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There is an example for $n=6$, a quite exceptional one. It comes from the smallest finite sporadic simple group $G=M_{11}$ of order 7920. $G$ has a permutation representation on 12 points, corresponding to the (left) cosets of a subgroup isomorphic to $PSL_2(11)$. A subgroup isomorphic to $A_6$, the alternating group of degree 6, has 2 orbits on 12 points, both of length 6, and the images of these orbits under $G$ are the 22 blocks of the design.