Relativistic electrons are an important space weather hazard, being a major source of radiation damage to satellites and posing a risk to humans in space. We use approximately 20 years of data from the US Global Positioning System (GPS) satellite NS41 to determine the characteristics of the geomagnetic storms that lead to the largest relativistic electron fluxes in GPS orbit. The largest CME-driven events are associated with the solar wind having negative excursions of the IMF Bz with minimum values of ~-14 nT two hours prior to zero epoch, defined as the time of the minimum in the Dst index and strong Dst minima, reaching ~-130 nT at zero epoch. In contrast, events driven by high speed solar wind streams (HSSs) are associated with smaller negative excursions of IMF Bz with minimum values of ~-4 nT two hours prior to zero epoch and moderate Dst minima, reaching ~-60 nT at zero epoch. Compared with HSS-driven events, peak E = 2.0 MeV fluxes associated with CME-driven events are larger by factors of 1.3 at L=4.5 and 2.4 at L=6.5. Both the CME- and HSS-driven events are associated with enhancements in the solar wind number density and pressure prior to zero epoch. Following zero epoch the solar wind number density and pressure become low and substorm activity is enhanced for several days.