Show that there exist Coxeter systems $(W, S)$ and $(W′ , S′)$ with $|S|\not =|S′|$ such that $W$ is isomorphic to $W′$ as abstract groups.
[Hint: Consider the dihedral group of order 12.]
Attempt: We can say $W$ is generated by $a,b$ such that $a^2=1, b^2=1,(ab)^6=1$. Now if we pick for example $\rho=(ab)^3$ we have that $W$ is generated by $a,b,\rho$ such that $a^2=1,b^2=1,\rho^2=1,(ab)^6=1, (a\rho)^2=1, (b\rho)^2=1 $. But this is not a free group with Coxeter relations because $\rho=(ab)^3$, and if we remove this condition using a fresh new $\rho$ we obtain a bigger group than $W$. So I am confused.
Thanks!