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

The exact description of intruder states in $^{112}$Cd nucleus by using  a mixing formalism based on $SU(1,1)$ transitional Hamiltonian and  $O(6)$-Casimir operator