Semidihedral group:SD16

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VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

The semidihedral group $SD_{16}$ (also denoted $QD_{16}$) is the semidihedral group (also called quasidihedral group) of order $16$. Specifically, it has the following presentation:

$SD_{16} := \langle a,x \mid a^8 = x^2 = e, xax^{-1} = a^3 \rangle$.

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

Function	Value	Similar groups	Explanation for function value
underlying prime of p-group	2
order (number of elements, equivalently, cardinality or size of underlying set)	16	groups with same order
prime-base logarithm of order	4	groups with same prime-base logarithm of order
max-length of a group	4		max-length of a group equals prime-base logarithm of order for group of prime power order
chief length	4		chief length equals prime-base logarithm of order for group of prime power order
composition length	4		composition length equals prime-base logarithm of order for group of prime power order
exponent of a group	8	groups with same order and exponent of a group | groups with same exponent of a group	cyclic subgroup of order 8.
prime-base logarithm of exponent	3	groups with same order and prime-base logarithm of exponent | groups with same prime-base logarithm of order and prime-base logarithm of exponent | groups with same prime-base logarithm of exponent
nilpotency class	3	groups with same order and nilpotency class | groups with same prime-base logarithm of order and nilpotency class | groups with same nilpotency class
derived length	2	groups with same order and derived length | groups with same prime-base logarithm of order and derived length | groups with same derived length	the derived subgroup is contained in the cyclic subgroup and is hence abelian
Frattini length	3	groups with same order and Frattini length | groups with same prime-base logarithm of order and Frattini length | groups with same Frattini length
minimum size of generating set	2	groups with same order and minimum size of generating set | groups with same prime-base logarithm of order and minimum size of generating set | groups with same minimum size of generating set
subgroup rank of a group	2	groups with same order and subgroup rank of a group | groups with same prime-base logarithm of order and subgroup rank of a group | groups with same subgroup rank of a group	All proper subgroups are cyclic, dihedral, semidihedral or Klein four-groups.
rank of a p-group	2	groups with same order and rank of a p-group | groups with same prime-base logarithm of order and rank of a p-group | groups with same rank of a p-group	there exist Klein four-subgroups.
normal rank of a p-group	1	groups with same order and normal rank of a p-group | groups with same prime-base logarithm of order and normal rank of a p-group | groups with same normal rank of a p-group	all abelian normal subgroups are cyclic.
characteristic rank of a p-group	1	groups with same order and characteristic rank of a p-group | groups with same prime-base logarithm of order and characteristic rank of a p-group | groups with same characteristic rank of a p-group	All abelian characteristic subgroups are cyclic.

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.

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 $x, a^2x, a^4x, a^6x$. These are all conjugate subgroups. Isomorphic to cyclic group:Z2. (4)
4. Subgroups of order four generated by the elements $ax, a^3x$. These are conjugate subgroups. Note that $\langle ax\rangle$ contains $e, ax, a^4, a^5x$. 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: $\langle a^4,x \rangle, \langle a^4,a^2x \rangle$. These are both conjugate subgroups. Isomorphic to Klein four-group. (2)
7. Cyclic subgroup $\langle a \rangle$ of order eight. Isomorphic to cyclic group:Z8. (1)
8. Subgroup $\langle a^2, x \rangle$ of order eight. Isomorphic to dihedral group:D8. (1)
9. Subgroup $\langle a^2, ax \rangle$ 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 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 $G$:

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.