What is the presentation of $\mathbb{Z}_n$

Question

I am given $\langle a,b | ab=ba, b^6=1 \rangle$ and I am supposed to compute the group that has this presentation. After racking my brains for a long time, the only thing I can come up with is $\mathbb{Z}_3$ since it's an abelian group under addition with two generators, one of which has order 6 and the other infinite order. I tried searching for the representation of $\mathbb{Z}_3$ to try to confirm my answer, but I am not able to find it anywhere. So I guess my question comes in two parts, how do you represent a group like $\mathbb{Z}_n$ and have I come to the correct conclusion in this case?

Answer 1

To summarise the comments: $ab = ba$ means the group is abelian. Since no other relator mentions the generator $a$, that means $\langle a \rangle$ is a "direct product" component of the group: the group is $$\mathbb{Z} \times \overline{\langle b \mid b^6 = 1 \rangle}$$ which is therefore $$\mathbb{Z} \times \mathbb{Z}_6$$