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Coxeter groups include groups like E6, G2 etc which when defined over finite fields are simple finite groups. Are there coxeter representation for such simple finite groups (like E6(q))?

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I think you confuse finite groups of Lie type and their Weyl groups; only the latter are necessarily Coxeter groups. Anyway Coxeter groups (of order${}>2$) are never simple groups, since they have an index $2$ (therefore normal) subgroup. – Marc van Leeuwen Jul 29 '12 at 20:29
It might help to describe what you mean by "coxeter representation". If you mean a presentation on involutions like $$\langle s_i : s_i^2 = (s_i s_j)^{m_{ij}} = 1 \rangle$$ then no, a finite simple group of order greater than 2 does not have such a presentation. A (very) few of the finite groups of Lie type do have such presentation, but I think they are more like coincidences. – Jack Schmidt Jul 29 '12 at 22:39

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