Collineation groups of finite projective planes are studied which do not leave invariant any point, line or triangle and contain a non-trivial perspectivity. In many instances the collineation group G can be determined, and it can be proved that the underlying projective plane contains a desarguesian subplane, whose order is related to the order of G. This can be done, because one has rather strong results about the structure of G, in particular about the first terms of a chief series of G. Projective planes are taken here as a suitable testing ground for methods, which can be used for many classes of designs to quickly find new examples and obtain some classification.