A Proposition from a book written by Benson Farb and Dan Margalit
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5$\begingroup$ This is not a good question. You should a) include the content in the actual question, rather than a screen capture, b) an actual reference rather than "a book" and c) explain some context of what you know, what you've tried and a specific question that has an answer. $\endgroup$– David Roberts ♦Commented Sep 5, 2017 at 1:45
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The question is terribly put , but the answer is: $S_{0, 4}$ is the four times punctured sphere. You can think of this sphere as the ideal simplex in $\mathbb{H}^3$ (it is a theorem of mine that this is always possible). A simplex always has a Klein four-group worth of symmetries, hence the $\mathbb{Z}/2 \mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ factor. The quotient is $S_{0, 2, 2, 2},$ which is the quotient of a punctured torus by the elliptic involution. The mapping class group of a punctured torus is $PSL(2, \mathbb{Z})$ and every element commutes with the elliptic action.
answered Sep 5, 2017 at 2:04