  A Beauville surface is a rigid complex surface of general type, isogenous to
a higher product by the free action of a finite group, called a Beauville
group. Here we consider which characteristically simple groups can be Beauville
groups. We show that if G is a cartesian power of a simple group L_2(q),
L_3(q), U_3(q), Sz(2^e), R(3^e), or of a sporadic simple group, then G is a
Beauville group if and only if it has two generators and is not isomorphic to
A_5.
