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A sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups. All the other finite simple groups form 18 infinite families numbered by q - power of prime number and n - natural number. Sporadic groups attach attention due to their sporadic/exceptional nature - similar to exceptional Lie groups. The first sporadic groups were found by Mathieu in 1860s. The last sporadic group J4 was discovered in 1975 by Janko.
27
votes
Small subgroups of the monster
No, but almost.
$\newcommand\Dic{\mathit{Dic}}$It turns out that all isomorphism types of groups of order up to 36 occur as subgroups of the Monster with a single exception: no subgroup of the monster …
6
votes
Accepted
Is a point stabilizer in the Mathieu group $M_{20}$ half-transitive?
No it is not: half-transitive means that all orbits have equal size (and the groups acts non-trivially), but this is not the case here. One can verify this e.g. using GAP:
gap> M21:=MathieuGroup(21);
…