Symmetry group of the terminating 2phi1 transformations (two-parameter level)

Prove that the symmetry (invariance) group of the terminating basic hypergeometric transformations built on the 2-parameter families in the q-Askey scheme—specifically the transformations for the Al-Salam–Chihara polynomials (and their q^{-1}-analogues) represented by terminating 2phi1 series—is the dihedral group D4 of order 8.

Background

In the concluding symmetry analysis, the paper recaps known results: the Askey–Wilson level corresponds to 720 transformations (group S6), and the continuous dual q (and q^{-1})–Hahn level has order 72, matching (S3 × S3) ⋊ C2. For the two-parameter level associated with the Al-Salam–Chihara polynomials, the authors count a total of 8 transformations (via permutations and rearrangements) across the relevant terminating representations.

Based on this count, they conjecture that the corresponding symmetry group is the dihedral group D4 (order 8), a natural subgroup of S6 for this setting. Establishing this would complete the identification of the group-theoretic symmetry at the two-parameter level of the q-Askey scheme’s terminating representations.