QuasiDihedralGroup - Maple Help

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GroupTheory

 SemiDihedralGroup
 construct a semi-dihedral group as a permutation group or a finitely presented group
 QuasiDihedralGroup
 construct a quasi-dihedral group as a permutation group or a finitely presented group

 Calling Sequence SemiDihedralGroup( n, formopt ) QuasiDihedralGroup( n, formopt )

Parameters

 n - algebraic; understood to be an integer greater than $1$ formopt - equation; (optional) equation of the form form = "fpgroup" or form = "permgroup" (default)

Description

 • The semi-dihedral of degree $n$ is a non-abelian group of order $8n$ which contains a cyclic subgroup of order $4n$ for $n>1$. It is defined by a presentation of the form

$⟨xy,|,{x}^{n}={y}^{2},,,{x}^{y}={x}^{-1}⟩$

 • The SemiDihedralGroup( n ) command returns a semi-dihedral group, either as a permutation group (the default) or as a finitely presented group.
 • You can specify the form of the group returned explicitly by passing one of the options 'form' = "permgroup" or 'form' = "fpgroup".
 • If the parameter n is not a positive integer, then a symbolic group representing the semi-dihedral group of order 8*n is returned.
 • If $n$ is a power of $2$, the resulting group is a quasi-dihedral group. In other words, a quasi-dihedral group is a semi-dihedral $2$-group. (This is analogous to the fact that a quaternion group is a dicyclic $2$-group.) A semi-dihedral group is nilpotent only if it is quasi-dihedral.
 • The QuasiDihedralGroup( n ) command returns a quasi-dihedral group of order ${2}^{n-1}$, provided that n is an integer greater than $1$. If n is a non-numeric algebraic expression, then a symbolic group representing the quasi-dihedral group of order ${2}^{n-1}$ is returned.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{SemiDihedralGroup}\left(21\right)$
 ${G}{≔}{{\mathbf{SD}}}_{{21}}$ (1)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${168}$ (2)
 > $\mathrm{ClassNumber}\left(G\right)$
 ${48}$ (3)
 > $\mathrm{IsNilpotent}\left(G\right)$
 ${\mathrm{false}}$ (4)
 > $\mathrm{IsSupersoluble}\left(G\right)$
 ${\mathrm{true}}$ (5)
 > $\mathrm{GroupOrder}\left(\mathrm{DerivedSubgroup}\left(G\right)\right)$
 ${21}$ (6)
 > $\mathrm{IsCyclic}\left(\mathrm{DerivedSubgroup}\left(G\right)\right)$
 ${\mathrm{true}}$ (7)

The center of a semi-dihedral group is always cyclic, but the order depends upon whether $n$ is odd or even. For odd $n$, the center has order $4$.

 > $\mathrm{IsCyclic}\left(\mathrm{Center}\left(G\right)\right)$
 ${\mathrm{true}}$ (8)
 > $\mathrm{GroupOrder}\left(\mathrm{Center}\left(G\right)\right)$
 ${4}$ (9)

For even $n$, the center has order $2$.

 > $\mathrm{GroupOrder}\left(\mathrm{Center}\left(\mathrm{SemiDihedralGroup}\left(20\right)\right)\right)$
 ${2}$ (10)
 > $\mathrm{seq}\left(\mathrm{GroupOrder}\left(\mathrm{Center}\left(\mathrm{SemiDihedralGroup}\left(n\right)\right)\right),n=2..20\right)$
 ${2}{,}{4}{,}{2}{,}{4}{,}{2}{,}{4}{,}{2}{,}{4}{,}{2}{,}{4}{,}{2}{,}{4}{,}{2}{,}{4}{,}{2}{,}{4}{,}{2}{,}{4}{,}{2}$ (11)

The permutation representation used in Maple is always transitive, but imprimitive.

 > $\mathrm{IsTransitive}\left(G\right)$
 ${\mathrm{true}}$ (12)
 > $\mathrm{IsPrimitive}\left(G\right)$
 ${\mathrm{false}}$ (13)
 > $\mathrm{Blocks}\left(G\right)$
 $\left\{\left\{{1}{,}{3}{,}{5}{,}{7}{,}{9}{,}{11}{,}{13}{,}{15}{,}{17}{,}{19}{,}{21}{,}{23}{,}{25}{,}{27}{,}{29}{,}{31}{,}{33}{,}{35}{,}{37}{,}{39}{,}{41}{,}{43}{,}{45}{,}{47}{,}{49}{,}{51}{,}{53}{,}{55}{,}{57}{,}{59}{,}{61}{,}{63}{,}{65}{,}{67}{,}{69}{,}{71}{,}{73}{,}{75}{,}{77}{,}{79}{,}{81}{,}{83}\right\}{,}\left\{{2}{,}{4}{,}{6}{,}{8}{,}{10}{,}{12}{,}{14}{,}{16}{,}{18}{,}{20}{,}{22}{,}{24}{,}{26}{,}{28}{,}{30}{,}{32}{,}{34}{,}{36}{,}{38}{,}{40}{,}{42}{,}{44}{,}{46}{,}{48}{,}{50}{,}{52}{,}{54}{,}{56}{,}{58}{,}{60}{,}{62}{,}{64}{,}{66}{,}{68}{,}{70}{,}{72}{,}{74}{,}{76}{,}{78}{,}{80}{,}{82}{,}{84}\right\}\right\}$ (14)
 > $\mathrm{orseq}\left(\mathrm{IsPrimitive}\left(\mathrm{SemiDihedralGroup}\left(n\right)\right),n=2..20\right)$
 ${\mathrm{false}}$ (15)
 > $\mathrm{andseq}\left(\mathrm{IsTransitive}\left(\mathrm{SemiDihedralGroup}\left(n\right)\right),n=2..20\right)$
 ${\mathrm{true}}$ (16)

Use the form = "fpgroup" option to construct a finitely presented semi-dihedral group.

 > $G≔\mathrm{SemiDihedralGroup}\left(6,'\mathrm{form}'="fpgroup"\right)$
 ${G}{≔}{{\mathbf{SD}}}_{{6}}$ (17)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${48}$ (18)

Note that dihedral and semi-dihedral groups of the same order are non-isomorphic.

 > $\mathrm{AreIsomorphic}\left(\mathrm{SemiDihedralGroup}\left(4\right),\mathrm{DihedralGroup}\left(16\right)\right)$
 ${\mathrm{false}}$ (19)
 > $\mathrm{Display}\left(\mathrm{CharacterTable}\left(\mathrm{SemiDihedralGroup}\left(5\right)\right)\right)$

 C 1a 2a 2b 2c 4a 4b 4c 4d 5a 5b 10a 10b 20a 20b 20c 20d |C| 1 1 5 5 1 1 5 5 2 2 2 2 2 2 2 2 $\mathrm{χ__1}$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $\mathrm{χ__2}$ $1$ $-1$ $-1$ $1$ $-I$ $I$ $I$ $-I$ $1$ $1$ $-1$ $-1$ $I$ $-I$ $I$ $-I$ $\mathrm{χ__3}$ $1$ $-1$ $-1$ $1$ $I$ $-I$ $-I$ $I$ $1$ $1$ $-1$ $-1$ $-I$ $I$ $-I$ $I$ $\mathrm{χ__4}$ $1$ $-1$ $1$ $-1$ $-I$ $I$ $-I$ $I$ $1$ $1$ $-1$ $-1$ $I$ $-I$ $I$ $-I$ $\mathrm{χ__5}$ $1$ $-1$ $1$ $-1$ $I$ $-I$ $I$ $-I$ $1$ $1$ $-1$ $-1$ $-I$ $I$ $-I$ $I$ $\mathrm{χ__6}$ $1$ $1$ $-1$ $-1$ $-1$ $-1$ $1$ $1$ $1$ $1$ $1$ $1$ $-1$ $-1$ $-1$ $-1$ $\mathrm{χ__7}$ $1$ $1$ $-1$ $-1$ $1$ $1$ $-1$ $-1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $\mathrm{χ__8}$ $1$ $1$ $1$ $1$ $-1$ $-1$ $-1$ $-1$ $1$ $1$ $1$ $1$ $-1$ $-1$ $-1$ $-1$ $\mathrm{χ__9}$ $2$ $-2$ $0$ $0$ $-2I$ $2I$ $0$ $0$ ${\left(-1\right)}^{3}{5}}-{\left(-1\right)}^{2}{5}}-1$ ${\left(-1\right)}^{2}{5}}-{\left(-1\right)}^{3}{5}}$ $-{\left(-1\right)}^{3}{5}}+{\left(-1\right)}^{2}{5}}+1$ ${\left(-1\right)}^{3}{5}}-{\left(-1\right)}^{2}{5}}$ ${\left(-1\right)}^{1}{10}}+{\left(-1\right)}^{9}{10}}$ ${\left(-1\right)}^{3}{10}}+{\left(-1\right)}^{7}{10}}$ $-{\left(-1\right)}^{3}{10}}-{\left(-1\right)}^{7}{10}}$ $-{\left(-1\right)}^{1}{10}}-{\left(-1\right)}^{9}{10}}$ $\mathrm{χ__10}$ $2$ $-2$ $0$ $0$ $-2I$ $2I$ $0$ $0$ ${\left(-1\right)}^{2}{5}}-{\left(-1\right)}^{3}{5}}$ ${\left(-1\right)}^{3}{5}}-{\left(-1\right)}^{2}{5}}-1$ ${\left(-1\right)}^{3}{5}}-{\left(-1\right)}^{2}{5}}$ $-{\left(-1\right)}^{3}{5}}+{\left(-1\right)}^{2}{5}}+1$ $-{\left(-1\right)}^{3}{10}}-{\left(-1\right)}^{7}{10}}$ $-{\left(-1\right)}^{1}{10}}-{\left(-1\right)}^{9}{10}}$ ${\left(-1\right)}^{1}{10}}+{\left(-1\right)}^{9}{10}}$ ${\left(-1\right)}^{3}{10}}+{\left(-1\right)}^{7}{10}}$ $\mathrm{χ__11}$ $2$ $-2$ $0$ $0$ $2I$ $-2I$ $0$ $0$ ${\left(-1\right)}^{3}{5}}-{\left(-1\right)}^{2}{5}}-1$ ${\left(-1\right)}^{2}{5}}-{\left(-1\right)}^{3}{5}}$ $-{\left(-1\right)}^{3}{5}}+{\left(-1\right)}^{2}{5}}+1$ ${\left(-1\right)}^{3}{5}}-{\left(-1\right)}^{2}{5}}$ $-{\left(-1\right)}^{1}{10}}-{\left(-1\right)}^{9}{10}}$ $-{\left(-1\right)}^{3}{10}}-{\left(-1\right)}^{7}{10}}$ ${\left(-1\right)}^{3}{10}}+{\left(-1\right)}^{7}{10}}$ ${\left(-1\right)}^{1}{10}}+{\left(-1\right)}^{9}{10}}$ $\mathrm{χ__12}$ $2$ $-2$ $0$ $0$ $2I$ $-2I$ $0$ $0$ ${\left(-1\right)}^{2}{5}}-{\left(-1\right)}^{3}{5}}$ ${\left(-1\right)}^{3}{5}}-{\left(-1\right)}^{2}{5}}-1$ ${\left(-1\right)}^{3}{5}}-{\left(-1\right)}^{2}{5}}$ $-{\left(-1\right)}^{3}{5}}+{\left(-1\right)}^{2}{5}}+1$ ${\left(-1\right)}^{3}{10}}+{\left(-1\right)}^{7}{10}}$ ${\left(-1\right)}^{1}{10}}+{\left(-1\right)}^{9}{10}}$ $-{\left(-1\right)}^{1}{10}}-{\left(-1\right)}^{9}{10}}$ $-{\left(-1\right)}^{3}{10}}-{\left(-1\right)}^{7}{10}}$ $\mathrm{χ__13}$ $2$ $2$ $0$ $0$ $-2$ $-2$ $0$ $0$ ${\left(-1\right)}^{3}{5}}-{\left(-1\right)}^{2}{5}}-1$ ${\left(-1\right)}^{2}{5}}-{\left(-1\right)}^{3}{5}}$ ${\left(-1\right)}^{3}{5}}-{\left(-1\right)}^{2}{5}}-1$ ${\left(-1\right)}^{2}{5}}-{\left(-1\right)}^{3}{5}}$ ${\left(-1\right)}^{3}{5}}-{\left(-1\right)}^{2}{5}}$ $-{\left(-1\right)}^{3}{5}}+{\left(-1\right)}^{2}{5}}+1$ $-{\left(-1\right)}^{3}{5}}+{\left(-1\right)}^{2}{5}}+1$ ${\left(-1\right)}^{3}{5}}-{\left(-1\right)}^{2}{5}}$ $\mathrm{χ__14}$ $2$ $2$ $0$ $0$ $-2$ $-2$ $0$ $0$ ${\left(-1\right)}^{2}{5}}-{\left(-1\right)}^{3}{5}}$ ${\left(-1\right)}^{3}{5}}-{\left(-1\right)}^{2}{5}}-1$ ${\left(-1\right)}^{2}{5}}-{\left(-1\right)}^{3}{5}}$ ${\left(-1\right)}^{3}{5}}-{\left(-1\right)}^{2}{5}}-1$ $-{\left(-1\right)}^{3}{5}}+{\left(-1\right)}^{2}{5}}+1$ ${\left(-1\right)}^{3}{5}}-{\left(-1\right)}^{2}{5}}$ ${\left(-1\right)}^{3}{5}}-{\left(-1\right)}^{2}{5}}$ $-{\left(-1\right)}^{3}{5}}+{\left(-1\right)}^{2}{5}}+1$ $\mathrm{χ__15}$ $2$ $2$ $0$ $0$ $2$ $2$ $0$ $0$ ${\left(-1\right)}^{3}{5}}-{\left(-1\right)}^{2}{5}}-1$ ${\left(-1\right)}^{2}{5}}-{\left(-1\right)}^{3}{5}}$ ${\left(-1\right)}^{3}{5}}-{\left(-1\right)}^{2}{5}}-1$ ${\left(-1\right)}^{2}{5}}-{\left(-1\right)}^{3}{5}}$ ${\left(-1\right)}^{2}{5}}-{\left(-1\right)}^{3}{5}}$ ${\left(-1\right)}^{3}{5}}-{\left(-1\right)}^{2}{5}}-1$ ${\left(-1\right)}^{3}{5}}-{\left(-1\right)}^{2}{5}}-1$ ${\left(-1\right)}^{2}{5}}-{\left(-1\right)}^{3}{5}}$ $\mathrm{χ__16}$ $2$ $2$ $0$ $0$ $2$ $2$ $0$ $0$ ${\left(-1\right)}^{2}{5}}-{\left(-1\right)}^{3}{5}}$ ${\left(-1\right)}^{3}{5}}-{\left(-1\right)}^{2}{5}}-1$ ${\left(-1\right)}^{2}{5}}-{\left(-1\right)}^{3}{5}}$ ${\left(-1\right)}^{3}{5}}-{\left(-1\right)}^{2}{5}}-1$ ${\left(-1\right)}^{3}{5}}-{\left(-1\right)}^{2}{5}}-1$ ${\left(-1\right)}^{2}{5}}-{\left(-1\right)}^{3}{5}}$ ${\left(-1\right)}^{2}{5}}-{\left(-1\right)}^{3}{5}}$ ${\left(-1\right)}^{3}{5}}-{\left(-1\right)}^{2}{5}}-1$

 > $G≔\mathrm{QuasiDihedralGroup}\left(2\right)$
 ${G}{≔}{{\mathrm{QD}}}_{{2}}$ (20)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${16}$ (21)
 > $\mathrm{seq}\left(\mathrm{GroupOrder}\left(\mathrm{QuasiDihedralGroup}\left(n\right)\right),n=2..8\right)$
 ${16}{,}{32}{,}{64}{,}{128}{,}{256}{,}{512}{,}{1024}$ (22)
 > $\mathrm{IsRegularPGroup}\left(\mathrm{QuasiDihedralGroup}\left(11\right)\right)$
 ${\mathrm{false}}$ (23)
 > $A≔\mathrm{QuasiDihedralGroup}\left(3,'\mathrm{form}'="fpgroup"\right)$
 ${A}{≔}{{\mathrm{QD}}}_{{3}}$ (24)
 > $B≔\mathrm{SemiDihedralGroup}\left(4,'\mathrm{form}'="fpgroup"\right)$
 ${B}{≔}{{\mathbf{SD}}}_{{4}}$ (25)
 > $\mathrm{AreIsomorphic}\left(A,B\right)$
 ${\mathrm{true}}$ (26)