Is there any finite group $G$ of order $n=m^2$ satisfying the following conditions?

(a) There is no subgroup of order $m$;

(b) There exist subsets $A, B$ such that |A|=|B|=m and $G=AB$.

Considering http://math.stackexchange.com/questions/882859/groups-of-order-n2-that-have-no-subgroup-of-order-n?lq=1,

$n$ must be $\geq 24^2=576$ ($m=24, 28, 30, \cdots$). Also, if answer of this problem is positive, then we obtain a counterexample for the problem Factorization of a finite group by two subsets.

Thanks in advance for your feedback.