Although almost all triangular polyhedra in the space are rigid, there are known examples of flexible ones, for instance Bricard's octahedra. The smallest known flexible polyhedron without self-intersections is the one constructed by Steffen. The only possible combinatorial structure for a smaller embedded flexible polyhedron is the one of a pentagonal bipyramid with a tetrahedron glued along one face. Since this operation does not change flexibility, we focus on flexible pentagonal bipyramids. In this talk, we show how Galois groups can be assigned to motions of flexible pentagonal bipyramids using the distance between the two non-equatorial vertices. Up to symmetry, we show that there are only two possible Galois groups and we construct flexible instances proving that both of them indeed exist.