In this thesis a new characterisation of the (S_3, A_n)- and (S_3, S_n)-amalgams of characteristic 2 and critical distance 3 is obtained. It is shown that such an amalgam exists only in the exceptional cases when n=3, 5 or 8. Let C denote the class of all finite groups X of even order and such that O^2(X) is the unique minimal normal subgroup of X. It is the secondary purpose of this thesis to begin an investigation into the structure of the (S_3, C)-amalgams of characteristic 2 and critical distance 3. It is hoped that the results obtained may shed some light on the reason why so few of these amalgams are known to exist.