# Semidihedral group:SD16

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## Contents

## Definition

The semidihedral group (also denoted ) is the semidihedral group (also called **quasidihedral group**) of order . Specifically, it has the following presentation:

.

The group can also be defined as the general semilinear group of degree one over the field of nine elements.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 16#Arithmetic functions

Note that the order can also be computed using the formula for the order of where and : The order of with is , which becomes .

### Arithmetic functions of a counting nature

## Group properties

### Important properties

Property | Satisfied | Explanation | Comment |
---|---|---|---|

group of prime power order | Yes | ||

nilpotent group | Yes | prime power order implies nilpotent | |

supersolvable group | Yes | via nilpotent: finite nilpotent implies supersolvable | |

solvable group | Yes | via nilpotent: nilpotent implies solvable | |

abelian group | No | ||

metacyclic group | Yes | has a cyclic subgroup of order 8, with cyclic quotient group | |

metabelian group | Yes | follows from being metacyclic; also, from having class three. |

### Other properties

Property | Meaning | Satisfied? | Explanation | Comment |
---|---|---|---|---|

group in which every normal subgroup is characteristic | every normal subgroup of the group is a characteristic subgroup of it. | Yes | ||

maximal class group | finite p-group that is non-abelian and whose nilpotency class is one less than its prime-base logarithm of order. | Yes | ||

UL-equivalent group | nilpotent group whose upper central series and lower central series coincide. | Yes | follows from being a maximal class group | |

stem group | center is contained in the derived subgroup. | Yes | ||

directly indecomposable group | nontrivial and cannot be expressed as an internal direct product of proper nontrivial subgroups. | Yes | ||

centrally indecomposable group | nontrivial and cannot be expressed as an internal central product of proper nontrivial subgroups. | Yes | ||

splitting-simple group | nontrivial and cannot be expressed as an internal semidirect product of proper nontrivial subgroups. | No | it is an internal semidirect product of the normal subgroup and the group . | |

finite group with periodic cohomology | finite group in which every abelian subgroup is cyclic. For the finite p-group case, this is equivalent to the rank being one. | No | ||

Schur-trivial group | the Schur multiplier is trivial. | Yes | ||

ambivalent group | every element is conjugate to its inverse. | No | are not conjugate. | |

ACIC-group | every automorph-conjugate subgroup is characteristic | No | is automorph-conjugate but not characteristic. | |

T-group | every subnormal subgroup is normal | No | is subnormal but not normal. | Also, not a Dedekind group. In fact, for nilpotent groups, being a T-group is equivalent to being a Dedekind group. |

## Subgroups

`Further information: Subgroup structure of semidihedral group:SD16`

There are a couple of interesting facts about this group:

- Every subgroup of this group is an automorph-conjugate subgroup.
- All the maximal subgroups are characteristic subgroups -- in fact, they are all isomorph-free subgroups.

## Subgroup-defining functions

Subgroup-defining function | Subgroup type in list | Page on subgroup embedding | Isomorphism class | Comment |
---|---|---|---|---|

Center | (2) | cyclic group:Z2 | ||

Commutator subgroup | (4) | cyclic group:Z4 | ||

Frattini subgroup | (4) | cyclic group:Z4 | ||

Socle | (2) | cyclic group:Z4 | ||

Join of abelian subgroups of maximum order | (7) | cyclic group:Z8 | ||

Join of abelian subgroups of maximum rank | (8) | dihedral group:D8 | ||

Join of elementary abelian subgroups of maximum order | (8) | dihedral group:D8 |

## GAP implementation

### Group ID

This finite group has order 16 and has ID 8 among the groups of order 16 in GAP's SmallGroup library. For context, there are 14 groups of order 16. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(16,8)`

For instance, we can use the following assignment in GAP to create the group and name it :

`gap> G := SmallGroup(16,8);`

Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:

`IdGroup(G) = [16,8]`

or just do:

`IdGroup(G)`

to have GAP output the group ID, that we can then compare to what we want.

### Hall-Senior number

This group of prime power order has order 16 and has Hall-Senior number 13 among the groups of order 16. This information can be used to construct the group in GAP using the Gap3CatalogueGroup function as follows:

`Gap3CatalogueGroup(16,13)`

WARNING: There is some disagreement between the GAP 3 catalogue numbers and the Hall-Senior numbers for some abelian groups, but it does not affect this group.

For instance, we can use the following assignment in GAP to create the group and name it :

`gap> G := Gap3CatalogueGroup(16,13);`

Conversely, to check whether a given group is in fact the group we want, we can use GAP's Gap3CatalogueIdGroup function:

`Gap3CatalogueIdGroup(G) = [16,13]`

or just do:

`Gap3CatalogueIdGroup(G)`

to have GAP output the group ID, that we can then compare to what we want.

### Description by presentation

The group can be defined using a presentation as follows:

gap> F := FreeGroup(2);; gap> G := F/[F.1^8,F.2^2,F.2*F.1*F.2*F.1^(-3)]; <fp group on the generators [ f1, f2 ]> gap> IdGroup(G); [ 16, 8 ]

### Other descriptions

Description | Functions used |
---|---|

SylowSubgroup(GL(2,3),2) |
SylowSubgroup, GL |