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 
https://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 https://mathoverflow.net/questions/177747/factorization-of-a-finite-group-by-two-subsets.

Thanks in advance for your feedback.