The pentavalent three-geodesic-transitive graphs

A complete classification is given of pentavalent 3-geodesic-transitive graphs which are not 3-arc-transitive, which shows that a pentavalent 3-geodesic-transitive but not 3-arc-transitive graph is one of the following graphs: (2×6)-grid¯, H(5,2), the icosahedron, the incidence graph of the 2-(11,5,2)-design, the Wells graph and the Sylvester graph.