The Chevalley group $G_2\left(q\right)$ , for a prime power $q$, is a generically simple group of Lie type. The groups $G_2\left(q\right)$ were studied by Dickson in 1905.

The ChevalleyG2( q ) command returns a permutation group isomorphic to the Chevalley group $G_2\left(q\right)$ , for prime powers $q\le 13$. For non-numeric values of the argument q, or for prime powers $q$ larger than $13$, a symbolic group representing the group $G_2\left(q\right)$ is returned.

Note that the group $G_2\left(2\right)$ is not simple, but its derived subgroup is simple (isomorphic to the simple unitary group $PSU\left(3\,3\right)$ .

For values of $q$ for which $G_2\left(q\right)$ is available as a permutation group, the generating permutations have orders $2$ and $3$ in each case.

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

$G≔\mathrm{ChevalleyG2}\left(2\right)$

$G≔⟨\left(1\,2\right)\left(3\,5\right)\left(4\,7\right)\left(6\,10\right)\left(8\,12\right)\left(9\,13\right)\left(11\,16\right)\left(14\,20\right)\left(17\,23\right)\left(19\,25\right)\left(21\,28\right)\left(22\,29\right)\left(26\,30\right)\left(27\,31\right)\left(32\,33\right)\left(34\,36\right)\left(35\,37\right)\left(38\,40\right)\left(39\,42\right)\left(41\,45\right)\left(43\,47\right)\left(44\,48\right)\left(46\,50\right)\left(52\,55\right)\left(53\,56\right)\left(57\,59\right)\left(60\,62\right)\left(61\,63\right)\,\left(1\,3\,6\right)\left(2\,4\,8\right)\left(5\,9\,14\right)\left(7\,11\,17\right)\left(10\,15\,21\right)\left(12\,18\,24\right)\left(13\,19\,26\right)\left(16\,22\,25\right)\left(20\,27\,32\right)\left(28\,33\,35\right)\left(29\,34\,30\right)\left(36\,38\,41\right)\left(37\,39\,43\right)\left(40\,44\,49\right)\left(42\,46\,51\right)\left(45\,50\,54\right)\left(47\,52\,56\right)\left(48\,53\,57\right)\left(55\,58\,60\right)\left(59\,61\,62\right)⟩$

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

$12096$

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

$\mathrm{false}$

$\mathrm{cs}≔\mathrm{CompositionSeries}\left(G\right)$

$\mathrm{cs}≔⟨\left(1\,2\right)\left(3\,5\right)\left(4\,7\right)\left(6\,10\right)\left(8\,12\right)\left(9\,13\right)\left(11\,16\right)\left(14\,20\right)\left(17\,23\right)\left(19\,25\right)\left(21\,28\right)\left(22\,29\right)\left(26\,30\right)\left(27\,31\right)\left(32\,33\right)\left(34\,36\right)\left(35\,37\right)\left(38\,40\right)\left(39\,42\right)\left(41\,45\right)\left(43\,47\right)\left(44\,48\right)\left(46\,50\right)\left(52\,55\right)\left(53\,56\right)\left(57\,59\right)\left(60\,62\right)\left(61\,63\right)\,\left(1\,3\,6\right)\left(2\,4\,8\right)\left(5\,9\,14\right)\left(7\,11\,17\right)\left(10\,15\,21\right)\left(12\,18\,24\right)\left(13\,19\,26\right)\left(16\,22\,25\right)\left(20\,27\,32\right)\left(28\,33\,35\right)\left(29\,34\,30\right)\left(36\,38\,41\right)\left(37\,39\,43\right)\left(40\,44\,49\right)\left(42\,46\,51\right)\left(45\,50\,54\right)\left(47\,52\,56\right)\left(48\,53\,57\right)\left(55\,58\,60\right)\left(59\,61\,62\right)⟩▹⟨\left(1\,18\,2\,15\right)\left(3\,27\,32\,10\,9\,29\,17\,7\right)\left(4\,23\,22\,13\,6\,33\,31\,5\right)\left(8\,12\,11\,26\,38\,40\,30\,16\right)\left(14\,35\,52\,59\,47\,46\,36\,25\right)\left(19\,34\,50\,43\,57\,55\,37\,20\right)\left(21\,39\,42\,28\right)\left(41\,45\,44\,61\,62\,60\,63\,48\right)\left(49\,53\,58\,56\right)\left(51\,54\right)\,\left(1\,35\,31\,9\,8\,23\,25\,19\right)\left(2\,18\,17\,13\,41\,44\,29\,22\right)\left(3\,24\,4\,21\right)\left(5\,28\,56\,61\,52\,51\,38\,16\right)\left(6\,32\,20\,15\,14\,34\,7\,11\right)\left(10\,43\,46\,33\right)\left(26\,30\,54\,37\,48\,58\,39\,27\right)\left(36\,50\,49\,62\,59\,55\,63\,53\right)\left(40\,57\,60\,47\right)\left(42\,45\right)⟩▹⟨⟩$

$\mathrm{seq}\left(\mathrm{IsSimple}\left(H\right)\,H=\mathrm{cs}\right)$

$\mathrm{false},\mathrm{true},\mathrm{false}$

$\mathrm{ClassifyFiniteSimpleGroup}\left(\mathrm{cs}\left[2\right]\right)$

$⟨\mathbf{\text{CFSG:}}\text{Steinberg Group}{}^{2}A_2\left(3\right)\=\mathrm{PSU}\left(3\,3\right)⟩$

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

$\mathrm{true}$

$G≔\mathrm{ChevalleyG2}\left(7\right)\:$

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

$664376138496$

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

$\mathrm{true}$

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

$72$

$G≔\mathrm{ChevalleyG2}\left(13\right)\:$

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

$3914077489672896$

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

$\mathrm{true}$

$G≔\mathrm{ChevalleyG2}\left(q\right)$

$G≔{\mathit{G}}_2\left(q\right)$

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

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

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

$q^6\left(q^6-1\right)\left(q^2-1\right)$

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

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

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

$\mathrm{false}$