Find a Group with following presentation

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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<x,y,z| x^2, y^2, z^2, (xz)^2, (xy)^3, (yz)^3 > –  Detectives Nov 19 '12 at 7:51
Oh, sry please use $\langle\rangle$ instead of $<>$ ... and use LaTeX. –  martini Nov 19 '12 at 7:52
I'm almost certain that someone can. –  Arthur Fischer Nov 19 '12 at 7:53
I would recommend choosing x,y,z to be reflections, in such a way that the products xz, xy, and yz will be rotations with the right order. –  Ted Nov 19 '12 at 7:55
–  Qiaochu Yuan Nov 19 '12 at 8:43

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.
Yes, the $(2,3,3)$ group I gave is a subgroup of index $2,$ not the whole group. –  Geoff Robinson Nov 19 '12 at 22:14