  By means of the LSDA+U method and the Green function method, we investigate
the electronic and magnetic properties of the new material of
Sr$_8$CaRe$_3$Cu$_4$O$_{24}$. Our LSDA+U calculation shows that this system is
an insulator with a net magnetic moment of 1.01 $\mu_{\rm B}$/f.u., which is in
good agreement with the experiment. Magnetic moments are mainly located at Cu
atoms, and the magnetic moments of neighboring Cu sites align anti-parallel. It
is the non-magnetic Re atoms that induce an orbital order of $d$ electrons of
Cu atoms, which is responsible for the strong exchange interaction and the high
magnetic transition temperature. Based on the LSDA+U results, we introduce an
effective model for the spin degrees of freedom, and investigate the
finite-temperature properties by the Green function method. The obtained
results are consistent with the experimental results, indicating that the
spin-alternating Heisenberg model is suitable for this compound.
