It is proved that the direct product of the J-trivial monoid S(xyx) with any noncommutative group of finite exponent is non-finitely based. This result provides new, simpler examples of two finitely based finite monoids for which their direct product is non-finitely based. It follows that the direct product of the monoid S(xyx) with any group of finite exponent is finitely based if and only if the group is commutative.