As an exercise I am trying to prove that the group $$G = \langle a,b,c \mid ac = ba, ab=ca, bc=ab\rangle$$ is infinite and non-abelian. Moreover, the author claims that its center has finite index.
I have tried to find a group which is (a subgroup of) this abstract group, but to no advance. I have seen similar posts on the forum, but their answers were rather specific to the considered group, and did not provide a general approach.
Best Answer
Its abelianisation is
$$\langle a,b,c\mid a=b=c\rangle^{{\rm ab}}\cong \Bbb Z$$
and $G\twoheadrightarrow S_3$ (see the comments).