  Standard perfect shuffles involve splitting a deck of $2n$ cards into two
stacks and interlacing the cards from the stacks. There are two ways that this
interlacing can be done, commonly referred to as an in shuffle and an out
shuffle, respectively. In 1983, Diaconis, Graham, and Kantor determined the
permutation group generated by in and out shuffles on a deck of $2n$ cards for
all $n$. Diaconis et al. concluded their work by asking whether similar results
can be found for so-called generalized perfect shuffles. For these new
shuffles, we split a deck of $mn$ cards into $m$ stacks and similarly interlace
the cards with an in $m$-shuffle or out $m$-shuffle (denoted $I_m$ and $O_m$,
respectively). In this paper, we find the structure of the group generated by
these two shuffles for a deck of $m^k$ cards, together with $m^y$-shuffles, for
all possible values of $m$, $k$, and $y$. The group structure is completely
determined by $k/\gcd(y,k)$ and the parity of $y/\gcd(y,k)$. In particular, the
group structure is independent of the value of $m$.
