  In this paper, we used a transitional Hamiltonian which has
$U(5)\leftrightarrow(6)$ transition to improve theoretical predictions for
energy spectra and quadrupole transition rates of $^{112}$Cd nucleus. To this
aim, the transitional Hamiltonian in the affine $SU(1,1)$ algebra has been
extended by adding the $O(6)$-Casimir operator and mixing Hamiltonian to
increase exactness in the description of 4$^+_2$ and 2$^+_3$ intruder levels of
this nucleus. We also considered the wave functions of both regular and
intruder states, as a combination in the $N$ and $N+2$ boson spaces. The
results confirm the advantages of using such mixing approaches and describing
the energy and transition rates with high accuracy
