A cubic (trivalent) graph Γ is said to be 4‐arc‐transitive if its automorphism group acts transitively on the 4‐arcs of Γ (where a 4‐arc is a sequencev0,v1, …v4of vertices of Γ such thatvi−1is adjacent tovifor 1 ⩽i⩽ 4, andvi−1≠vi+1for 1 ⩽i<4). In his investigations into graphs of this sort, Biggs defined a family of groups 4+(am), form= 3,4,5…, each presented in terms of generators and relations under the additional assumption that the vertices of a circuit of lengthmare cyclically permuted by some automorphism. In this paper it is shown that whenevermis a proper multiple of 6, the group 4+(am) is infinite. The proof is obtained by constructing transitive permutation representations of arbitrarily large degree.