  In this Letter, we propose a growing network model that can generate
scale-free networks with a tunable community strength. The community strength,
$C$, is directly measured by the ratio of the number of external edges to
internal ones; a smaller $C$ corresponds to a stronger community structure.
According to the criterion obtained based on the master stability function, we
show that the synchronizability of a community network is significantly weaker
than that of the original Barab\'asi-Albert network. Interestingly, we found an
unreported linear relationship between the smallest nonzero eigenvalue and the
community strength, which can be analytically obtained by using the
combinatorial matrix theory. Furthermore, we investigated the Kuramoto model
and found an abnormal region ($C\leq 0.002$), in which the network has even
worse synchronizability than the uncoupled case (C=0). On the other hand, the
community effect will vanish when $C$ exceeds 0.1. Between these two extreme
regions, a strong community structure will hinder global synchronization.
