  We investigate a detail of a dodecamer cluster ordering in a double-exchange
spin ice model on a kagom\'e lattice. In frustrated systems, ordinary spin
orderings are suppressed and macroscopic degeneracy remains down to low
temperatures. In some frustrated systems, the degeneracy is lifted due to
residual interactions and cluster orderings are stabilized. In the present
model, the spin ice state is first formed at intermediate temperatures, and
further entropies are released at lower temperatures as the dodecamer phase
emerges. Since the spin symmetry is not broken in the dodecamer phase, there
still exists macroscopic degeneracy. At further low temperatures, a possible
spin ordering due to inter-dodecamer interactions is proposed. We discuss that
such a multiple-site clustering larger than a bond-pair might be generic to
frustrated systems where macroscopic degeneracy is lifted by residual
interactions.
