Semidihedral group:SD16

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Definition

The semidihedral group $S{D}_{16}$ is the semidihedral group of order $16$. Specifically, it has the following presentation:

$S{D}_{16}:=⟨a,x\mid a^8=x^2=e,xa{x}^{-1}=a^3⟩$.

Subgroups

Further information: Subgroup structure of semidihedral group:SD16

Here is a list of subgroups.

1. The trivial subgroup. Isomorphic to trivial group. (1).
2. The subgroup of order two generated by $a^4$. This is the center of the whole group. Isomorphic to cyclic group:Z2. (1)
3. Subgroups of order two generated by the elements $x,a^{2}x,a^{4}x,a^{6}x$. These are all conjugate subgroups. Isomorphic to cyclic group:Z2. (4)
4. Subgroups of order four generated by the elements $ax,a^{3}x$. These are conjugate subgroups. Note that $⟨ax⟩$ contains $e,ax,a^4,a^{5}x$. Isomorphic to cyclic group:Z4. (2)
5. Subgroup of order four generated by the element $a^2$. This is a characteristic subgroup and is the commutator subgroup of the whole group. Isomorphic to cyclic group:Z4. (1)
6. Subgroups of order four: $⟨a^4,x⟩,⟨a^4,a^{2}x⟩$. These are both conjugate subgroups. Isomorphic to Klein four-group. (2)
7. Cyclic subgroup $⟨a⟩$ of order eight. Isomorphic to cyclic group:Z8. (1)
8. Subgroup $⟨a^2,x⟩$ of order eight. Isomorphic to dihedral group:D8. (1)
9. Subgroup $⟨a^2,ax⟩$ of order eight. Isomorphic to quaternion group. (1)
10. 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

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

gap> G := SmallGroup(16,8);

Conversely, to check whether a given group $G$ 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.