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Having said this, Scott Carnahan answered this MO question by stating that this game gives "a representation of $M_{13}$ on a finite state machine" and this is what makes $M_{13}$ remarkable. In some sense Scott's answer explains why $M_{13}$ is a big deal, but it's not quite in the direction I'm looking for...

Having said this, Scott Carnahan answered this MO question by stating that this game gives "a representation of $M_{13}$ on a finite state machine" and this is what makes $M_{13}$ remarkable. In some sense Scott's answer explains why $M_{13}$ is a big deal, but it's not quite in the direction I'm looking for...

Back to transitivity: What I would like is a characterization of $M_{13}$ via some group-theoretic properties. For instance as a subset of a permutation group, via the notion of transitivity: So, for instance, we can characterize $M_{12}$ and $M_{13}$ as the only subgroups of $S_n$ that don't contain $A_n$ and that are 5-transitive. Here's an analogous statement that (if true) would be the sort of thing I'm looking for.

Back to transitivity: What I would like is a characterization of $M_{13}$ via some group-theoretic properties. For instance as a subset of a permutation group, via the notion of transitivity: So, for instance, we can characterize $M_{12}$ and $M_{24}$ as the only subgroups of $S_n$ that don't contain $A_n$ and that are 5-transitive. Here's an analogous statement that (if true) would be the sort of thing I'm looking for.

Having said this, Scott Carnahaned answer this MO question by stating that this game gives "a representation of $M_{13}$ on a finite state machine" and this is what makes $M_{13}$ remarkable. In some sense Scott's answer explains why $M_{13}$ is a big deal, but it's not quite in the direction I'm looking for...

Having said this, Scott Carnahan answered this MO question by stating that this game gives "a representation of $M_{13}$ on a finite state machine" and this is what makes $M_{13}$ remarkable. In some sense Scott's answer explains why $M_{13}$ is a big deal, but it's not quite in the direction I'm looking for...