Cayley graph related to the permutohedron of order 3 with k-faces labelled with weak orderings

Geometrically this is the face lattice of the hexagon (without the empty face).
It has  A000670(3) = 13 nodes and 18 edges.

This is like a Hasse diagram with the outer nodes finer than those on the inside.

Reading the permutation matrices by row gives the permutations in the Cayley graph, which also appear as labels a|b|c|d.
(Reading them by column gives the permutations in the permutohedron.)

There is also: left action last ⋅ first

This is: right action first ⋅ last

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