Von Dyck groups that are conjugated.

Question

Let us consider the Von Dyck groups $$ D(a,b,c)=\langle x,y,z\mid x^{a}=y^{b}=z^{c}=xyz=1\rangle $$ and $$ D(a'.b',c')=\langle x,y,z\mid x^{a'}=y^{b'}=z^{c'}=xyz=1\rangle. $$ Suppose $$ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}<1,\quad \frac{1}{a'}+\frac{1}{b'}+\frac{1}{c'}<1. $$ I have read that in this case $$ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}= \frac{1}{a'}+\frac{1}{b'}+\frac{1}{c'} $$ implies that $D(a,b,c)$ and $D(a',b',c')$ are conjugated. How could we prove it?

Answer 1

I don't think this can be correct. $D(3,6,8)$ has abelianization $C_6$, whereas $D(4,4,8)$ has abelianization $C_4^2$, so the groups are not isomorphic.