The Ree groups ${}^{2}F_4\left(q\right)$ , for an odd power $q$ of $2$, are a series of (typically) simple groups of Lie type, first constructed by R. Ree. They are defined only for $q=2^{2e+1}$ an odd power of $2$ (where, here, $0\le e$).

The Ree2F4( q ) command constructs a symbolic group representing the large Ree group ${}^{2}F_4\left(q\right)$ .

The large Ree groups ${}^{2}F_4\left(q\right)$ are simple for all odd powers $q$ of $2$ greater than $2$, but ${}^{2}F_4\left(2\right)$ is not simple. Its derived subgroup is the simple Tits group.

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

$G≔\mathrm{Ree2F4}\left(2\right)$

$G≔{}^2\mathit{F}_4\left(2\right)$

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

$35942400$

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

$\mathrm{false}$

$\mathrm{IsSimple}\left(\mathrm{Ree2F4}\left(8\right)\right)$

$\mathrm{true}$

$\mathrm{MinPermRepDegree}\left(\mathrm{Ree2F4}\left(8\right)\right)$

$1210323465$

$\mathrm{IsSimple}\left(\mathrm{Ree2F4}\left(q\right)\right)$

$\left\{\begin{array}{cc}\mathrm{false}& q=2\\ \mathrm{true}& \mathrm{otherwise}\end{array}\right.$

$\mathrm{ClassNumber}\left(\mathrm{Ree2F4}\left(q\right)\right)$

$q^4+4q^2+17$