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