The Ree groups ${}^{2}G_2\left(q\right)$ , for an odd power $q$ of $3$, are a series of (typically) simple groups of Lie type, first constructed by R. Ree. They are defined only for $q=3^{2e+1}$ an odd power of $3$ (where, here, $0\le e$).

The Ree2G2( q ) command constructs a permutation group isomorphic to ${}^{2}G_2\left(q\right)$ , for q equal to either $3$ or $27$.

If the argument q is not numeric, or if it is an odd power of $3$ greater than $27$, then a symbolic group representing ${}^{2}G_2\left(q\right)$ is returned.

The Ree group ${}^{2}G_2\left(3\right)$ is not simple, but mRee( q ) is simple for admissible values of $3<q$. The derived subgroup of ${}^{2}G_2\left(3\right)$ is simple, isomorphic to the group $PSL\left(2\&comma;8\right)$ .

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\&colon;$

$G≔\mathrm{Ree2G2}\left(3\right)$

$G≔{}^2\mathit{G}_2\left(3\right)$

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

$1512$

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

$\mathrm{false}$

$L≔\mathrm{DerivedSubgroup}\left(G\right)$

$L≔\left[{}^2\mathit{G}_2\left(3\right)\&comma;{}^2\mathit{G}_2\left(3\right)\right]$

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

$\mathrm{true}$

$\mathrm{ClassifyFiniteSimpleGroup}\left(L\right)$

$⟨\mathbf{\text{CFSG:}}\text{Chevalley Group}{A}_1\left(8\right)\&equals;\mathrm{PSL}\left(2\&comma;8\right)⟩$

$G≔\mathrm{Ree2G2}\left(27\right)$

$G≔{}^2\mathit{G}_2\left(27\right)$

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

$\mathrm{true}$

$\mathbf{use}\mathrm{GraphTheory}\mathbf{in}\mathrm{DrawGraph}\left(\mathrm{GruenbergKegelGraph}\left(G\right)\right)\mathbf{end\; use}$

$G≔\mathrm{Ree2G2}\left(243\right)$

$G≔{}^2\mathit{G}_2\left(243\right)$

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

Error, (in GroupTheory:-Generators) cannot compute the generators of a symbolic group

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

$49825657439340552$

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

$\mathrm{true}$

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

$14348908$

$\mathrm{IsSimple}\left(\mathrm{Ree2G2}\left(q\right)\right)$

$\left\{\begin{array}{cc}\mathrm{false}& q=3\\ \mathrm{true}& \mathrm{otherwise}\end{array}\right.$

$\mathrm{ClassNumber}\left(\mathrm{Ree2G2}\left(q\right)\right)$

$q+8$

The GroupTheory[Ree2G2] command was introduced in Maple 2021.

For more information on Maple 2021 changes, see Updates in Maple 2021.