This is a follow-up to Puzzle about 6 infinite cylinders in space
Given six identical infinite (no caps) cylinders is there a beautiful arrangement in space such that each touches each other.
What you need to know is that beauty to a complete philistine such as myself means symmetry.
A symmetry in turn is for the purpose of this puzzle a nondistorting map of space onto itself that exchanges some cylinders but leaves the configuration as a whole in place.
Bonus: There are philistines and philistines. For a distinguished philistine such as myself mere beauty does not cut it. Only perfection will do.
A perfection within the confines of this puzzle is a symmetry or group of symmetries that acts transitively on the cylinders. Or in English: by possibly repeated possibly mixed application we can send any cylinder to any other.
Example: The finite cylinder solutions given here are symmetric but not perfect. The second, 7-pencil solution can be made into a perfect 6-pencil by leaving out the central, upright one.
As this is presumably rather hard partial answers (like considering only certain given symmetries) are welcome.