# reflection groups over finite fields and coxeter groups

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