The Steinberg group ${}^{2}E_6\left(q\right)$ , for a prime power $q$, is a simple group of Lie type.

The Steinberg2E6( q ) command returns a symbolic group representing the Steinberg group ${}^{2}E_6\left(q\right)$ .

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$G≔\mathrm{Steinberg2E6}\left(2\right)$

$G≔{}^{2}E_6\left(2\right)$

$\mathrm{type}\left(G\,'\mathrm{Group}'\right)$

$\mathrm{true}$

$\mathrm{type}\left(G\,'\mathrm{PermutationGroup}'\right)$

$\mathrm{false}$

$\mathrm{GroupOrder}\left(G\right)$

$76532479683774853939200$

$\mathrm{MinPermRepDegree}\left(G\right)$

$3968055$

$\mathrm{IsSimple}\left(G\right)$

$\mathrm{true}$

$\mathrm{IsSimple}\left(\mathrm{Steinberg2E6}\left(4096\right)\right)$

$\mathrm{true}$

$\mathrm{GroupOrder}\left(\mathrm{Steinberg2E6}\left(27\right)\right)$

$4423616655215750021498369285918192558448382923930662108203473523560591980763988390542885164349041907752671641600$

$\mathrm{ClassNumber}\left(\mathrm{Steinberg2E6}\left(32\right)\right)$

$1109537246$

$G≔\mathrm{Steinberg2E6}\left(q\right)$

$G≔{}^{2}E_6\left(q\right)$

$\mathrm{ClassNumber}\left(G\right)$

$q^6+q^5+2q^4+4q^3+\left(\left\{\begin{array}{cc}11q^2+11q+16& \mathrm{irem}\left(q\,6\right)=1\\ 12q^2+14q+30& \mathrm{irem}\left(q\,6\right)=2\\ 11q^2+11q+15& \mathrm{irem}\left(q\,6\right)=3\\ 10q^2+10q+14& \mathrm{irem}\left(q\,6\right)=4\\ 13q^2+15q+34& \mathrm{irem}\left(q\,6\right)=5\end{array}\right.\right)$

$\mathrm{IsSimple}\left(G\right)$

$\mathrm{true}$