If we don't know the property that "A Group whose order is the square of a prime is abelian group" Then also we can fill the entire table based on the given info.
Here , group is given . So, It will satisfy closure, associativity,identity, inverse properties of a group.
So, Here it is given ,
$1) \;e*e = e$ ,$e*a = a ,\; e*b=b\; , e*c=c$ [Left Identity property]
So,
$2)\;e*e\;,a*e = a ,\; b*e=b\; , c*e=c$ [Right Identity property]
$3)\; a*e = a\; , \; a*a=b,\; a*b=c,\;a*c=e$
Now, based on the given info,
$1)\; b*a = (a*a)*a = a*(a*a) = a*b = c$
$2)\; b*b = (a*a)*b =a*(a*b) = a*c = e$
$3)\; b*c = (a*a)*c = a*(a*c) = a*e = a$
$4)\; c*a= (a*b)*a = (a*(a*a))*a = (a*a)*(a*a) = b*b = e$
$5) \;c*b = (a*b)*b = (b*a)*b =b*(a*b) = b*c = a$
$6)\;c*c=(a*b)*(a*b)=a*(b*a)*b =a*(a*b)*b = (a*a)*(b*b) =b*e=b$