Label the vertices of the octahedron from to in "order". What does it it mean in order? It mean we have all along the same square with in opposite sides - not sure if that makes sense. Now we need to replace the octahedron in such a way so that "order" is preserved. There are six ways to place the vertex and then four possible places to choose for . After that is doen the "order" must be contained so along that square and are determined. This leaves us with for which there are two locations on opposite sides. Multiply out the numbers and get that the symmetry group has order 48. You can find it here