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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

Parameters

Description

Examples

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,|,xn=y2,,,xy=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 2n1, 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 2n1 is returned.

Examples

withGroupTheory:

GSemiDihedralGroup21

GroupTheory:-SemiDihedralGroup21,form=permgroup

(1)

GroupOrderG

168

(2)

ClassNumberG

48

(3)

IsNilpotentG

false

(4)

IsSupersolubleG

true

(5)

GroupOrderDerivedSubgroupG

21

(6)

IsCyclicDerivedSubgroupG

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.

IsCyclicCenterG

true

(8)

GroupOrderCenterG

4

(9)

For even n, the center has order 2.

GroupOrderCenterSemiDihedralGroup20

2

(10)

seqGroupOrderCenterSemiDihedralGroupn,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.

IsTransitiveG

true

(12)

IsPrimitiveG

false

(13)

BlocksG

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)

orseqIsPrimitiveSemiDihedralGroupn,n=2..20

false

(15)

andseqIsTransitiveSemiDihedralGroupn,n=2..20

true

(16)

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

GSemiDihedralGroup6,'form'=fpgroup

GroupTheory:-SemiDihedralGroup6,form=fpgroup

(17)

GroupOrderG

48

(18)

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

AreIsomorphicSemiDihedralGroup4,DihedralGroup16

false

(19)

DisplayCharacterTableSemiDihedralGroup5

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

2I

2I

0

0

−135−1251

−125−135

−135−125

−135+−125+1

−1110+−1910

−1310−1710

−1110−1910

−1310+−1710

χ__10

2

−2

0

0

2I

2I

0

0

−125−135

−135−1251

−135+−125+1

−135−125

−1310−1710

−1110+−1910

−1310+−1710

−1110−1910

χ__11

2

−2

0

0

2I

2I

0

0

−135−1251

−125−135

−135−125

−135+−125+1

−1110−1910

−1310+−1710

−1110+−1910

−1310−1710

χ__12

2

−2

0

0

2I

2I

0

0

−125−135

−135−1251

−135+−125+1

−135−125

−1310+−1710

−1110−1910

−1310−1710

−1110+−1910

χ__13

2

2

0

0

−2

−2

0

0

−135−1251

−125−135

−125−135

−135−1251

−135−125

−135+−125+1

−135−125

−135+−125+1

χ__14

2

2

0

0

−2

−2

0

0

−125−135

−135−1251

−135−1251

−125−135

−135+−125+1

−135−125

−135+−125+1

−135−125

χ__15

2

2

0

0

2

2

0

0

−135−1251

−125−135

−125−135

−135−1251

−125−135

−135−1251

−125−135

−135−1251

χ__16

2

2

0

0

2

2

0

0

−125−135

−135−1251

−135−1251

−125−135

−135−1251

−125−135

−135−1251

−125−135

GQuasiDihedralGroup2

GroupTheory:-QuasiDihedralGroup2,form=permgroup

(20)

GroupOrderG

16

(21)

seqGroupOrderQuasiDihedralGroupn,n=2..8

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

(22)

IsRegularPGroupQuasiDihedralGroup11

false

(23)

AQuasiDihedralGroup3,'form'=fpgroup

GroupTheory:-QuasiDihedralGroup3,form=fpgroup

(24)

BSemiDihedralGroup4,'form'=fpgroup

GroupTheory:-SemiDihedralGroup4,form=fpgroup

(25)

AreIsomorphicA,B

true

(26)

See Also

GroupTheory[CharacterTable]

GroupTheory[ClassNumber]

GroupTheory[GroupOrder]