Generalized quaternion group:Q16

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This particular group is a finite group of order: 16

Definition

The group $Q_{16}$, sometimes termed the generalized quaternion group of order $16$, is a generalized quaternion group. It can be described by the following presentation:

$G:=⟨a,b,c\mid a^4=b^2=c^2=abc⟩$.

Note that $c=ab=b{a}^{-1}$ from these relations, and $ba{b}^{-1}=a^{-1}$. This in turn forces that $b^2=b(b^2)b^{-1}=b{a}^4{b}^{-1}=a^{-4}=b^{-2}$, forcing $b^2=a^4=c^2=abc$ to have order two. We shall denote this element of order two, which is clearly central, as $z$.

Arithmetic functions

Function	Value	Explanation
order	16
exponent	8	Cyclic subgroup of order eight.
nilpotency class	3
derived length	2
Frattini length	3
Fitting length	1
minimum size of generating set	2	$a$ and $b$.
subgroup rank	2
max-length	4
rank as p-group	1	All abelian subgroups are cyclic.
normal rank	1	All abelian normal subgroups is cyclic.
characteristic rank of a p-group	1	All abelian characteristic subgroups are cyclic.

Subgroups

Further information: Subgroup structure of generalized quaternion group:Q16

1. The trivial subgroup. Isomorphic to trivial group. (1)
2. The center, which is a subgroup of order two, generated by $z=a^4=b^2=c^2$. Isomorphic to cyclic group:Z2. (1)
3. The cyclic subgroup of order four generated by $a^2$. Isomorphic to cyclic group:Z4. (1)
4. The four cyclic subgroups of order four, namely: $⟨b⟩$, $⟨ab⟩$, $⟨a^{2}b⟩$ and $⟨a^{3}b⟩$. These come in two conjugacy classes of 2-subnormal subgroups, one conjugacy class comprising $⟨ab⟩$ and $⟨a^{3}b⟩$ and the other comprising $⟨b⟩$ and $⟨a^{2}b⟩$. Isomorphic to cyclic group:Z4. (4)
5. The cyclic subgroup of order eight, generated by $a$. This is characteristic; in fact, it equals the centralizer of commutator subgroup. Isomorphic to cyclic group:Z8. (1)
6. Two quaternion groups of order eight, namely $⟨a^2,b⟩$ and $⟨a^2,ab⟩$. Isomorphic to quaternion group. (2)
7. The whole 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	(3)		cyclic group:Z4
Frattini subgroup	(3)		cyclic group:Z4
Socle	(2)		cyclic group:Z2
Join of abelian subgroups of maximum order	(5)		cyclic group:Z8
Join of abelian subgroups of maximum rank	(7)		whole group
Join of elementary abelian subgroups of maximum order	(2)		cyclic group:Z2

GAP implementation

Group ID

The generalized quaternion group of order $16$ has ID $9$. In other words, it can be described using the SmallGroup function as:

SmallGroup(16,9)