  A $3$-$(v,\{4,6\},1)$ design is a configuration of $v$ points and a
collection of $4$- and $6$-element subsets called blocks, that jointly contain
every 3-element subset exactly once. Using an exhaustive computer search on
$v\leq 28$ points we investigate the $3$-$(v,\{4,6\},1)$ designs that have a
transitive automorphism group and where the blocks of size 6 form a 2-class
symmetric design. A 2-class symmetric design with parameters
$(v,k{;}\lambda_1,\lambda_2{;}\delta_1,\delta_2)$ is a set-system on $v$ points
and $v$ blocks of size $k$, where every pair of points are in $\lambda_1$ or
$\lambda_2$ blocks and every pair of blocks intersect in $\delta_1$ or
$\delta_2$ points. The 2-class symmetric designs include biplanes,
semi-biplanes, and 2-class symmetric partially balanced incomplete block
designs.
