In order to construct a Cayley graph, you need to have a set of generators; for example, you could take the generators {α, β} or the generators {T_1, …, T_6} mentioned in the post. Realizing a group as a Galois group over Q doesn’t really help, because it doesn’t equip the group with a set of generators.
There are Cayley graphs for the Rubik's cube, however is it possible to generate Cayley graphs from Galois groups and rings?
In order to construct a Cayley graph, you need to have a set of generators; for example, you could take the generators {α, β} or the generators {T_1, …, T_6} mentioned in the post. Realizing a group as a Galois group over Q doesn’t really help, because it doesn’t equip the group with a set of generators.
Really appreciated the excellent analysis and discussion of Rubik’s cube and its generalization.