  In this paper, we consider the lengths of cycles that can be embedded on the
edges of the generalized pancake graphs which are the Cayley graph of the
generalized symmetric group $S(m,n)$, generated by prefix reversals. The
generalized symmetric group $S(m,n)$ is the wreath product of the cyclic group
of order $m$ and the symmetric group of order $n!$. Our main focus is the
underlying \emph{undirected} graphs, denoted by $\mathbb{P}_m(n)$. In the cases
when the cyclic group has one or two elements, these graphs are isomorphic to
the pancake graphs and burnt pancake graphs, respectively. We prove that when
the cyclic group has three elements, $\mathbb{P}_3(n)$ has cycles of all
possible lengths, thus resembling a similar property of pancake graphs and
burnt pancake graphs. Moreover, $\mathbb{P}_4(n)$ has all the even-length
cycles. We utilize these results as base cases and show that if $m>2$ is even,
$\mathbb{P}_m(n)$ has all cycles of even length starting from its girth to a
Hamiltonian cycle. Moreover, when $m>2$ is odd, $\mathbb{P}_m(n)$ has cycles of
all lengths starting from its girth to a Hamiltonian cycle. We furthermore show
that the girth of $\mathbb{P}_m(n)$ is $\min\{m,6\}$ if $m\geq3$, thus
complementing the known results for $m=1,2.$
