  The question of characterizing the (finite) representable relation algebras
in a ``nice" way is open. The class $\mathbf{RRA}$ is known to be not finitely
axiomatizable in first-order logic. Nevertheless, it is conjectured that
``almost all'' finite relation algebras are representable.
  All finite relation algebras with three or fewer atoms are representable. So
one may ask, Over what cardinalities of sets are they representable? This
question was answered completely by Andr\'eka and Maddux (``Representations for
small relation algebras,'' \emph{Notre Dame J. Form. Log.}, \textbf{35}
(1994)); they determine the spectrum of every finite relation algebra with
three or fewer atoms.
  In the present paper, we restrict attention to cyclic group representations,
and completely determine the cyclic group spectrum for all seven symmetric
integral relation algebras on three atoms. We find that in some instances, the
spectrum and cyclic spectrum agree; in other instances, the spectra disagree
for finitely many $n$; finally, for other instances, the spectra disagree for
infinitely many $n$. The proofs employ constructions, SAT solvers, and the
probabilistic method.
