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
: 
: 
which becomes 
. See 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 |
