Groups of Order 6

Theorem
There exist exactly $2$ groups of order $6$, up to isomorphism:


 * $C_6$, the cyclic group of order $6$


 * $S_3$, the symmetric group on $3$ letters.

Proof
From Existence of Cyclic Group of Order n we have that one such group of order $6$ is $C_6$ the cyclic group of order $6$:

This is exemplified by the additive group of integers modulo $6$, whose Cayley table can be presented as:

Then we have the symmetric group on $3$ letters.

From Order of Symmetric Group, this has order $6$.

It can be exemplified by the symmetry group of the equilateral triangle, whose Cayley table can be presented as:

It remains to be shown that these are the only $2$ groups of order $6$.