Space Groups and Semidirect Products

In Application 3.15 we considered the “full” cubic group G* of order 48, contrasted with the group G of the rigid cube in Application 3.14, which has order 24. Mathematical treatments usually regard the cube as solid and hence use the smaller group of symmetries; however, since a cubic group arising in crystallography allows the inversion center (or, equivalently, reflection in a plane parallel to two opposite faces of the cube), it will be helpful to see how the character table for the larger group can be formed from the table for the smaller one. Our first project in this chapter will be to prove a theorem that determines the connection between the two tables. The proof will be organized by means of the following three lemmas.