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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.

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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 $q$.

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