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This group, denoted , 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:
- It is the projective special unitary group of degree three for the quadratic extension field:F4 over field:F2.
- It is the special unitary group of degree three for the quadratic extension field:F4 over field:F2.
This is one of the 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 is simple, but not a sporadic simple group -- it is isomorphic to projective special linear group:PSL(3,4).
Arithmetic functions of a counting nature
|number of subgroups||68||See subgroup structure of Mathieu group:M9|
|number of conjugacy classes||6|
|number of conjugacy classes of subgroups||14||See subgroup structure of Mathieu group:M9|
|T-group||No||The -Sylow subgroup is abelian normal, has subgroups that are not normal.|
|Monolithic group||Yes||Unique minimal normal subgroup is -Sylow subgroup.|
|One-headed group||No||Three maximal normal subgroups of order .|
|Rational-representation group||No||Its quotient, quaternion group, is not a rational-representation group.|
|Any two elements generating the same cyclic subgroup are automorphic||Yes||Follows from being a rational group|
|Every element is automorphic to its inverse||Yes||Follows from being an ambivalent group|
|Frobenius group||Yes||Frobenius kernel is -Sylow subgroup, complement is -Sylow subgroup|
|Camina group||Yes||Derived subgroup is of order , three non-identity cosets are conjugacy classes.|
Linear representation theory
Further information: linear representation theory of Mathieu group:M9
|Degrees of irreducible representations over a splitting field (such as or )|| 1,1,1,1,2,8|
maximum: 8, lcm: 8, number: 6, sum of squares: 72
|Ring generated by character values||-- ring of integers|
|Field generated by character values||-- field of rational numbers. So the group is a rational group.|
This finite group has order 72 and has ID 41 among the groups of order 72 in GAP's SmallGroup library. For context, there are 50 groups of order 72. It can thus be defined using GAP's SmallGroup function as:
For instance, we can use the following assignment in GAP to create the group and name it :
gap> G := SmallGroup(72,41);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [72,41]
or just do:
to have GAP output the group ID, that we can then compare to what we want.