ExceptionalGroup - Maple Help

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GroupTheory

 ExceptionalGroup

 Calling Sequence ExceptionalGroup( name )

Parameters

 name - : string : an exceptional group name in {"G2(2)", "G2(3)", "G2(4)", "G2(5)", "R(3)", "R(27)", "Sz(8)", "Sz(32)", "3D4(2)", "3D4(3)", "F4(2)"}

Description

 • Twisted or exceptional groups of Lie type are a class of finite simple groups. These are the Chevalley groups ${G}_{2}\left(q\right)$, Ree groups $R\left(q\right)$, Suzuki groups $Sz\left(q\right)$, and Steinberg-Tits triality Groups ${}^{3}D_{4}\left(q\right)$ where q is a power of a prime.
 • Note that the group ${G}_{2}\left(2\right)$is not simple, but its derived subgroup is simple (isomorphic to the simple unitary group $PSU\left(3,3\right)$).
 • The ExceptionalGroup command returns a permutation group isomorphic to the exceptional group whose name is passed as argument.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{ExceptionalGroup}\left("Sz\left(8\right)"\right)$
 ${G}{≔}{Sz}\left({8}\right)$ (1)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${29120}$ (2)
 > $G≔\mathrm{ExceptionalGroup}\left("3D4\left(2\right)"\right)$
 ${G}{≔}{}^{{3}}{D}_{{4}}\left({2}\right)$ (3)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${211341312}$ (4)
 > $G≔\mathrm{ExceptionalGroup}\left("G2\left(2\right)"\right)$
 ${G}{≔}{{G}}_{{2}}\left({2}\right)$ (5)
 > $\mathrm{IsSimple}\left(G\right)$
 ${\mathrm{false}}$ (6)
 > $L≔\mathrm{DerivedSubgroup}\left(G\right)$
 ${L}{≔}\left[{{G}}_{{2}}\left({2}\right){,}{{G}}_{{2}}\left({2}\right)\right]$ (7)
 > $\mathrm{GroupOrder}\left(L\right)$
 ${6048}$ (8)
 > $\mathrm{IsSimple}\left(L\right)$
 ${\mathrm{true}}$ (9)

Compatibility

 • The GroupTheory[ExceptionalGroup] command was introduced in Maple 17.