  In this paper, we investigate the properties of a random walk on the
alternating group $A_n$ generated by $3$-cycles of the form $(i,n-1,n)$ and
$(i,n,n-1)$. We call this the transpose top-$2$ with random shuffle. We find
the spectrum of the transition matrix of this shuffle. We show that the mixing
time is of order $\left(n-\frac{3}{2}\right)\log n$ and prove that there is a
total variation cutoff for this shuffle.
