Suppose we are given a group $G$ in terms of generators $t_1, ..., t_n$ which are order 2 in $S_m$ (however we don't assume anything other than that these elements generate $G$ and have order 2). Since $G$ is finite and generated by transpositions, it must have a root system. What is the best known algorithm for finding the root system?

Suppose we are given a group $G$ in terms of generators $t_1, ..., t_n$ which are order 2 in $S_m$ (however we don't assume anything other than that these elements generate $G$ and have order 2). What is the most efficient way to determine:

1. If $G$ is abstractly isomorphic to a Coxeter group
2. Assuming yes, a Coxeter system for $G$
3. Assuming no, a presentation of $G$ as a quotient of a Coxeter group