  Let $A$ be a semistable principally polarized abelian variety of dimension
$d$ defined over the rationals. Let $\ell$ be a prime and let
$\bar{\rho}_{A,\ell} : G_{\mathbb{Q}} \rightarrow
\mathrm{GSp}_{2d}(\mathbb{F}_\ell)$ be the representation giving the action of
$G_{\mathrm{Q}} :=\mathrm{Gal}(\bar{\mathrm{Q}}/\mathrm{Q})$ on the
$\ell$-torsion group $A[\ell]$. We show that if $\ell \ge \max(5,d+2)$, and if
image of $\bar{\rho}_{A,\ell}$ contains a transvection then
$\bar{\rho}_{A,\ell}$ is either reducible or surjective.
  With the help of this we study surjectivity of $\bar{\rho}_{A,\ell}$ for
semistable principally polarized abelian threefolds, and give an example of a
genus $3$ hyperelliptic curve $C/\mathbb{Q}$ such that $\bar{\rho}_{J,\ell}$ is
surjective for all primes $\ell \ge 3$, where $J$ is the Jacobian of $C$.
