  A connected graph of order $n$ admitting a semiregular automorphism of order
$n/k$ is called a $k$-multicirculant. Highly symmetric multicirculants of small
valency have been extensively studied, and several classification results exist
for cubic vertex- and arc-transitive multicirculants. In this paper we study
the broader class of cubic vertex-transitive graphs of order $n$ admitting an
automorphism of order $n/3$ or larger that may not be semiregular. In
particular, we show that any such graph is either a $k$-multicirculant for some
$k \leq 3$, or it belongs to an infinite family of graphs of girth $6$.
