the data you like to know is encoded in the character value; once the character is known, the eigenvalues of each group element can be found, see e.g. http://math.stackexchange.com/questions/427529/computing-eigenvalues-from-characters
GAP can compute these using the function EigenvaluesChar, see Sect 72.8-19 of its manual: http://gap-system.org/Manuals/doc/ref/chap72.html