  Quantum Fourier transformations are an essential component of many quantum
algorithms, from prime factoring to quantum simulation. While the standard
abelian QFT is well-studied, important variants corresponding to
\emph{nonabelian} groups of interest have seen less development. In particular,
fast nonabelian Fourier transformations are important components for both
quantum simulations of field theories as well as approaches to the nonabelian
hidden subgroup problem. In this work, we present fast quantum Fourier
transformations for a number of nonabelian groups of interest for high energy
physics, $\mathbb{BT}$, $\mathbb{BO}$, $\Delta(27)$, $\Delta(54)$, and
$\Sigma(36\times3)$. For each group, we derive explicit quantum circuits and
estimate resource scaling for fault-tolerant implementations. Our work shows
that the development of a fast Fourier transformation can substantively reduce
simulation costs by up to three orders of magnitude for the finite groups that
we have investigated.
