Mathieu group:M9

Definition
This group, denoted $$M_9$$, and termed the Mathieu group of degree nine, is defined in the following equivalent ways:


 * It is the external semidirect product of elementary abelian group:E9 (a two-dimensional vector space over field:F3) by quaternion group where the latter acts on the former via the faithful irreducible representation of quaternion group (a two-dimensional irreducible representation) over field:F3 (this representation can be thought of as the embedding Q8 in GL(2,3)).
 * It is the subgroup of the symmetric group of degree nine given by the following generating set:

$$\! M_9 := \langle (1,4,9,8)(2,5,3,6), (1,6,5,2)(3,7,9,8) \rangle$$.

This is one of the member of family::Mathieu groups, but is not one of the five sporadic simple Mathieu groups. Rather, it is among the two Mathieu groups (the other being Mathieu group:M10) that are not simple. The Mathieu group $$M_{21}$$ is simple, but not a sporadic simple group -- it is isomorphic to projective special linear group:PSL(3,4).
 * It is the member of family::projective special unitary group $$PSU(3,2)$$ of degree three for the quadratic extension field:F4 over field:F2.
 * It is the member of family::special unitary group $$SU(3,2)$$ of degree three for the quadratic extension field:F4 over field:F2.