  We consider a random walk on the hyperoctahedral group $B_n$ generated by the
signed permutations of the forms $(i,n)$ and $(-i,n)$ for $1\leq i\leq n$. We
call this the flip-transpose top with random shuffle on $B_n$. We find the
spectrum of the transition probability matrix for this shuffle. We prove that
the mixing time for this shuffle is of order $n\log n$. We also show that this
shuffle exhibits the cutoff phenomenon. In the appendix, we show that a similar
random walk on the demihyperoctahedral group $D_n$ also has a cutoff at
$\left(n-\frac{1}{2}\right)\log n$.
