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I am working through an abstract algebra exercise book for my exam. It has solutions, but sometimes only references are given to books which I may not have access to. So I think, that the exercise is a bit more advanced and not suitable for an exam. Can somebody tell me, if the following exercise is a rather long and difficult one or it may be solved during an exam? I know this is vague, but I have really no idea how to solve this (thus if it is doable, a hint would be nice).

Let $x = (1,2,3,4,5,6,7,8,9,10,11)$ and $y = (5,6,4,10)(11,8,3,7)$ in $S_{11}$. Show that $$|M_{11}| := |\langle x,y\rangle| = 8\cdot 9 \cdot 10 \cdot 11$$

I took only an abstract algebra I course, so I know the basics up to Sylow theorems and classification of finite abelian groups.

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    My immediate reaction is that you could not attempt this question without knowing something about the definition and properties of $M_{11}$, and then it would depend on exactly what you knew. – 2017-01-31
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    Does this actually have anything to do with $M_{11}$? One can address what you have written by proving $|\langle x,y\rangle|=2^4\cdot 3^2\cdot 5\cdot 11$. I'm not sure if this helps you. – 2017-01-31
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    It might not be too hard to prove with $G := \langle x,y \rangle$ that $|G| \ge 7920$, but how would you prove (without using a computer) for example that $G \ne A_{11}$? One possibility might be to show that $G$ preserves a Steiner system $S(4,5,11)$, but that would be hard work, because there are $66$ blocks in this system, and you first have to find one. – 2017-02-01

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