Generalized quaternion group:Q16

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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 $z$.

We can thus use an alternative presentation that requires only two generators:

$G:=⟨a,b\mid a^4=b^2=abab⟩$

Equivalently, it is the dicyclic group of order $16$.

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
rank of a p-group	1	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	all abelian subgroups are cyclic.
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.

Arithmetic functions of a counting nature

Function	Value	Similar groups	Explanation
number of conjugacy classes	7	groups with same order and number of conjugacy classes | groups with same prime-base logarithm of order and number of conjugacy classes | groups with same number of conjugacy classes	As $Q_{2^n},n=4$: $2^{n-2}+3=2^2+3=7$
number of equivalence classes under rational conjugacy	6	groups with same order and number of equivalence classes under rational conjugacy | groups with same prime-base logarithm of order and number of equivalence classes under rational conjugacy | groups with same number of equivalence classes under rational conjugacy
number of conjugacy classes of rational elements	5	groups with same order and number of conjugacy classes of rational elements | groups with same prime-base logarithm of order and number of conjugacy classes of rational elements | groups with same number of conjugacy classes of rational elements

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	$a,b$ don't commute.
metacyclic group	Yes	$⟨a⟩$ is cyclic of order eight, quotient group is cyclic of order two.
metabelian group	Yes	follows from being metacyclic.

Other properties

Property	Meaning	Satisfied?	Explanation	Comment
finite group with periodic cohomology	finite group in which every abelian subgroup is cyclic	Yes
Schur-trivial group	the Schur multiplier is trivial	Yes	follows from having periodic cohomology
maximal class group	finite p-group of class more than one whose class is one less than the prime-base logarithm of order	Yes	class is 3, prime-base logarithm of order is 4.
UL-equivalent group	upper central series and lower central series coincide.
stem group	the center is contained in the derived subgroup	Yes	follows from being a non-abelian UL-equivalent group.
directly indecomposable group	nontrivial and cannot be expressed as an internal direct product of nontrivial subgroups	Yes
centrally indecomposable group	nontrivial and cannot be expressed as a internal central product of proper nontrivial subgroups	Yes
splitting-simple group	nontrivial and cannot be expressed as an internal semidirect product of nontrivial subgroups	Yes

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

Cayley table

As the dicyclic group of order 16

With the presentation $⟨a,x\mid a^8=e,x^2=a^4,xa{x}^{-1}=a^{-1}⟩$, this group has Cayley table:

	$e$	$x$	$a$	$ax$	$a^2$	$a^{2}x$	$a^3$	$a^{3}x$	$a^4$	$a^{4}x$	$a^5$	$a^{5}x$	$a^6$	$a^{6}x$	$a^7$	$a^{7}x$
$e$	$e$	$x$	$a$	$ax$	$a^2$	$a^{2}x$	$a^3$	$a^{3}x$	$a^4$	$a^{4}x$	$a^5$	$a^{5}x$	$a^6$	$a^{6}x$	$a^7$	$a^{7}x$
$x$	$x$	$a^4$	$a^{7}x$	$a^3$	$a^{6}x$	$a^2$	$a^{5}x$	$a$	$a^{4}x$	$e$	$a^{3}x$	$a^7$	$a^{2}x$	$a^6$	$ax$	$a^5$
$a$	$a$	$ax$	$a^2$	$a^{2}x$	$a^3$	$a^{3}x$	$a^4$	$a^{4}x$	$a^5$	$a^{5}x$	$a^6$	$a^{6}x$	$a^7$	$a^{7}x$	$e$	$x$
$ax$	$ax$	$a^5$	$x$	$a^4$	$a^{7}x$	$a^3$	$a^{6}x$	$a^2$	$a^{5}x$	$a$	$a^{4}x$	$e$	$a^{3}x$	$a^7$	$a^{2}x$	$a^6$
$a^2$	$a^2$	$a^{2}x$	$a^3$	$a^{3}x$	$a^4$	$a^{4}x$	$a^5$	$a^{5}x$	$a^6$	$a^{6}x$	$a^7$	$a^{7}x$	$e$	$x$	$a$	$ax$
$a^{2}x$	$a^{2}x$	$a^6$	$ax$	$a^5$	$x$	$a^4$	$a^{7}x$	$a^3$	$a^{6}x$	$a^2$	$a^{5}x$	$a$	$a^{4}x$	$e$	$a^{3}x$	$a^7$
$a^3$	$a^3$	$a^{3}x$	$a^4$	$a^{4}x$	$a^5$	$a^{5}x$	$a^6$	$a^{6}x$	$a^7$	$a^{7}x$	$e$	$x$	$a$	$ax$	$a^2$	$a^{2}x$
$a^{3}x$	$a^{3}x$	$a^7$	$a^{2}x$	$a^6$	$ax$	$a^5$	$x$	$a^4$	$a^{7}x$	$a^3$	$a^{6}x$	$a^2$	$a^{5}x$	$a$	$a^{4}x$	$e$
$a^4$	$a^4$	$a^{4}x$	$a^5$	$a^{5}x$	$a^6$	$a^{6}x$	$a^7$	$a^{7}x$	$e$	$x$	$a$	$ax$	$a^2$	$a^{2}x$	$a^3$	$a^{3}x$
$a^{4}x$	$a^{4}x$	$e$	$a^{3}x$	$a^7$	$a^{2}x$	$a^6$	$ax$	$a^5$	$x$	$a^4$	$a^{7}x$	$a^3$	$a^{6}x$	$a^2$	$a^{5}x$	$a$
$a^5$	$a^5$	$a^{5}x$	$a^6$	$a^{6}x$	$a^7$	$a^{7}x$	$e$	$x$	$a$	$ax$	$a^2$	$a^{2}x$	$a^3$	$a^{3}x$	$a^4$	$a^{4}x$
$a^{5}x$	$a^{5}x$	$a$	$a^{4}x$	$e$	$a^{3}x$	$a^7$	$a^{2}x$	$a^6$	$ax$	$a^5$	$x$	$a^4$	$a^{7}x$	$a^3$	$a^{6}x$	$a^2$
$a^6$	$a^6$	$a^{6}x$	$a^7$	$a^{7}x$	$e$	$x$	$a$	$ax$	$a^2$	$a^{2}x$	$a^3$	$a^{3}x$	$a^4$	$a^{4}x$	$a^5$	$a^{5}x$
$a^{6}x$	$a^{6}x$	$a^2$	$a^{5}x$	$a$	$a^{4}x$	$e$	$a^{3}x$	$a^7$	$a^{2}x$	$a^6$	$ax$	$a^5$	$x$	$a^4$	$a^{7}x$	$a^3$
$a^7$	$a^7$	$a^{7}x$	$e$	$x$	$a$	$ax$	$a^2$	$a^{2}x$	$a^3$	$a^{3}x$	$a^4$	$a^{4}x$	$a^5$	$a^{5}x$	$a^6$	$a^{6}x$
$a^{7}x$	$a^{7}x$	$a^3$	$a^{6}x$	$a^2$	$a^{5}x$	$a$	$a^{4}x$	$e$	$a^{3}x$	$a^7$	$a^{2}x$	$a^6$	$ax$	$a^5$	$x$	$a^4$

GAP implementation

Group ID

This finite group has order 16 and has ID 9 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(16,9)

For instance, we can use the following assignment in GAP to create the group and name it $G$:

gap> G := SmallGroup(16,9);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [16,9]

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 14 among the groups of order 16. This information can be used to construct the group in GAP using the Gap3CatalogueGroup function as follows:

Gap3CatalogueGroup(16,14)

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 $G$:

gap> G := Gap3CatalogueGroup(16,14);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's Gap3CatalogueIdGroup function:

Gap3CatalogueIdGroup(G) = [16,14]

or just do:

Gap3CatalogueIdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.

Description by presentation

gap> F := FreeGroup(2);
<free group on the generators [ f1, f2 ]>
gap> G := F/[F.1^8,F.2^2*F.1^(-4),F.2*F.1*F.2^(-1)*F.1];
<fp group on the generators [ f1, f2 ]>
gap> IdGroup(G);
[ 16, 9 ]

Other descriptions

Description	Functions used
SylowSubgroup(SL(2,7),2)	SylowSubgroup, SL