three classes of (monomial type) hyperovals, the hyperconics (f(t) = t2), proper translation hyperovals (f(t) = t4) and the Segre hyperovals (f(t) = t6). There are also classes corresponding to the Payne hyperovals and the Cherowitzo hyperovals (for more details see . In the collineation groups stabilizing each of these hyperovals have been determined. Note that in the original determination of the collineation group for the Payne hyperovals the case of q = 32 had to be treated separately and relied heavily on computer results. In an alternative version of the proof is given which does not depend on computer computations. In 1991, O'Keefe and Penttila discovered a new hyperoval in this plane by means of a detailed investigation of the divisibility properties of the orders of automorphism groups of hypothetical hyperovals . One of its o-polynomials is given by: f(x) = x4 + x16 + x28 + η11(x6 + x10 + x14 + x18 + x22 + x26) + η20(x8 + x20) + η6(x12 + x24), where η is a primitive root of GF(32) satisfying η5 = η2 + 1. The full automorphism group of this hyperoval has order 3. cleverly structured an exhaustive computer search for all hyperovals in this plane. The result was that the above listing is complete, there are just six classes of hyperovals in PG(2,32). PG(2,64) By extending the ideas in to PG(2,64), were able to search for hyperovals whose automorphism group admitted a collineation of order 5. They found two and showed that no other hyperoval exists in this plane that has such an automorphism. This settled affirmatively a long open question of B. Segre who wanted to know if there were any hyperovals in this plane besides the hyperconics. The hyperovals are: f(x) = x8 + x12 + x20 + x22 + x42 + x52 + η21(x4+x10+x14+x16+x30+x38+x44+x48+x54+x56+x58+x60+x62) + η42(x2 + x6 + x26 + x28 + x32 + x36 + x40), which has an automorphism group of order 15, and f(x) = x24 + x30 + x62 + η21(x4 +x8+x10+x14+x16+x34+x38 +x40 +x44+x46+x52+x54+x58+x60) + η42(x6+ x12+ x18+ x20+ x26+ x32 + x36+ x42+ x48+x50), which has an automorphism group of order 60, where η is a primitive element of GF(64) satisfying η6 = η + 1. In it is shown that these are Subiaco hyperovals. By refining the computer search program, extended the search to hyperovals admitting an automorphism of order 3, and