The flag-transitive symmetric designs with 45 points, blocks of size 12, and 3 blocks on every point pair
We construct two flag-transitive designs with the parameters of the title, and prove that these are the only two examples. One is point-primitive and related to unitary geometry, while the other is point-imprimitive and constructed from a 1-dimensional affine space. This classification may be contrasted with a construction by Mathon and Spence in 1996 of more than 3,700 designs with these parameters, almost a third of which had a trivial automorphism group.
symmetric design flag-transitive design automorphism group