Generalized quaternion group:Q16

View a complete list of particular groups (this is a very huge list!)[SHOW MORE]

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:

$G := \langle a,b,c \mid a^4 = b^2 = c^2 = abc \rangle$.

Note that $c = ab = ba^{-1}$ from these relations, and $bab^{-1} = a^{-1}$. This in turn forces that $b^2 = b(b^2)b^{-1} = ba^4b^{-1} = a^{-4} = b^{-2}$, 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:

$G := \langle a,b \mid a^4 = b^2 = abab \rangle$

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
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 $\langle a \rangle$ 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: $\langle b \rangle$, $\langle ab \rangle$, $\langle a^2b \rangle$ and $\langle a^3b\rangle$. These come in two conjugacy classes of 2-subnormal subgroups, one conjugacy class comprising $\langle ab \rangle$ and $\langle a^3b \rangle$ and the other comprising $\langle b \rangle$ and $\langle a^2b \rangle$. 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 $\langle a^2,b \rangle$ and $\langle a^2, ab \rangle$. 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

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 14 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