Genus two embedded minimal surfaces in ${\mathbb{S}}^3$ with bidihedral symmetry

The isometry group of the classical Lawson embedded minimal surface ${\xi }_{2,1}\subset {\mathbb{S}}^3$ of genus 2 is isomorphic to the product $S_3\times D_4$ of the permutation group of three elements and the dihedral group of order 8 (symmetries of a square). $S_3\times D_4$ has a subgroup of index 3 isomorphic to the bidihedral group $D_{4h}={\mathbb{Z}}_2\times D_4$, where $D_4$ is the dihedral group of order 8. We prove that ${\xi }_{2,1}$ is the unique closed embedded minimal surface of genus 2 in ${\mathbb{S}}^3$ whose isometry group contains $D_{4h}$.