New strongly regular graphs from orthogonal groups O+(6,2) and O−(6,2)

We prove the existence of strongly regular graphs with parameters (216, 40, 4, 8) and (540, 187, 58, 68). We also construct a strongly regular graph with parameters (540, 224, 88, 96) that was previously unknown. Further, we construct all distance-regular graphs with at most 600 vertices, admitting a transitive action of the orthogonal group O+(6,2) or O−(6,2). Furthermore, we show that under certain conditions an orbit matrix M of a strongly regular graph Γ can be used to define a new strongly regular graph Γ˜, where the vertices of the graph Γ˜ correspond to the orbits of Γ (the rows of M). We show that some of the obtained strongly regular graphs are related to each other in a way that one can be constructed from an orbit matrix of the other.