Cayley graph of S₄,

generated by

• blue: (12)
• green: (13)
• red: (14)

This is a Nauru graph, compare The many faces of the Nauru graph by 0xDE

The permutations of four elements 1,2,3,4 are represented by sequences (like in this file), but with the first digit omitted.

Reflected triples are centrally symmetric to each other in this arrangement. Referring to the structure itself (a torus tiled with twelve hexagons, four of each color) this means, that reflected triples are on opposite sides of magenta hexagons. The following equation of permutation compositions corresponds to that fact:
(14)∘(12)∘(14) = (12)∘(14)∘(12) = (24)

Triples linked by an edge differ in only one digit:
The left one, when the edge is blue;
the middle one, when it's green;