The Suzuki groups $\mathbf{Sz}\left(q\right)$ , of type $\mathrm{\²B₂}\left(q\right)$, for an odd power $q$ of $2$, are a series of (typically) simple groups of Lie type, first constructed by M. Suzuki. They are defined only for $q=2^{2e+1}$ an odd power of $2$ (where, here, $0\le e$).

The groups $\mathbf{Sz}\left(q\right)$ should not be confused with the "Suzuki group" of order $448345497600$, one of the sporadic finite simple groups. (See GroupTheory[SuzukiGroup].)

The Suzuki groups $\mathbf{Sz}\left(q\right)$ are notable among the finite simple groups in that they are the only finite non-abelian simple groups whose order is not divisible by $3$.

The Suzuki2B2( q ) command constructs a permutation group isomorphic to $\mathbf{Sz}\left(q\right)$ , for admissible values of $q$ up to $512$.

If the argument q is not numeric, or if it is an odd power of $2$ greater than $512$, then a symbolic group representing $\mathbf{Sz}\left(q\right)$ is returned.

(The Suzuki groups $\mathbf{Sz}\left(8\right)$ and $\mathbf{Sz}\left(\mathrm{32}\right)$ are also available by using the ExceptionalGroup command.)

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

$G≔\mathrm{Suzuki2B2}\left(2\right)$

$G≔\mathbf{Sz}\left(2\right)$

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

$20$

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

$\mathrm{false}$

$\left(\mathrm{IsSoluble}\mathbf{and}\mathrm{IsFrobeniusGroup}\right)\left(G\right)$

$\mathrm{true}$

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

$\mathrm{cs}≔\mathbf{Sz}\left(2\right)▹⟨\left(1\,5\,3\,2\,4\right)\,\left(2\,3\right)\left(4\,5\right)⟩▹\left[\mathbf{Sz}\left(2\right)\,\mathbf{Sz}\left(2\right)\right]▹⟨⟩$

$\mathrm{seq}\left(\mathrm{GroupOrder}\left(S\right)\,S=\mathrm{cs}\right)$

$20,10,5,1$

$\mathbf{use}\mathrm{plots}\,\mathrm{GraphTheory}\mathbf{in}\mathrm{display}\left(\mathrm{Array}\left(\left[\mathrm{DrawGraph}\left(\mathrm{CayleyGraph}\left(G\right)\right)\,\mathrm{DrawSubgroupLattice}\left(G\,\'\mathrm{labels}\'\=\'\mathrm{ids}\'\right)\right]\right)\right)\mathbf{end\; use}$

$G≔\mathrm{Suzuki2B2}\left(32\right)$

$G≔\mathbf{Sz}\left(32\right)$

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

$32537600$

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

$\mathrm{true}$

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

$\mathrm{true}$

$\mathrm{OrderClassPolynomial}\left(G\,x\right)$

$7936000x^{41}+15744000x^{31}+6507520x^{25}+1301504x^5+1016800x^4+31775x^2+x$

$\mathrm{Display}\left(\mathrm{CharacterTable}\left(\mathrm{Suzuki2B2}\left(8\right)\right)\right)$

$G≔\mathrm{Suzuki2B2}\left(q\right)$

$G≔\mathbf{Sz}\left(q\right)$

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

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

$\mathrm{IsSimple}\left(G\right)\mathbf{assuming}2<q$

$\mathrm{true}$

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

$\mathrm{true}$

$G≔\mathrm{Suzuki2B2}\left(2^{101}\right)$

$G≔\mathbf{Sz}\left(2535301200456458802993406410752\right)$

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

${\left(2\right)}^{202}\left(5\right)\left(809\right)\left(9491060093\right)\left(5218735279937\right)\left(600503817460697\right)\left(53425037363873248657\right)\left(7432339208719\right)\left(341117531003194129\right)$

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

$6427752177035961102167848369364650410088811975131171341205505$

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

$\mathrm{true}$

The GroupTheory[Suzuki2B2] command was introduced in Maple 2020.

For more information on Maple 2020 changes, see Updates in Maple 2020.