More on the Exceptional Outer Automorphism of S₆

As anyone who’s read the Wikipedia article about the Exceptional Outer Automorphism of S₆ knows, the Outer Automorphism of S₆ switches:

• Single swaps with triple swaps
• Single 3-cycles with double 3-cycles
• Single 6-cycles with combination 3-cycle+2-cycles

and preserves all the other conjugacy classes (i.e. the identity, 4-cycles, double swaps, combination 4-cycle+2-cycles, 5-cycles).

But what I find really fascinating¹ is what it does to subgroups of S₆.

Consider how the rotations of a cube permute its six faces. This forms a subgroup of S₆. Apply the Exceptional Outer Automorphism, and it just turns into all the permutations of four objects, namely the four diagonals. Allow reflections in the first group, and chirality corresponds to swaps of the other two objects. And if we color pairs of opposite edges in identical colors and ask about the permutations of those, it corresponds to the even subgroup of the aforementioned S₄*S₂ subgroup.

How about the permutations of the flags of a triangle under symmetries of the triangle? Turns out that cleanly corresponds to the permutations of the triangle’s vertices. What if we allow 60-degree rotations on the triangle, making it a hexagon whose edges get permuted? Once again, central inversion corresponds to swaps of two of the other objects.

Take a dodecahedron and color pairs of opposite faces in six colors. Rotating the dodecahedron permutes these colors. Take the subgroup that corresponds to those permutations and it maps cleanly onto A₅, as the permutations of the five cubes inside the dodecahedron.

A subgroup generated by three non-intersecting swaps corresponds to a subgroup generated by the other representation of the Klein 4-group in S₄, plus an additional swap. Its even subgroup {e, (12)(34), (12)(56), (34)(56)} corresponds to just that Klein 4-group in S₄.

An A₃² subgroup corresponds to another A₃² subgroup. Turning both of them into their Cayley graph with the 3-cycles as generators explains why the 3 by 3 torus graph is its own complement.

According to A000638 on the Online Encyclopedia of Integer Sequences, there are 56 subgroups of S₆ up to conjugacy. 18 of them map to the same conjugacy class under the Exceptional Outer Automorphism of S₆, and the other 38 make 19 pairs.

In conclusion, the Exceptional Outer Automorphism of S₆ is my favorite isomorphism.

1. One of the least formal things I’ve written on this blog. ↩︎
2. This footnote should also be treated as an exponent. ↩︎