# Mathieu group

## 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 --