# 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:

.

## 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

## Group properties

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

abelian group | No | ||

group of prime power order | Yes | ||

nilpotent group | Yes | ||

group in which every normal subgroup is characteristic | Yes | ||

maximal class group | Yes | ||

directly indecomposable group | Yes | ||

centrally indecomposable group | Yes | ||

splitting-simple group | No |

## Subgroups

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

Here is a list of subgroups.

- The trivial subgroup. Isomorphic to trivial group. (1).
- The subgroup of order two generated by . This is the center of the whole group. Isomorphic to cyclic group:Z2. (1)
- Subgroups of order two generated by the elements . These are all conjugate subgroups. Isomorphic to cyclic group:Z2. (4)
- Subgroups of order four generated by the elements . These are conjugate subgroups. Note that contains . Isomorphic to cyclic group:Z4. (2)
- Subgroup of order four generated by the element . This is a characteristic subgroup and is the commutator subgroup of the whole group. Isomorphic to cyclic group:Z4. (1)
- Subgroups of order four: . These are both conjugate subgroups. Isomorphic to Klein four-group. (2)
- Cyclic subgroup of order eight. Isomorphic to cyclic group:Z8. (1)
- Subgroup of order eight. Isomorphic to dihedral group:D8. (1)
- Subgroup of order eight. Isomorphic to quaternion group. (1)
- The whole group. (1)

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.

## Quotient groups

- The group itself. Obtained as the quotient by the trivial subgroup. (1)
- The quotient by the center, or the inner automorphism group. Isomorphic to dihedral group:D8. (1)
- The quotient by the commutator subgroup, or the abelianization. Isomorphic to Klein four-group. (1)
- The quotient by the cyclic subgroup of order eight. Isomorphic to cyclic group:Z2. (1)
- The quotient by the dihedral subgroup of order eight. Isomorphic to cyclic group:Z2. (1)
- The quotient group by the quaternion group. Isomorphic to cyclic group:Z2. (1)
- The trivial group. (1)

## 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.