Let G be a group of order 3*2^n. Prove there exists a non-complete non-cyclic Cayley graph of G such that there is a unique shortest path between every pair of vertices, or otherwise prove no such graph exists.
No Cayley graph of a group of order 3⋅2n3⋅2n can have a unique path between every pair of vertices.Also, there are trivial solutions with odd cycles and complete graphs which must be excluded. (So the answer to the prompt as originally stated is wrong too)