\(\text{PSp}_4(3)\) as a symmetric (36, 15, 6)-design
Let \(G\) be a group isomorphic to \(\text{PSp}_4(3) \cong U_4(2)\) and let \(\mathcal P\) be the conjugacy class of subgroups of \(G\) isomorphic to \(S_6 \cong \text{Sp}_4(2)'\). Then \(S \in {\mathcal P}\) has three orbits \(\{S\}, B\) and \(C\) on \(\mathcal P\) of length \(1,15\) and \(20\), respectively. It is shown that \(({\mathcal P}, {\mathcal B}, I)\) is a symmetric \(2\)-\((36,15,6)\) design with set of blocks \({\mathcal B} = \{B^g \mid g \in G\}\) where a point \(R\) is incident with a block \(B^g\) if \(R\) is contained in \(B^g\). The automorphism group \(\Aut(G)\) acts on this design flag-transitively.
scientific article; zbMATH DE number 1319596