I'm given the problem where one can perform perfect shuffles (i.e. you split the deck into halves and then interweave them) on a deck of $52$ cards (both in and out shuffles) and I am supposed to determine whether all $52!$ possible deck orderings are possible through a composition of such shuffles. I know that given only in or out shuffles you cannot do so since they are cyclic and of order $8$ and $52$ but I really have no idea how to even begin to tackle this problem of composing them. Was hoping for any hints or thoughts on as to how I should attempt this problem? Thanks!

EDIT: An out shuffle is when you interweave leaving the top card on the top while an in shuffle is when you interweave by putting the top card of the bottom half on top of the whole deck.
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Not an answer, just a suggestion.

You have two permutations, and you want to find out if they generate the entire group of all permutations. 

If $a$ is one permutation, and $b$ is the other, then $ab^{-1}=(1\,27)(2\,28)(3\,29)\cdots(26\,52)$. We can see that $a$ and $c=ab^{-1}$ generates the same subgroup as $a$ and $b$.

$c=ab^{-1}$ is probably easier to deal with, since its order is $2$.
Given an initial configuration of a deck of $52$ cards, perfectly shuffling them $52$ times will take you through exactly $52$ of the $52!$ permutations on the deck. In order to hit every one of them, you must perfectly shuffle the deck and then switch the top two cards before perfectly shuffling it again. Of course you'd be doing this for several lifetimes. In the same vein, perfectly out-shuffling $52$ times, and then in-shuffling once before out-shuffling again $52$ times will indeed have the same effect as composing (or switching just the top two cards) the permutations. But again we could never experience this in hundreds of lifetimes. A computer simulation could demonstrate it perhaps. The (n-1) k-cycle has $8$ elements in its one-orbit for $n=52$, whereas the (n+1) k-cycle has $52$. This is how order is determined. There is no multiplication that can be performed on these orders to determine whether they span the full symmetric set. We can see the composition. Go through all $52$ configurations stemming from the initial position, make one alteration, repeat. Time permitting, we will see every member of S sub $52$.