The existence of symmetric designs with parameters (189, 48, 12) (Q1374195)

Since 1994, I have searched for the possible existence of symmetric designs with parameters (189, 48, 12) (of order 36). Finally, the following result was proved: Let \(G\) be the wreathed product of a Frobenius group \(F_{21}\) of order 21 with a cyclic group \(\mathbb{Z}_2\) of order 2. Suppose that \(G\) acts as an automorphism group on a symmetric design with parameters (198, 48, 12). Then there exists (up to isomorphism) exactly one such design \(D\) which is self-dual. The full automorphism group of \(D\) is our group \(G\) of order 882 and acts in three orbits of lengths 21, 21, and 147 for points and blocks. The group \(G\) has three conjugacy classes of subgroups of order 7. The elements of order 7 in one class act fixed-point-free and in the other two classes have exactly 7 fixed points each. The group \(G\) has three conjugacy classes of subgroups of order 3. The elements of order 3 in one class act fixed-point-free and in the other two classes have exactly 3 and 6 fixed points, respectively. An involution in \(G\) has exactly 29 fixed points. The 2-rank of \(D\) is 63.