TODO this would give a better motivation for the Mathieu group
Higher transitivity: mathoverflow.net/questions/5993/highly-transitive-groups-without-assuming-the-classification-of-finite-simple-g
Might be a bit complex: math.stackexchange.com/questions/698327/classification-of-triply-transitive-finite-groups
en.wikipedia.org/w/index.php?title=Mathieu_group&oldid=1034060469#Multiply_transitive_groups is a nice characterization of 4 of the Mathieu groups.
Apparently only Mathieu group and Mathieu group .
www.maths.qmul.ac.uk/~pjc/pps/pps9.pdf mentions:Hmm, is that 54, or more likely 5 and 4?
The automorphism group of the extended Golay code is the 54-transitive Mathieu group . This is one of only two finite 5-transitive groups other than symmetric and alternating groups
scite.ai/reports/4-homogeneous-groups-EAKY21 quotes link.springer.com/article/10.1007%2FBF01111290 which suggests that is is also another one of the Mathieu groups, math.stackexchange.com/questions/698327/classification-of-triply-transitive-finite-groups#comment7650505_3721840 and en.wikipedia.org/wiki/Mathieu_group_M12 mentions .
math.stackexchange.com/questions/700235/is-there-an-easy-proof-for-the-classification-of-6-transitive-finite-groups says there aren't any non-boring ones.
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