Link structure explained

There are fifteen math students at Princeton in my year. We have devised a way of linking to each other in such a way that the following are true:

• Each person links to two other people.
• Each person is linked to by two other people.
• The largest distance between any two people is 4 clicks.
• The group of symmetries of the linking digraph is Z/5Z.

Following one of the two links results in a Hamiltonian 15-cycle of the students (ordered alphabetically by last name), so it is called "next". The other splits into a 5-cycle and a 10-cycle and is thus more mysterious, so it is called "magic".

Note that it is not possible to achieve 3 clicks while maintaining in- and out-degree 2, although a basic calculation might suggest otherwise. (One can reach 2 people after the first click, then possibly 4 with the next, and possibly 8 with yet one more. In total, this equals exactly the 14 other people that must be reachable.)

The overall (di)graph might be the unique graph with the above properties. We're not exactly sure.