  The paper is an investigation of the structure of block-transitive
automorphism groups of a 3-design with small block size. Let $G$ be a
block-transitive automorphism group of a nontrivial $3$-$(v,k,\lambda)$ design
$\mathcal{D}$ with $k\le 6$. We prove that if $G$ is point-primitive then $G$
is of affine or almost simple type. If $G$ is point-imprimitive then
$\mathcal{D}$ is a $3$-$(16,6,\lambda)$ design with $\lambda\in\{4, 12, 16, 24,
28, 48, 56, 64, 84, 96, 112, 140\}$, and $rank(G)=3$.
