From the (sharpened) Golod/Shafarevich inequality we know that for finite $p-$groups, where $r$ is the minimal number of relations and $d$ is the minimal number of generators, that $r > \frac{d^2}{4}$
For the symmetric group of degree $3$, $S_3$, I know that $d = 2$ and $r = 2$. But can someone explain to me what two generators we can use and demonstrate that these generators and relations (defined abstractly without already using the permutations) give us the multiplication table for $S_3$? Also, back to $p-$groups, the above inequality for $d = 4$ would give us that $r\ge 5$.
Can someone give me an example of a $2-$group with $d = 4$ and $r = 5$?
$$\langle a,b \mid a^2=1, a^{-1}ba=b^2 \rangle$$
is a presentation of $S_3$.
Note that the relations imply $$b = a^{-2}ba^2=a^{-1}(a^{-1}ba)a = a^{-1}b^2a = (a^{-1}ba)^2 = b^4$$
giving $b^3=1$, and now it is easy to see that $G \cong S_3$.
Added later: More generally, for any $k \ge 1$, $$\langle a,b \mid a^2=1, a^{-1}b^ka = b^{k+1} \rangle$$ is a presentation of the dihedral group of order $2(2k+1)$. The proof is left to the reader.