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Reduced numbers play an important role in the study of modular group action on the PSL(2,Z)-subset of Q(  m √ )/Q. For this purpose, we define new notions of equivalent, cyclically equivalent, and similar G-circuits in PSL(2,Z)-orbits of real quadratic fields. In particular, we classify PSL(2,Z)-orbits of Q(  m √ )/Q  ∪ k∈NQ(  k2m √ ) containing G-circuits of length 6 and determine that the number of equivalence classes of G-circuits of length 6 is ten. We also employ the icosahedral group to explore cyclically equivalence classes of circuits and similar G-circuits of length 6 corresponding to each of these ten circuits. +is study also helps us in classifying reduced numbers lying in the PSL(2,Z)-orbits.

Icosahedral Group and Classification of PSL(2, Z)-Orbits of Real Quadratic Fields