TITLE: Order from Coxeter-Dynkin diagram
QUESTION [5 upvotes]: How is the order of a Coxeter group determined from its Coxeter-Dynkin diagram?

REPLY [3 votes]: Several answers to this question are given in Section 12 of a paper called "Generalized cluster complexes and Coxeter combinatorics" (https://arxiv.org/abs/math/0505085).  Specifically, there are four (more or less crazy) ways to determine invariants including exponents recursively from the diagram (if you know all invariants for connected subgraphs of the diagram).  Once you know the exponents, you know the order of $W$.
These methods are much too involved to describe here, but several of them have to do with the Fuss-Catalan number associated to $W$.  This is a polynomial in a parameter $m$ and has a product formula in terms of the Coxeter number and exponents.  You can factor the Fuss-Catalan number (polynomial) completely and (if you know the Coxeter number $h$) read off the exponents.