Lie symmetry algebra of the dispersionless Davey-Stewartson (dDS) system is shown to be infinite dimensional. The structure of the algebra turns out to be Kac-Moody-Virasoro one, which is typical for integrable evolution equations in 2+1 dimensions. Symmetry group transformations are constructed using a direct (global) approach. They are split into both connected and discrete ones. Several exact solutions are obtained as an application of the symmetry properties.