# Triple-Transitivity/"Specify three know all" property of exotic transitive $S_5\subset S_6$

Let the exotic transitive subgroup $$S_5\subset S_6$$ act on $$\{1,2,\dots,6\}$$. For $$1\leq i,j\leq 6$$, define subsets: $$X_{ji}:=\{\sigma\in S_5\,\mid \sigma(j)=i\}.$$

Does the following properties hold (note 3 implies the other two):

1. Exotic $$S_5$$ is doubly-transitive in the sense that for all distinct $$1\leq j_1,j_2\leq 6$$ and (necessarily) distinct $$1\leq i_1,i_2\leq 6$$ there exists $$\sigma\in S_5$$ such that $$\sigma(j_1)=i_1\text{ and }\sigma(j_2)=i_2.$$ That is, we have $$X_{j_1i_1}\cap X_{j_2i_2}\neq \emptyset$$.
2. Exotic $$S_5$$ is also triply-transitive in the sense that for all distinct $$1\leq j_1,j_2,j_3\leq 6$$ and (necessarily) distinct $$1\leq i_1,i_2,i_3\leq 6$$ there exists $$\sigma\in S_5$$ such that: $$\sigma(j_m)=i_m\qquad (1\leq m\leq 3).$$ That is, we have $$X_{j_1i_1}\cap X_{j_2i_2}\cap X_{j_3i_3}\neq \emptyset$$.
3. Once we know three values of any $$\sigma\in S_5$$, we in fact know all six. That is, for suitably distinct $$1\leq j_1,j_2,j_3\leq 6$$ and distinct $$1\leq i_1,i_2,i_3\leq 6$$, the intersection $$X_{j_1i_1}\cap X_{j_2i_2}\cap X_{j_3i_3}$$ is a singleton.

I am led to this question by the following observations. $$X_{11}$$ is a stabiliser subgroup and the $$X_{1j}$$ are cosets. Also $$\sqcup_{j=1}^6 X_{1j}$$ is a partition of $$S_5$$ and so the cardinality of each must be $$|S_5|/6=20$$. Now consider the intersections $$X_{11}\cap X_{2j}$$. We know that $$X_{11}\cap X_{21}$$ is empty and therefore we have five such sets $$X_{11}\cap X_{2j}$$. We have the sum of the $$|X_{11}\cap X_{2j}|$$ is twenty and if each had equal cardinality there four would be five lots of four. Say $$|X_{11}\cap X_{22}|=4$$. Now, looking at $$|X_{11}\cap X_{22}\cap X_{3k}|$$, there are four of them, and if each of them has equal cardinality then they have to be singletons, with e.g. $$X_{11}\cap X_{22}\cap X_{33}=\{e\}$$.

I presume this could be sorted out with some GAP code.

The group you call the exotic $$S_5$$ is otherwise (and better) known as $${\rm PGL}(2,5)$$, and the three properties you described are collectively known as sharp triple transitivity. The group $${\rm PSL}(2,q)$$ acts sharply triply transitively on the $$q+1$$ points of the projective line for all prime powers $$q$$.