Groups of order 6

There are, up to isomorphism, two groups of order 6, indicated in the table below:

There are many ways of demonstrating that there are only two groups of order six, including simply looking at the possible multiplication tables. One of the general approaches is via the classification of groups of order a product of two distinct primes. Since $$6 = 3 \cdot 2$$ and $$2 \mid (3 - 1)$$, the number $$6$$ falls in the two isomorphism classes case in that classification.