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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 $8 n$ which contains a cyclic subgroup of order $4 n$ for $n > 1$ . It is defined by a presentation of the form

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

•	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

>	$with (GroupTheory) &colon;$

>	$G ≔ SemiDihedralGroup (21)$

$G ≔ {SD}_{21}$

(1)

>	$GroupOrder (G)$

$168$

(2)

>	$ClassNumber (G)$

$48$

(3)

>	$IsNilpotent (G)$

$false$

(4)

>	$IsSupersoluble (G)$

$true$

(5)

>	$GroupOrder (DerivedSubgroup (G))$

$21$

(6)

>	$IsCyclic (DerivedSubgroup (G))$

$true$

(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 $4$ .

>	$IsCyclic (Center (G))$

$true$

(8)

>	$GroupOrder (Center (G))$

$4$

(9)

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

>	$GroupOrder (Center (SemiDihedralGroup (20)))$

$2$

(10)

>	$seq (GroupOrder (Center (SemiDihedralGroup (n))), n = 2 .. 20)$

$2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2$

(11)

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

>	$IsTransitive (G)$

$true$

(12)

>	$IsPrimitive (G)$

$false$

(13)

>	$Blocks (G)$

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

(14)

>	$orseq (IsPrimitive (SemiDihedralGroup (n)), n = 2 .. 20)$

$false$

(15)

>	$andseq (IsTransitive (SemiDihedralGroup (n)), n = 2 .. 20)$

$true$

(16)

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

>	$G ≔ SemiDihedralGroup (6,'form' = fpgroup)$

$G ≔ {SD}_{6}$

(17)

>	$GroupOrder (G)$

$48$

(18)

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

>	$AreIsomorphic (SemiDihedralGroup (4), DihedralGroup (16))$

$false$

(19)

>	$Display (CharacterTable (SemiDihedralGroup (5)))$

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

$χ__1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$χ__2$	$1$	$−1$	$−1$	$1$	$−I$	$I$	$I$	$−I$	$1$	$1$	$−1$	$−1$	$I$	$−I$	$I$	$−I$
$χ__3$	$1$	$−1$	$−1$	$1$	$I$	$−I$	$−I$	$I$	$1$	$1$	$−1$	$−1$	$−I$	$I$	$−I$	$I$
$χ__4$	$1$	$−1$	$1$	$−1$	$−I$	$I$	$−I$	$I$	$1$	$1$	$−1$	$−1$	$I$	$−I$	$I$	$−I$
$χ__5$	$1$	$−1$	$1$	$−1$	$I$	$−I$	$I$	$−I$	$1$	$1$	$−1$	$−1$	$−I$	$I$	$−I$	$I$
$χ__6$	$1$	$1$	$−1$	$−1$	$−1$	$−1$	$1$	$1$	$1$	$1$	$1$	$1$	$−1$	$−1$	$−1$	$−1$
$χ__7$	$1$	$1$	$−1$	$−1$	$1$	$1$	$−1$	$−1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$χ__8$	$1$	$1$	$1$	$1$	$−1$	$−1$	$−1$	$−1$	$1$	$1$	$1$	$1$	$−1$	$−1$	$−1$	$−1$
$χ__9$	$2$	$−2$	$0$	$0$	$- 2 I$	$2 I$	$0$	$0$	${(−1)}^{\frac{3}{5}} - {(−1)}^{\frac{2}{5}} - 1$	${(−1)}^{\frac{2}{5}} - {(−1)}^{\frac{3}{5}}$	$- {(−1)}^{\frac{3}{5}} + {(−1)}^{\frac{2}{5}} + 1$	${(−1)}^{\frac{3}{5}} - {(−1)}^{\frac{2}{5}}$	${(−1)}^{\frac{1}{10}} + {(−1)}^{\frac{9}{10}}$	${(−1)}^{\frac{3}{10}} + {(−1)}^{\frac{7}{10}}$	$- {(−1)}^{\frac{3}{10}} - {(−1)}^{\frac{7}{10}}$	$- {(−1)}^{\frac{1}{10}} - {(−1)}^{\frac{9}{10}}$
$χ__10$	$2$	$−2$	$0$	$0$	$- 2 I$	$2 I$	$0$	$0$	${(−1)}^{\frac{2}{5}} - {(−1)}^{\frac{3}{5}}$	${(−1)}^{\frac{3}{5}} - {(−1)}^{\frac{2}{5}} - 1$	${(−1)}^{\frac{3}{5}} - {(−1)}^{\frac{2}{5}}$	$- {(−1)}^{\frac{3}{5}} + {(−1)}^{\frac{2}{5}} + 1$	$- {(−1)}^{\frac{3}{10}} - {(−1)}^{\frac{7}{10}}$	$- {(−1)}^{\frac{1}{10}} - {(−1)}^{\frac{9}{10}}$	${(−1)}^{\frac{1}{10}} + {(−1)}^{\frac{9}{10}}$	${(−1)}^{\frac{3}{10}} + {(−1)}^{\frac{7}{10}}$
$χ__11$	$2$	$−2$	$0$	$0$	$2 I$	$- 2 I$	$0$	$0$	${(−1)}^{\frac{3}{5}} - {(−1)}^{\frac{2}{5}} - 1$	${(−1)}^{\frac{2}{5}} - {(−1)}^{\frac{3}{5}}$	$- {(−1)}^{\frac{3}{5}} + {(−1)}^{\frac{2}{5}} + 1$	${(−1)}^{\frac{3}{5}} - {(−1)}^{\frac{2}{5}}$	$- {(−1)}^{\frac{1}{10}} - {(−1)}^{\frac{9}{10}}$	$- {(−1)}^{\frac{3}{10}} - {(−1)}^{\frac{7}{10}}$	${(−1)}^{\frac{3}{10}} + {(−1)}^{\frac{7}{10}}$	${(−1)}^{\frac{1}{10}} + {(−1)}^{\frac{9}{10}}$
$χ__12$	$2$	$−2$	$0$	$0$	$2 I$	$- 2 I$	$0$	$0$	${(−1)}^{\frac{2}{5}} - {(−1)}^{\frac{3}{5}}$	${(−1)}^{\frac{3}{5}} - {(−1)}^{\frac{2}{5}} - 1$	${(−1)}^{\frac{3}{5}} - {(−1)}^{\frac{2}{5}}$	$- {(−1)}^{\frac{3}{5}} + {(−1)}^{\frac{2}{5}} + 1$	${(−1)}^{\frac{3}{10}} + {(−1)}^{\frac{7}{10}}$	${(−1)}^{\frac{1}{10}} + {(−1)}^{\frac{9}{10}}$	$- {(−1)}^{\frac{1}{10}} - {(−1)}^{\frac{9}{10}}$	$- {(−1)}^{\frac{3}{10}} - {(−1)}^{\frac{7}{10}}$
$χ__13$	$2$	$2$	$0$	$0$	$−2$	$−2$	$0$	$0$	${(−1)}^{\frac{3}{5}} - {(−1)}^{\frac{2}{5}} - 1$	${(−1)}^{\frac{2}{5}} - {(−1)}^{\frac{3}{5}}$	${(−1)}^{\frac{3}{5}} - {(−1)}^{\frac{2}{5}} - 1$	${(−1)}^{\frac{2}{5}} - {(−1)}^{\frac{3}{5}}$	${(−1)}^{\frac{3}{5}} - {(−1)}^{\frac{2}{5}}$	$- {(−1)}^{\frac{3}{5}} + {(−1)}^{\frac{2}{5}} + 1$	$- {(−1)}^{\frac{3}{5}} + {(−1)}^{\frac{2}{5}} + 1$	${(−1)}^{\frac{3}{5}} - {(−1)}^{\frac{2}{5}}$
$χ__14$	$2$	$2$	$0$	$0$	$−2$	$−2$	$0$	$0$	${(−1)}^{\frac{2}{5}} - {(−1)}^{\frac{3}{5}}$	${(−1)}^{\frac{3}{5}} - {(−1)}^{\frac{2}{5}} - 1$	${(−1)}^{\frac{2}{5}} - {(−1)}^{\frac{3}{5}}$	${(−1)}^{\frac{3}{5}} - {(−1)}^{\frac{2}{5}} - 1$	$- {(−1)}^{\frac{3}{5}} + {(−1)}^{\frac{2}{5}} + 1$	${(−1)}^{\frac{3}{5}} - {(−1)}^{\frac{2}{5}}$	${(−1)}^{\frac{3}{5}} - {(−1)}^{\frac{2}{5}}$	$- {(−1)}^{\frac{3}{5}} + {(−1)}^{\frac{2}{5}} + 1$
$χ__15$	$2$	$2$	$0$	$0$	$2$	$2$	$0$	$0$	${(−1)}^{\frac{3}{5}} - {(−1)}^{\frac{2}{5}} - 1$	${(−1)}^{\frac{2}{5}} - {(−1)}^{\frac{3}{5}}$	${(−1)}^{\frac{3}{5}} - {(−1)}^{\frac{2}{5}} - 1$	${(−1)}^{\frac{2}{5}} - {(−1)}^{\frac{3}{5}}$	${(−1)}^{\frac{2}{5}} - {(−1)}^{\frac{3}{5}}$	${(−1)}^{\frac{3}{5}} - {(−1)}^{\frac{2}{5}} - 1$	${(−1)}^{\frac{3}{5}} - {(−1)}^{\frac{2}{5}} - 1$	${(−1)}^{\frac{2}{5}} - {(−1)}^{\frac{3}{5}}$
$χ__16$	$2$	$2$	$0$	$0$	$2$	$2$	$0$	$0$	${(−1)}^{\frac{2}{5}} - {(−1)}^{\frac{3}{5}}$	${(−1)}^{\frac{3}{5}} - {(−1)}^{\frac{2}{5}} - 1$	${(−1)}^{\frac{2}{5}} - {(−1)}^{\frac{3}{5}}$	${(−1)}^{\frac{3}{5}} - {(−1)}^{\frac{2}{5}} - 1$	${(−1)}^{\frac{3}{5}} - {(−1)}^{\frac{2}{5}} - 1$	${(−1)}^{\frac{2}{5}} - {(−1)}^{\frac{3}{5}}$	${(−1)}^{\frac{2}{5}} - {(−1)}^{\frac{3}{5}}$	${(−1)}^{\frac{3}{5}} - {(−1)}^{\frac{2}{5}} - 1$

>	$G ≔ QuasiDihedralGroup (2)$

$G ≔ {QD}_{2}$

(20)

>	$GroupOrder (G)$

$16$

(21)

>	$seq (GroupOrder (QuasiDihedralGroup (n)), n = 2 .. 8)$

$16, 32, 64, 128, 256, 512, 1024$

(22)

>	$IsRegularPGroup (QuasiDihedralGroup (11))$

$false$

(23)

>	$A ≔ QuasiDihedralGroup (3,'form' = fpgroup)$

$A ≔ {QD}_{3}$

(24)

>	$B ≔ SemiDihedralGroup (4,'form' = fpgroup)$

$B ≔ {SD}_{4}$

(25)

>	$AreIsomorphic (A, B)$

$true$

(26)

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