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GroupTheory

 Suzuki2B2

 Calling Sequence Suzuki2B2( q )

Parameters

 q - : {posint,algebraic} : an odd power of $2$, or an expression

Description

 • 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.)

Examples

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

The smallest of the Suzuki groups is a non-simple group of order $20$ that is, in fact, a soluble Frobenius group.

 > $G≔\mathrm{Suzuki2B2}\left(2\right)$
 ${\mathrm{GroupTheory}}{:-}{\mathrm{PermutationGroup}}{}\left(\left\{{\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}{,}{\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}\right\}{,}{\mathrm{degree}}{=}{5}\right)$ (1)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${20}$ (2)
 > $\mathrm{IsSimple}\left(G\right)$
 ${\mathrm{false}}$ (3)
 > $\left(\mathrm{IsSoluble}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{and}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{IsFrobeniusGroup}\right)\left(G\right)$
 ${\mathrm{true}}$ (4)
 > $\mathrm{cs}≔\mathrm{CompositionSeries}\left(G\right)$
 ${\mathrm{CompositionSeries}}{}\left({\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}\right)$ (5)
 > $\mathrm{seq}\left(\mathrm{GroupOrder}\left(S\right),S=\mathrm{cs}\right)$
 ${20}{,}{10}{,}{5}{,}{1}$ (6)
 > $\mathbf{use}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{plots},\mathrm{GraphTheory}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{display}\left(\mathrm{Array}\left(\left[\mathrm{DrawGraph}\left(\mathrm{CayleyGraph}\left(G\right)\right),\mathrm{DrawSubgroupLattice}\left(G,'\mathrm{labels}'='\mathrm{ids}'\right)\right]\right)\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end use}$  For values of $q$ larger than $2$, the group $\mathbf{Sz}\left(q\right)$ is simple.

 > $G≔\mathrm{Suzuki2B2}\left(32\right)$
 ${\mathrm{GroupTheory}}{:-}{\mathrm{PermutationGroup}}{}\left(\left\{{\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}{,}{\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}\right\}{,}{\mathrm{degree}}{=}{1025}\right)$ (7)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${32537600}$ (8)
 > $\mathrm{IsSimple}\left(G\right)$
 ${\mathrm{true}}$ (9)
 > $\mathrm{IsCNGroup}\left(G\right)$
 ${\mathrm{true}}$ (10)
 > $\mathrm{OrderClassPolynomial}\left(G,x\right)$
 ${7936000}{}{{x}}^{{41}}{+}{15744000}{}{{x}}^{{31}}{+}{6507520}{}{{x}}^{{25}}{+}{1301504}{}{{x}}^{{5}}{+}{1016800}{}{{x}}^{{4}}{+}{31775}{}{{x}}^{{2}}{+}{x}$ (11)
 > $\mathrm{Display}\left(\mathrm{CharacterTable}\left(\mathrm{Suzuki2B2}\left(8\right)\right)\right)$

 C 1a 2a 4a 4b 5a 7a 7b 7c 13a 13b 13c |C| 1 455 1820 1820 5824 4160 4160 4160 2240 2240 2240 $\mathrm{χ__1}$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $\mathrm{χ__2}$ $14$ $-2$ $2I$ $-2I$ $-1$ $0$ $0$ $0$ $1$ $1$ $1$ $\mathrm{χ__3}$ $14$ $-2$ $-2I$ $2I$ $-1$ $0$ $0$ $0$ $1$ $1$ $1$ $\mathrm{χ__4}$ $35$ $3$ $-1$ $-1$ $0$ $0$ $0$ $0$ $-{\left(-1\right)}^{2}{13}}-{\left(-1\right)}^{10}{13}}+{\left(-1\right)}^{3}{13}}+{\left(-1\right)}^{11}{13}}$ $-{\left(-1\right)}^{4}{13}}-{\left(-1\right)}^{6}{13}}+{\left(-1\right)}^{7}{13}}+{\left(-1\right)}^{9}{13}}$ $-{\left(-1\right)}^{8}{13}}-{\left(-1\right)}^{12}{13}}+{\left(-1\right)}^{1}{13}}+{\left(-1\right)}^{5}{13}}$ $\mathrm{χ__5}$ $35$ $3$ $-1$ $-1$ $0$ $0$ $0$ $0$ $-{\left(-1\right)}^{8}{13}}-{\left(-1\right)}^{12}{13}}+{\left(-1\right)}^{1}{13}}+{\left(-1\right)}^{5}{13}}$ $-{\left(-1\right)}^{2}{13}}-{\left(-1\right)}^{10}{13}}+{\left(-1\right)}^{3}{13}}+{\left(-1\right)}^{11}{13}}$ $-{\left(-1\right)}^{4}{13}}-{\left(-1\right)}^{6}{13}}+{\left(-1\right)}^{7}{13}}+{\left(-1\right)}^{9}{13}}$ $\mathrm{χ__6}$ $35$ $3$ $-1$ $-1$ $0$ $0$ $0$ $0$ $-{\left(-1\right)}^{4}{13}}-{\left(-1\right)}^{6}{13}}+{\left(-1\right)}^{7}{13}}+{\left(-1\right)}^{9}{13}}$ $-{\left(-1\right)}^{8}{13}}-{\left(-1\right)}^{12}{13}}+{\left(-1\right)}^{1}{13}}+{\left(-1\right)}^{5}{13}}$ $-{\left(-1\right)}^{2}{13}}-{\left(-1\right)}^{10}{13}}+{\left(-1\right)}^{3}{13}}+{\left(-1\right)}^{11}{13}}$ $\mathrm{χ__7}$ $64$ $0$ $0$ $0$ $-1$ $1$ $1$ $1$ $-1$ $-1$ $-1$ $\mathrm{χ__8}$ $65$ $1$ $1$ $1$ $0$ ${\left(-1\right)}^{2}{7}}-{\left(-1\right)}^{5}{7}}$ ${\left(-1\right)}^{4}{7}}-{\left(-1\right)}^{3}{7}}$ ${\left(-1\right)}^{6}{7}}-{\left(-1\right)}^{1}{7}}$ $0$ $0$ $0$ $\mathrm{χ__9}$ $65$ $1$ $1$ $1$ $0$ ${\left(-1\right)}^{6}{7}}-{\left(-1\right)}^{1}{7}}$ ${\left(-1\right)}^{2}{7}}-{\left(-1\right)}^{5}{7}}$ ${\left(-1\right)}^{4}{7}}-{\left(-1\right)}^{3}{7}}$ $0$ $0$ $0$ $\mathrm{χ__10}$ $65$ $1$ $1$ $1$ $0$ ${\left(-1\right)}^{4}{7}}-{\left(-1\right)}^{3}{7}}$ ${\left(-1\right)}^{6}{7}}-{\left(-1\right)}^{1}{7}}$ ${\left(-1\right)}^{2}{7}}-{\left(-1\right)}^{5}{7}}$ $0$ $0$ $0$ $\mathrm{χ__11}$ $91$ $-5$ $-1$ $-1$ $1$ $0$ $0$ $0$ $0$ $0$ $0$

For non-numeric arguments, a symbolic group is returned.

 > $G≔\mathrm{Suzuki2B2}\left(q\right)$
 ${\mathrm{GroupTheory}}{:-}{\mathrm{Suzuki2B2}}{}\left({q}\right)$ (12)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${{q}}^{{2}}{}\left({{q}}^{{2}}{+}{1}\right){}\left({q}{-}{1}\right)$ (13)
 > $\mathrm{IsSimple}\left(G\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}2
 ${\mathrm{true}}$ (14)
 > $\mathrm{IsCNGroup}\left(G\right)$
 ${\mathrm{true}}$ (15)

A symbolic group is also returned if the numeric argument q exceeds $512$.

 > $G≔\mathrm{Suzuki2B2}\left({2}^{101}\right)$
 ${\mathrm{GroupTheory}}{:-}{\mathrm{Suzuki2B2}}{}\left({2535301200456458802993406410752}\right)$ (16)
 > $\mathrm{ifactor}\left(\mathrm{GroupOrder}\left(G\right)\right)$
 ${\left({2}\right)}^{{202}}{}\left({5}\right){}\left({809}\right){}\left({9491060093}\right){}\left({5218735279937}\right){}\left({600503817460697}\right){}\left({53425037363873248657}\right){}\left({7432339208719}\right){}\left({341117531003194129}\right)$ (17)
 > $\mathrm{MinimumPermutationRepresentationDegree}\left(G\right)$
 ${6427752177035961102167848369364650410088811975131171341205505}$ (18)
 > $\mathrm{IsSimple}\left(G\right)$
 ${\mathrm{true}}$ (19)
 > 

Compatibility

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