Classification of 5-transitive groups

Apparently only Mathieu group $M_{1}2$ and Mathieu group $M_{2}4$.

www.maths.qmul.ac.uk/~pjc/pps/pps9.pdf mentions:

The automorphism group of the extended Golay code is the 54-transitive Mathieu group $M_{24}$. This is one of only two finite 5-transitive groups other than symmetric and alternating groups

Hmm, is that 54, or more likely 5 and 4?

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 M_12.