# Groups of derangements: what is known about subgroups of a symmetric group $S_{n}$ that contain only derangements (plus the identity)?

A derangement is a permutation that has no fixed points.

My question is . . .

What is known about subgroups of a symmetric group $$S_{n}$$ that contain only derangements (plus the identity)?

It is clear that elements of such groups would need to be a product of $$\frac{n}{k}$$ disjoint $$k$$-cycles.

It would be simple to see cyclic groups, but this is certainly not all possible.

For example, if $$n=6$$, I can generate in GAP a subgroup generated by $$(163)(245)$$ and $$(15)(23)(46)$$ which contains only derangements plus the identity.

Is any more known?