What is the symmetry group $S_3$ asked in this question my professor gave us?

Question

We are following the book Abstract Algebra: An Introduction by W. Hungerford in my Group Theory class and my professor asked this exact question on the quiz:

$S_3 = \langle a, b \mid a^3 = b^2 = 1, bab = a^2\rangle$

a) List the elements of $S_3$

I know that $S_3$ is the symmetry group of 3, which equals the group of all permutations of the elements in the set ={1,2,3}. But I'm not sure how the cyclic (?) notation is trying to show. I know $S_3$ is not a cyclic group. Is the group we're talking about here a subgroup of the symmetry group of 3, or is it a newly defined group? Are they equivalent?

b) List all the subgroups of $S_3$.

c) Give 3 reasons why $S_3$ is not isomorphic to $\mathbb Z/6\mathbb Z$

One of the reasons could be that the cardinalities are not equal, however, since I don't understand what the notation stands for I wasn't able to solve any one of the questions.

I would greatly appreciate help.

Answer 1

For c)

1. $S_3$ is nonabelian and $\mathbb{Z}_6$ is abelian (most obvious)
2. $\mathbb{Z}_6$ is cyclic and $S_3$ is not
3. $\mathbb{Z}_6$ is (isomorphic to) the direct sum of its two cyclic subgroups and $S_3$ is not

While these may be acceptable for quiz answers, 2. and 3. are so close to 1. that I'd not necessarily consider them distinct answers. It would be interesting to know what the professor was looking for.