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Objectives: To determine the exact number of equivalence classes of G-circuits of length q ≥ 2.Methods/Statistical Analysis: To classify G-orbits of Q( √ m)/Q containing G-circuits of length 6. Findings: The equivalence classes of G-circuits of length 6 is ten in number and determine the exact number of G-orbits and structure of G-orbits corresponding to each of ten equivalence classes of Gcircuits. Furthermore, we describe some generalized G-circuits of length 2t corresponding to each of these ten equivalence classes and the structure of theseG-circuitswith conditions on t.Applications/Improvements:Weemploy Symmetries of Icosahedral group to explore cyclically equivalence classes of Gcircuits and similar G-circuits of length 6 corresponding to each of these ten equivalence classes. This study helps us in classifying reduced numbers lying in PSL(2, Z)-orbits. These results are verified by some suitable example.

Symmetries of Icosahedral group and classification of G-circuits of length six