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A Coxeter group is a group defined by a presentation by involutions $r_i$ with relators $(r_ir_j)^{m_{ij}}=1$ for certain family $(m_{ij})$ of integers greater than 1.

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Subgroups of finite reflection groups that do not fix a point

You have to consider the Coxeter Graph: The subgroups you are looking for are those that you get by removing one edge...
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