Some new designs with prescribed automorphism groups

We establish the existence of simple designs with parameters $2$-$(55,10,4)$, $3$-$(20,5,4)$, $3$-$(21,7,30)$, $4$-$(15,5,2)$, $4$-$(16,8,45)$, $5$-$(16,7,10)$, and $5$-$(17,8,40)$, which have previously been unknown. For the corresponding $t$, $v$, and $k$, we study the set of all $\lambda$ for which simple $t$-$(v,k,\lambda)$ designs exist.