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two ( for ), and six exceptional groups ( and ). Every finite Coxeter group is the direct product of finitely many of these irreducible groups.

Weyl groups

Many, but not all of these, are Weyl groups, and every Weyl group can be realized as a Coxeter group. The Weyl groups are the families and and the exceptions and denoted in Weyl group notation as

The non-Weyl ones are the exceptions and and those members of the family that

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