= Suzuki group of Lie type
{c}
{title2=$Sz(q)={}^{2}B_2(q),\quad q=2^{2a+1}\ge8$}
For $q=2^{2a+1}\geq8$, the Suzuki group $Sz(q)={}^2B_2(q)$ is a twisted <finite group of Lie type>. In <characteristic two>, the algebraic <symplectic group over a field> $Sp_4$ 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.<On the simple groups of Suzuki and Ree by Robert A. Wilson>[Introduction and sections 3-5, the four-dimensional construction]{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 $20$.
Back to article page
