Lam, T.Y. and Leep, David B
"Combinatorial structure on the automorphism group of S_6
Expo. Math. 11 no. 4, pp. 289-308 (1993).
i don't know what exactly you are looking for, but all questions you've been asking so far are very well-known results and can be found in almost any decent book in group theory.
for example my favorite is "An Introduction to the Theory of Groups", 4th edition, by Joseph Rotman.
There are not that many authors in algebra who are as brilliant as Joseph Rotman (group and field theory), Louis Rowen (ring theory), Igor Shafarevich (algebraic geometry), ...