  We study a one-dimensional spin-$1/2$ model with three-spin interactions and
a transverse magnetic field $h$. The model has a $Z_2 \times Z_2$ symmetry, and
a duality between $h$ and $1/h$. The self-dual point at $h=1$ is a quantum
critical point with a continuous phase transition. We compute the critical
exponents $z$, $\beta$, $\gamma$ and $\nu$, and the central charge $c$
numerically using exact diagonalization (ED) for systems with periodic boundary
conditions. We find that both $z$ and $c$ are equal to $1$, implying that the
critical point is governed by a conformal field theory. The values obtained for
$\beta/\nu$, $\gamma/\nu$, and $\nu$ from ED suggest that the model exhibits
Ashkin-Teller criticality with an effective coupling that is intermediate
between the four-state Potts model and two decoupled transverse field Ising
models. An analysis on larger systems but with open boundaries using
density-matrix renormalization group calculations, however, shows that the
self-dual point may be in the same universality class as the four-state Potts
model. An energy level spacing analysis shows that the model is not integrable.
For a system with periodic boundary conditions, there are an exponentially
large number of exact mid-spectrum zero-energy eigenstates. A subset of these
eigenstates have wave functions which are independent of $h$ and have unusual
entanglement structure, suggesting that they are quantum many-body scars. The
number of such states scales at least linearly with system size. Finally, we
study the infinite-temperature autocorrelation functions close to one end of an
open system. We find that some of the autocorrelators relax anomalously in
time, with pronounced oscillations and very small decay rates if $h \gg 1$ or
$h \ll 1$. If $h$ is close to the critical point, the autocorrelators decay
quickly to zero except for an autocorrelator at the end site.
