triple factorization of non-abelian groups by two maximal subgroups

the triple factorization of a group $g$ has been studied recently showing that $g=aba$ for some proper subgroups $a$ and $b$ of $g$, the definition of rank-two geometry and rank-two coset geometry which is closely related to the triple factorization was defined and calculated for abelian groups. in this paper we study two infinite classes of non-abelian finite groups $d_{2n}$ and $psl(2,2^{n})$ for their triple factorizations by finding certain suitable maximal subgroups, which these subgroups are define with original generators of these groups. the related rank-two coset geometries motivate us to define the rank-two coset geometry graphs which could be of intrinsic tool on the study of triple factorization of non-abelian groups.