$ n $ product groups-periodicals

We call a group (infinite) $ n $periodic product for an integer $ n geq $ 3 if $ prod_ {i = 1} ^ n G cong G $ but for all integers k $ with $ 2 leq k leq n-1 $ we have $ prod_ {i = 1} ^ k G not free G $.

Is there an integer $ n geq $ 3 such as there is a $ n $-product-periodic group, and is there an integer $ m geq $ 3 such as there is no $ m $product-periodic group?