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Problem: Given two permutations $x=(1\space2\space3\space4\space5\space6\space7\space8\space9\space10\space11)$ and $y=(5\space6\space4\space10)(11\space8\space3\space7)$. Find the number of elements in the group $G$ generated by $x$ and $y$.

I observed that the number of elements in $G$ is a multiple of 44 and I know there are more elements but I don't know how many. Please help.

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    Hint: this is the Mathieu $M11$ group. – 2017-02-06
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    @MarcBogaerts, I never heard of that term. The Wikipedia page on Mathieu Group just states that it has 7920 elements without proof. – 2017-02-06
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    Maybe you should look elsewhere than wikipedia – 2017-02-06
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    Would you please give more information on G? – 2017-02-06
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    @theSongbird, G is the unique group generated by x and y by the usual operation of composition. – 2017-02-06
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    @Back2Black How did you figure out that 44 | org(G)? Since x is basically $e$ in $S_{11}$, wouldn't it all come down to finding the order of y? – 2017-02-06
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    The identical question was asked a week or two ago but I can't find it. It is not easy because you need somehow to prove that the two permutations do not generate $M_{11}$. You could do it easily on a computer. The only approach that I can think of is to show that the two permutations both permute the $66$ blocks of a Steiner system $S(4,5,11)$, but that would be difficult and tedious. – 2017-02-06
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    @theSongbird $x$ has order $11$ and $y$ has order $4$ so clearly $44$ divides the orde rof the group. Also, the commutator $[x,y]$ has order $5$ (and the $5$-cycles are blocks of the preserved Steiner System), $x^2y$ has order $8$, $xy^2$ has order $3$, so $1320$ divides $|G|$. – 2017-02-06
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    The question @DerekHolt was referring to should be [this one](http://math.stackexchange.com/questions/2122279/hint-on-an-exercise-of-mathieu-groups). – 2017-02-07

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