5 disjoint 2-cycles and the 3 sets of 5 pairwise disjoint triangles to exhibit an outer automorphism of $S_6$.
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