  Divide a deck of $kn$ cards into $k$ equal piles and place them from left to
right. The standard shuffle $\sigma$ is performed by picking up the top cards
one by one from left to right and repeating until all cards have been picked
up. For every permutation $\tau$ of the $k$ piles, use $\rho_{\tau}$ to denote
the induced permutation on the $kn$ cards. The shuffle group $G_{k,kn}$ is
generated by $\sigma$ and the $k!$ permutations $\rho_{\tau}$. It was
conjectured by Cohen et al in 2005 that the shuffle group $G_{k,kn}$ contains
$A_{kn}$ if $k\geq3$, $(k,n)\ne\{4,2^f\}$ for any positive integer $f$ and $n$
is not a power of $k$. Very recently, Xia, Zhang and Zhu reduced the proof of
the conjecture to that of the $2$-transitivity of the shuffle group and then
proved the conjecture under the condition that $k\ge4$ or $k\nmid n$. In this
paper, we proved that the group $G_{3,3n}$ is $2$-transitive for any positive
integer $n$ which is a multiple of $3$ but not a power of $3$. This result
leads to the complete classification of the shuffle groups $G_{k,kn}$ for all
$k\ge2$ and $n\ge1$.
