A joint experimental-computational study of the molecular structure and vibrational spectra of the XeF₆ molecule is reported. The vibrational frequencies, intensities, and in particular the isotopic frequency shifts of the vibrational spectra for ¹²⁹XeF₆ and ¹³⁶XeF₆ isotopologues recorded in the neon matrix agree very well with those obtained from relativistic coupled-cluster calculations for XeF₆ in the C₃ᵥ structure, thereby strongly supporting the observation of the C₃ᵥ conformer of the XeF₆ molecule in the neon matrix. A C₃ᵥ transition state connecting the C₃ᵥ and Oₕ local minima is located computationally. The calculated barrier of 220 cm⁻¹ between the C₃ᵥ minima and the transition state corroborates the experimental observation of the C₃ᵥ conformer and the absence of the Oₕ conformer in solid noble gas matrices. For comparison matrix-isolation spectra have also been recorded and analyzed for the ¹²⁹XeOF₄ and the ¹³⁶XeOF₄ isotopologues. The matrix-isolation complexation shifts obtained for the XeF₆·NCCH₃ relative to those of free matrix isolated XeF₆ and CH₃CN are in good agreement with those reported for crystalline XeF₆·NCCH₃.