A group under MAcountable whose square is countably compact but whose cube is not

We show under MAcountable the existence of a countable subgroup E of 2c such that the group H generated by E and G = {x ϵ 2c}: supp x is bounded in c is a group as in the title. We also show under MAcountable that for each k ϵ N there exists a countable family {{En: n ϵ N}} of countable subgroups of 2c such that if Hn = En + G, then for each subset F of N of size k, ΠnϵFHn is incountable compact, while for each subset F of N of size k + 1, ΠnϵFHn is not countably compact.