  In this paper, we consider the Strichartz inequality for a fourth-order
Schr\"odinger equation on $\mathbb{R}^3$. We show that extremizers exist using
a linear profile decomposition. Based on the existence of extremizers, we use
the associated Euler-Lagrange equation to show that the extremizers have
exponential decay and consequently must be analytic. Furthermore, we also prove
that extremizers must have the even property
$|\widehat{f}(\xi)|=|\widehat{f}(-\xi)|$ almost everywhere.
