Classification of locally dihedral amalgams

An amalgam of rank 2 is a triple A=(A1,A12,A2) of finite groups A1,A12,A2 such that A1∩A2=A12. The degree of A is the pair (d1,d2) where di is the index of A12 in Ai for i=1,2. Let the degree of A be (k,2) where k⩾3 and suppose that only the identity subgroup of A12 is normal in both A1 and A2, let K=CoreA1(A12) and suppose that A1/K≅D2k is the dihedral group of order 2k. Then under the above conditions A is called a locally D2k amalgam. Such amalgams were classified for k=3 by Djoković and Miller [D.Ž. Djoković, G.L. Miller, Regular groups of automorphisms of cubic graphs, J. Combin. Theory Ser. B 29 (1980) 195–230], classified for odd numbers k in [A.Q. Sami, Locally dihedral amalgams of odd type, J. Algebra 298 (2006) 630–644] and partially classified for k=4 by Djoković [D.Ž. Djoković, A class of finite group-amalgams, Proc. Amer. Math. Soc. 80 (1) (1980) 22–26]. In this paper we classify locally D2k amalgams for all even numbers k and describe them in terms of generators and relations. We find that if then A12 is an elementary abelian 2-group.