<s>[INST] Minimal generation of $S_n$ For a given $n$, the symmetric group $S_n$ is generated by permutations $p_1$, $p_2$, and $p_3$.  Two of them generate subgroups $P_{23}$, $P_{13}$, $P_{12}$, which have group orders $a$, $b$, and $c$.
What is the minimal $max(a,b,c)$ for various $n$?  
 [/INST] The answer is $8$ for all $n$ with $8 \le n  \le 26$ (as verret observed, it is $10$ for $n=7$), and I conjecture that it is $8$ for all larger $n$, with $S_n$ being a quotient of $$\langle x,y,z \mid x^2=y^2=z^2=(xy)^4=(yz)^4=(xz)^4=1 \rangle.$$
(It is possible that this is a known result.)
There are lots of solutions. Here are some for $n=6,7,8,9$ as requested.
$$n=6: (1, 2)(3, 4)(5, 6),\ 
    (1, 2)(3, 6),\ 
    (2, 3),$$
$$n=7: (1, 7)(2, 4)(5, 6),\ 
    (1, 2),\ 
    (1, 7)(2, 3)(4, 6),$$
$$n=8: (1, 8)(2, 3)(6, 7),\ 
    (1, 7)(2, 4)(3, 5),\ 
    (1, 2),$$
$$n=9: (1, 2)(3, 4),\ 
    (2, 4)(5, 6)(7, 8),\ 
    (3, 5)(4, 7)(6, 9).$$
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