[[Permutation group]]
# Multiply transitive permutation group

A [[permutation group]] $G \sube \Omega!$ is **$k$-transitive** iff it acts on ordered tuples in $^\Omega \mathrm{P}_{k}$ [[transitive group action|transitively]]. #m/def/group 

## Classification

- For $k \geq 6$, the only $k$-transitive groups are $\mathrm{S}_{n}$ with $n \geq k$ and $\mathrm{Alt}_{n}$ with $n \geq k+2$;
- The only 5-transitive groups that aren't 6-transitive are [[Mathieu group M24]], [[Mathieu group M12]], [[Symmetric group|$\mathrm S_{5}$]], and [[Alternating group|$\mathrm{Alt}_7$]].

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