Let x be a member of C. By the orbit equation, |C| = [G: G_x] where G_x = {g in G | gxg^-1 = x} (i.e., G_x is the "stabilizer" or "isotropy subgroup" of x). So |G|/|G_x| = |C| = |G|/3. Thus, |G_x| = 3.

Since x is in G_x, we have x^|G_x| = x³ = e. That is, x has order 3. (Note that if x = e, then C = {1} and hence |G| = 3. This is ruled out by hypothesis. Also, if x² = e, then 2 divides 3; a contradiction.)