The Mathieu groups $M_n$, for $n$ in $\left\{9\,10\,11\,12\,21\,22\,23\,24\right\}$ are a family of transitive permutation groups studied by Émile Mathieu in the late nineteenth century.  The simple groups in the family are examples of highly transitive groups. The Mathieu group $M_n$ is simple for $n$ in $\left\{11\,12\,21\,22\,23\,24\right\}$.

Note that while the Mathieu group $M_{21}$ of order $20160$ is simple, it is not sporadic, being isomorphic to the group $PSL\left(3\,4\right)$ .

The MathieuGroup( n ) command returns a permutation group isomorphic to the Mathieu group of degree n, where the degree n must be in { 9, 10, 11, 12, 21, 22, 23, 24 }. This is a sporadic finite simple group for n=11, 12, 22, 23, 24.

The Mathieu group $M_9$ is, in fact a soluble group.

The form = F option controls the form of the group returned. By default, a permutation group is returned; this is equivalent to passing the option form = "permgroup". A finitely presented group can be obtained by passing the option form = "fpgroup".

In the Standard Worksheet interface, you can insert this group into a document or worksheet by using the Group Constructors palette.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$\mathrm{MathieuGroup}\left(11\right)$

$M_{11}$

$G≔\mathrm{MathieuGroup}\left(23\right)$

$G≔M_{23}$

$\mathrm{type}\left(G\,'\mathrm{PermutationGroup}'\right)$

$\mathrm{true}$

$\mathrm{GroupOrder}\left(G\right)$

$10200960$

$\mathrm{Transitivity}\left(G\right)$

$4$

$G≔\mathrm{MathieuGroup}\left(12\right)$

$G≔M_{12}$

$\mathrm{Degree}\left(G\right)$

$12$

$\mathrm{Transitivity}\left(G\right)$

$5$

$G≔\mathrm{MathieuGroup}\left(9\right)\:$

$\mathrm{IsSoluble}\left(G\right)$

$\mathrm{true}$

$\mathrm{DerivedSeries}\left(G\right)$

$M_9▹\left[M_9\,M_9\right]▹\left[\left[M_9\,M_9\right]\,\left[M_9\,M_9\right]\right]▹⟨⟩$

$\mathrm{Display}\left(\mathrm{CharacterTable}\left(\mathrm{MathieuGroup}\left(10\right)\right)\right)$

$G≔\mathrm{MathieuGroup}\left(11\,'\mathrm{form}'=''fpgroup''\right)$

$G≔⟨a\,b\mid a^2\,b^4\,a{b}^{2}a{b}^{2}a{b}^{2}a{b}^{2}a{b}^{2}a{b}^2\,ababa{b}^{-1}aba{b}^{2}a{b}^{-1}aba{b}^{-1}a{b}^{-1}\,ababababababababababab⟩$

The GroupTheory[MathieuGroup] command was introduced in Maple 17.

For more information on Maple 17 changes, see Updates in Maple 17.

The GroupTheory[MathieuGroup] command was updated in Maple 2020.