Fully idempotent and coidempotent modules

‎We introduce a two generated weakly branch contracting automaton‎ ‎group $G$ which is generated by a two state automaton on a three‎ ‎letter alphabet‎. ‎Using its branch structure and the finiteness‎ ‎nature of a sequence of its factor groups we compute the order of‎ ‎some of these factors‎. ‎Furthermore some algebraic properties of $G$‎ ‎are detected‎.