On 2-arc-transitive representations of the groups of fourth-power-free order

A complete classification of 2 -arc-transitive representations of primitive or bi-primitive groups of fourth-power-free order is given. The list consists of the following graphs: K n , K n , n − n K 2 , K r , r with r an odd prime, Peterson graph O 2 , incidence and non-incidence graphs H D ( 11 , 5 , 2 ) and H D ( 11 , 6 , 2 ) of the Hadamard design on 11 points, some explicit coset graphs of two-dimensional projective groups PSL ( 2 , p l ) with 1 ≤ l ≤ 2 and of Janko simple group J 1 , and the standard double cover of these coset graphs. Moreover, a classification of primitive permutation groups of fourth-power-free order is given.