Does there exist a group $G$ (finite or infinite) with three subgroups $A, B, C \leq G$ satisfying the following three conditions?

1. $A = N_G(A)$, $B = N_G(B)$, $C = N_G(C)$;

2. $AB = BC = CA = G$;

3. $A \cap B = B \cap C = C \cap A = 1$.

(This question turned up in a more specific setting, but a negative answer to the existence of such a group would settle our case.)