Virtual invariant pairs: $S: BQ^3_{2} \quad U = \pmatrix{1 & 2 & 3 \cr 1 & 3 & 2 \cr 1 & 3 & 2} \quad D = \pmatrix{1 & 3 & 2 \cr 1 & 3 & 2 \cr 1 & 3 & 2}$\hfil\break $T: BQ^3_{1} \quad U = \pmatrix{1 & 2 & 3 \cr 1 & 3 & 2 \cr 1 & 3 & 2} \quad D = \pmatrix{1 & 2 & 3 \cr 1 & 3 & 2 \cr 1 & 3 & 2}$\hfil\break $S: BQ^3_{1} \quad U = \pmatrix{1 & 2 & 3 \cr 1 & 3 & 2 \cr 1 & 3 & 2} \quad D = \pmatrix{1 & 2 & 3 \cr 1 & 3 & 2 \cr 1 & 3 & 2}$\hfil\break $T: BQ^3_{5} \quad U = \pmatrix{1 & 3 & 2 \cr 1 & 3 & 2 \cr 1 & 3 & 2} \quad D = \pmatrix{1 & 3 & 2 \cr 1 & 3 & 2 \cr 1 & 3 & 2}$\hfil\break $S: BQ^3_{5} \quad U = \pmatrix{1 & 3 & 2 \cr 1 & 3 & 2 \cr 1 & 3 & 2} \quad D = \pmatrix{1 & 3 & 2 \cr 1 & 3 & 2 \cr 1 & 3 & 2}$\hfil\break $T: BQ^3_{1} \quad U = \pmatrix{1 & 2 & 3 \cr 1 & 3 & 2 \cr 1 & 3 & 2} \quad D = \pmatrix{1 & 2 & 3 \cr 1 & 3 & 2 \cr 1 & 3 & 2}$\hfil\break $S: BQ^3_{1} \quad U = \pmatrix{1 & 2 & 3 \cr 1 & 3 & 2 \cr 1 & 3 & 2} \quad D = \pmatrix{1 & 2 & 3 \cr 1 & 3 & 2 \cr 1 & 3 & 2}$\hfil\break $T: Q^3_{1} \quad U = \pmatrix{1 & 2 & 3 \cr 1 & 2 & 3 \cr 1 & 2 & 3} \quad D = \pmatrix{1 & 2 & 3 \cr 1 & 2 & 3 \cr 1 & 2 & 3}$\hfil\break $S: Q^3_{1} \quad U = \pmatrix{1 & 2 & 3 \cr 1 & 2 & 3 \cr 1 & 2 & 3} \quad D = \pmatrix{1 & 2 & 3 \cr 1 & 2 & 3 \cr 1 & 2 & 3}$\hfil\break $T: BQ^3_{1} \quad U = \pmatrix{1 & 2 & 3 \cr 1 & 3 & 2 \cr 1 & 3 & 2} \quad D = \pmatrix{1 & 2 & 3 \cr 1 & 3 & 2 \cr 1 & 3 & 2}$\hfil\break $S: Q^3_{2} \quad U = \pmatrix{1 & 3 & 2 \cr 1 & 2 & 3 \cr 1 & 2 & 3} \quad D = \pmatrix{1 & 2 & 3 \cr 1 & 2 & 3 \cr 1 & 2 & 3}$\hfil\break $T: BQ^3_{1} \quad U = \pmatrix{1 & 2 & 3 \cr 1 & 3 & 2 \cr 1 & 3 & 2} \quad D = \pmatrix{1 & 2 & 3 \cr 1 & 3 & 2 \cr 1 & 3 & 2}$\hfil\break $S: BQ^3_{2} \quad U = \pmatrix{1 & 2 & 3 \cr 1 & 3 & 2 \cr 1 & 3 & 2} \quad D = \pmatrix{1 & 3 & 2 \cr 1 & 3 & 2 \cr 1 & 3 & 2}$\hfil\break $T: BQ^3_{5} \quad U = \pmatrix{1 & 3 & 2 \cr 1 & 3 & 2 \cr 1 & 3 & 2} \quad D = \pmatrix{1 & 3 & 2 \cr 1 & 3 & 2 \cr 1 & 3 & 2}$\hfil\break $S: BQ^3_{2} \quad U = \pmatrix{1 & 2 & 3 \cr 1 & 3 & 2 \cr 1 & 3 & 2} \quad D = \pmatrix{1 & 3 & 2 \cr 1 & 3 & 2 \cr 1 & 3 & 2}$\hfil\break $T: Q^3_{1} \quad U = \pmatrix{1 & 2 & 3 \cr 1 & 2 & 3 \cr 1 & 2 & 3} \quad D = \pmatrix{1 & 2 & 3 \cr 1 & 2 & 3 \cr 1 & 2 & 3}$\hfil\break $S: BQ^3_{3} \quad U = \pmatrix{1 & 2 & 3 \cr 3 & 1 & 2 \cr 2 & 3 & 1} \quad D = \pmatrix{1 & 3 & 2 \cr 1 & 3 & 2 \cr 1 & 3 & 2}$\hfil\break $T: Q^3_{1} \quad U = \pmatrix{1 & 2 & 3 \cr 1 & 2 & 3 \cr 1 & 2 & 3} \quad D = \pmatrix{1 & 2 & 3 \cr 1 & 2 & 3 \cr 1 & 2 & 3}$\hfil\break $S: BQ^3_{4} \quad U = \pmatrix{1 & 2 & 3 \cr 1 & 2 & 3 \cr 2 & 1 & 3} \quad D = \pmatrix{1 & 2 & 3 \cr 1 & 2 & 3 \cr 2 & 1 & 3}$\hfil\break $T: Q^3_{1} \quad U = \pmatrix{1 & 2 & 3 \cr 1 & 2 & 3 \cr 1 & 2 & 3} \quad D = \pmatrix{1 & 2 & 3 \cr 1 & 2 & 3 \cr 1 & 2 & 3}$\hfil\break $S: Q^3_{1} \quad U = \pmatrix{1 & 2 & 3 \cr 1 & 2 & 3 \cr 1 & 2 & 3} \quad D = \pmatrix{1 & 2 & 3 \cr 1 & 2 & 3 \cr 1 & 2 & 3}$\hfil\break $T: BQ^3_{4} \quad U = \pmatrix{1 & 2 & 3 \cr 1 & 2 & 3 \cr 2 & 1 & 3} \quad D = \pmatrix{1 & 2 & 3 \cr 1 & 2 & 3 \cr 2 & 1 & 3}$\hfil\break $S: BQ^3_{5} \quad U = \pmatrix{1 & 3 & 2 \cr 1 & 3 & 2 \cr 1 & 3 & 2} \quad D = \pmatrix{1 & 3 & 2 \cr 1 & 3 & 2 \cr 1 & 3 & 2}$\hfil\break $T: Q^3_{1} \quad U = \pmatrix{1 & 2 & 3 \cr 1 & 2 & 3 \cr 1 & 2 & 3} \quad D = \pmatrix{1 & 2 & 3 \cr 1 & 2 & 3 \cr 1 & 2 & 3}$\hfil\break $S: Q^3_{1} \quad U = \pmatrix{1 & 2 & 3 \cr 1 & 2 & 3 \cr 1 & 2 & 3} \quad D = \pmatrix{1 & 2 & 3 \cr 1 & 2 & 3 \cr 1 & 2 & 3}$\hfil\break $T: BQ^3_{5} \quad U = \pmatrix{1 & 3 & 2 \cr 1 & 3 & 2 \cr 1 & 3 & 2} \quad D = \pmatrix{1 & 3 & 2 \cr 1 & 3 & 2 \cr 1 & 3 & 2}$\hfil\break $S: Q^3_{2} \quad U = \pmatrix{1 & 3 & 2 \cr 1 & 2 & 3 \cr 1 & 2 & 3} \quad D = \pmatrix{1 & 2 & 3 \cr 1 & 2 & 3 \cr 1 & 2 & 3}$\hfil\break $T: BQ^3_{5} \quad U = \pmatrix{1 & 3 & 2 \cr 1 & 3 & 2 \cr 1 & 3 & 2} \quad D = \pmatrix{1 & 3 & 2 \cr 1 & 3 & 2 \cr 1 & 3 & 2}$\hfil\break $S: Q^3_{3} \quad U = \pmatrix{1 & 3 & 2 \cr 3 & 2 & 1 \cr 2 & 1 & 3} \quad D = \pmatrix{1 & 2 & 3 \cr 1 & 2 & 3 \cr 1 & 2 & 3}$\hfil\break $T: BQ^3_{5} \quad U = \pmatrix{1 & 3 & 2 \cr 1 & 3 & 2 \cr 1 & 3 & 2} \quad D = \pmatrix{1 & 3 & 2 \cr 1 & 3 & 2 \cr 1 & 3 & 2}$\hfil\break $S: BQ^3_{7} \quad U = \pmatrix{2 & 3 & 1 \cr 2 & 3 & 1 \cr 2 & 3 & 1} \quad D = \pmatrix{3 & 1 & 2 \cr 3 & 1 & 2 \cr 3 & 1 & 2}$\hfil\break $T: Q^3_{1} \quad U = \pmatrix{1 & 2 & 3 \cr 1 & 2 & 3 \cr 1 & 2 & 3} \quad D = \pmatrix{1 & 2 & 3 \cr 1 & 2 & 3 \cr 1 & 2 & 3}$\hfil\break $S: Q^3_{1} \quad U = \pmatrix{1 & 2 & 3 \cr 1 & 2 & 3 \cr 1 & 2 & 3} \quad D = \pmatrix{1 & 2 & 3 \cr 1 & 2 & 3 \cr 1 & 2 & 3}$\hfil\break $T: BQ^3_{7} \quad U = \pmatrix{2 & 3 & 1 \cr 2 & 3 & 1 \cr 2 & 3 & 1} \quad D = \pmatrix{3 & 1 & 2 \cr 3 & 1 & 2 \cr 3 & 1 & 2}$\hfil\break $S: Q^3_{2} \quad U = \pmatrix{1 & 3 & 2 \cr 1 & 2 & 3 \cr 1 & 2 & 3} \quad D = \pmatrix{1 & 2 & 3 \cr 1 & 2 & 3 \cr 1 & 2 & 3}$\hfil\break $T: Q^3_{1} \quad U = \pmatrix{1 & 2 & 3 \cr 1 & 2 & 3 \cr 1 & 2 & 3} \quad D = \pmatrix{1 & 2 & 3 \cr 1 & 2 & 3 \cr 1 & 2 & 3}$\hfil\break