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$\begingroup$

Can anyone describe a group with following presentations? (rigorous proof is not needed) $ \langle x,y,z \mid x^2, y^2, z^2, (xz)^2, (xy)^3, (yz)^3\rangle $

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    See http://en.wikipedia.org/wiki/Coxeter_group .2012-11-19

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Since all generators are involutions, and all other relations are powers of a product of generators, this is a Coxeter group, and it suffices to translate those other relations into a Coxeter diagram. You get a linear diagram with three nodes and simple bonds, which corresponds the the symmetric group $S_4$. Concretely $x,y,z$ will give the adjacent transpositions $(1~2)$, $(2~3)$, and $(3~4)$ respectively.

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    Yes, the $(2,3,3)$ group I gave is a subgroup of index $2,$ not the whole group.2012-11-19