Here is a paragraph with some statements about the Gauge Symmetry Breaking from Georgi's book Lie Algebras in Particle Physics 2nd ed -- From Isospin to Unified Theories (Georgi, 1999) p.285.
His claim is too quick. Can some experts explain which symmetry breaking pattern he is thinking of? In particular, he uses three Higgs fields in three representations:
24 should be the adjoint representation of SU(5).
5 should be the fundamental representation of SU(5).
45 should be EITHER the adjoint representation of SO(10) or some (what kind?) representation of SU(5).
question: How do we use representation 24, 5, and 45 to break from $SO(10)$ to $SU(5)$ to $SU(3) \times SU(2) \times U(1)$?