Codex @codex  0

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.

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 $20$.