In general is not a group under multiplication. For example is not a group under multiplication, as 2*3=0. However the group of units mod 6 IS a group under multiplication.

I feel that the book is using the same notation as Gallian's Contemporary Abstract Algebra. Where denotes the group under addition.

So in this case, it's useful to compute the order of (2,2). The order of 2 in is 3 and the order of 2 in is 2. So the order of (2,2) is the least common multiple of 2 and 3, which is 6. As is a group of order 24, the factor group will be of order $24/6 = 4$. So just find a representative for each coset and you'll "compute the factor group"