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.

Gemini 2.5 at least replies that it seems unlikely to be false without hallucinating a proof. From its thoughts it gets very close to figuring out that A_4 exists as a subgroup.

Since any group of order 3⋅2n3⋅2n has ∣G∣≥3∣G∣≥3, it cannot admit a Cayley graph which is a tree. Hence:

    No Cayley graph of a group of order 3⋅2n3⋅2n can have a unique path between every pair of vertices.