On symmetric (69, 17, 4)-designs admitting an action of Frobenius groups (Q5934179)

To achieve the complete classification of symmetric \((69, 17, 4)\)-designs admitting an action of any Frobenius group of automorphisms, a positive result, other than non-existence results, is given. That is, when the Frobenius group of order 39 acts on a symmetric \((69, 17, 4)\)-design and the cyclic group of order 3 fixes 9 points, then there are up to isomorphism and duality two such designs \({\mathcal D}_1\) and \({\mathcal D}_2\). They are both self-dual and as their full automorphism groups appear \(\text{Aut }{\mathcal D}_1\cong \text{Frob}_{39}\times {\mathcal Z}_4\) and \(\text{Aut }{\mathcal D}_2\cong \text{Frob}_{39}\cdot{\mathcal Z}_4\).