Mathieu group

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The list

There are eight isomorphism classes of Mathieu groups, given by their degrees. The degree of a Mathieu group is the smallest size set on which it has a faithful permutation representation, which is precisely the representation of interest:

Degree Mathieu group Order of group Prime factorization of order Number of conjugacy classes Sporadic simple? Simple? Almost simple? Alternative descriptions
9 Mathieu group:M9 72 2^3 \cdot 3^2 6 No No No projective special unitary group PSU(3,2)
10 Mathieu group:M10 720 2^4 \cdot 3^2 \cdot 5 8 No No Yes  ?
11 Mathieu group:M11 7920 2^4 \cdot 3^2 \cdot 5 \cdot 11 10 Yes Yes Yes --
12 Mathieu group:M12 95040 2^6 \cdot 3^3 \cdot 5 \cdot 11 15 Yes Yes Yes --
21 Mathieu group:M21 20160 2^6 \cdot 3^2 \cdot 5 \cdot 7 10 No Yes Yes projective special linear group PSL(3,4)
22 Mathieu group:M22 443520 2^7 \cdot 3^2 \cdot 5 \cdot 7 \cdot 11 12 Yes Yes Yes --
23 Mathieu group:M23 10200960 2^7 \cdot 3^2 \cdot 5 \cdot 7 \cdot 11 \cdot 23 17 Yes Yes Yes --
24 Mathieu group:M24 244823040 2^{10} \cdot 3^3 \cdot 5 \cdot 7 \cdot 11 \cdot 23 26 Yes Yes Yes --