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For $q=3^{2a+1}\ge 27$, the small Ree group ${}^2{G}_2(q)$ is a twisted finite group of Lie type. In characteristic three, the algebraic group of type G2 has an exceptional endomorphism $\sigma$ with ${\sigma }^2=F_3$, the Frobenius endomorphism of an algebraic group. The fixed points of ${\sigma }^{2a+1}$ form ${}^2{G}_2(q)$. This explains both the twist and the odd exponent restriction.^[ref]

It is simple, has order $q^3(q^3+1)(q-1)$, and acts two-transitively on $q^3+1$ points. At $q=27$ the degree is $19684$ and the order is $10073444472$. The excluded parameter $q=3$ gives a nonsimple group. The characteristic-two family ${}^2{F}_4(q)$ is a different Ree family.

Robert A. Wilson works on finite groups, their group representations and computational constructions, especially sporadic simple groups. His research homepage documents this specialization and work on the Monster group.^[ref]

His work on Suzuki group of Lie type and Ree group of type G2 constructs these exceptional families through automorphisms and analyzes their actions and simplicity. His lectures on sporadic groups also connect Mathieu groups with the symmetries of the extended binary Golay code, illustrating how error-correcting structures reveal exceptional finite symmetry.^[ref][ref]

For $q=2^{2a+1}\ge 8$, the Suzuki group $Sz(q)={}^2{B}_2(q)$ is a twisted finite group of Lie type. In characteristic two, the algebraic symplectic group over a field $S{p}_4$ has an exceptional endomorphism $\sigma$ satisfying ${\sigma }^2=F_2$. Fixed points of ${\sigma }^{2a+1}$ give $Sz(q)$; its natural matrix representation has degree four.^[ref]

It is simple, of order $q^2(q^2+1)(q-1)$, and acts two-transitively on the $q^2+1$ points of a Suzuki ovoid. A point stabilizer has order $q^2(q-1)$ and contains a subgroup of order $q^2$ acting regularly on the remaining points. At $q=8$ the degree is $65$ and the order is $29120$. The parameter $q=2$ instead gives a nonsimple group of order $20$.

A Suzuki ovoid is a special set of $q^2+1$ points in the three-dimensional projective space over ${\mathbb{F}}_q$, for $q=2^{2a+1}$. It is an ovoid: no three of its points are collinear. Its projective symmetries include the Suzuki group of Lie type $Sz(q)$, acting two-transitively on these points.

For $q\ge 8$, fixing one point leaves a stabilizer of order $q^2(q-1)$, with a subgroup of order $q^2$ acting regularly on the other $q^2$ points. Thus the geometry realizes the natural permutation action of the group, rather than only its four-dimensional matrix action. For $q=8$, there are $65$ ovoid points and $|Sz(8)|=29120$.

The absence of collinear triples means that choosing two ovoid points determines a secant containing exactly those two ovoid points. There are therefore $\left(\genfrac{}{}{0ex}{}{q^2+1}{2}\right)$ such secants. This turns part of the geometry into an explicit pair-counting problem.^[ref]