  Let $C_t$ be a cycle of length $t$, and let $P_1,\ldots,P_t$ be $t$ polygon
chains. A polygon flower $F=(C_t; P_1,\ldots,P_t)$ is a graph obtained by
identifying the $i$th edge of $C_t$ with an edge $e_i$ that belongs to an
end-polygon of $P_i$ for $i=1,\ldots,t$. In this paper, we first give an
explicit formula for the sandpile group $S(F)$ of $F$, which shows that the
structure of $S(F)$ only depends on the numbers of spanning trees of $P_i$ and
$P_i/ e_i$, $i=1,\ldots,t$. By analyzing the arithmetic properties of those
numbers, we give a simple formula for the minimum number of generators of
$S(F)$, by which a sufficient and necessary condition for $S(F)$ being cyclic
is obtained. Finally, we obtain a classification of edges that generate the
sandpile group.
  Although the main results concern only a class of outerplanar graphs, the
proof methods used in the paper may be of much more general interest. We make
use of the graph structure to find a set of generators and a relation matrix
$R$, which has the same form for any $F$ and has much smaller size than that of
the (reduced) Laplacian matrix, which is the most popular relation matrix used
to study the sandpile group of a graph.
