Modular maximal-cyclic group:M16

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Definition

The group, sometimes denoted $M_{16}$ or $M_{4}(2)$, called the modular-maximal cyclic group of order 16, or the modular group of order 16, is defined as follows:

$M_{16}=\langle a,x\mid a^{8}=x^{2}=e,xax^{-1}=a^{5}\rangle$.

Here, $e$ denotes the identity element.

It is a certain group of order 16. It is in the family of modular maximal-cyclic groups.

Position in classifications

Get more information about groups of the same order at Groups of order 16#The list

Type of classification	Position/number in classification
GAP ID	$(16,6)$, i.e., $6^{th}$ among groups of order 16
Hall-Senior number	12 among groups of order 16
Hall-Senior symbol	$16\Gamma _{2}d$

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	2	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, 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	2	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	2	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	There is a unique (hence characteristic) Klein four-subgroup.

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	$a,x$ do not commute
metacyclic group	Yes	has cyclic subgroup $\langle a\rangle$ of order eight, quotient group isomorphic to $\langle x\rangle$ is cyclic of order two.
metabelian group	Yes	follows from being metacyclic.

Other properties

Property	Satisfied?	Explanation	Comment
finite group that is 1-isomorphic to an abelian group	Yes	via cocycle halving generalization of Baer correspondence	See element structure of groups of order 16#1-isomorphism
Schur-trivial group	Yes
directly indecomposable group	Yes	cannot be expressed as a direct product of proper subgroups; all proper subgroups are abelian, but the group is non-abelian.
splitting-simple group	No	it is an internal semidirect product of $\langle a\rangle$ (a normal subgroup) and $\langle x\rangle$.
centrally indecomposable group	Yes	it cannot be expressed as a central product of strictly smaller subgroups.
group of nilpotency class two	Yes
group in which every normal subgroup is characteristic	No	$\langle a\rangle$ is normal but not characteristic.
ambivalent group	No	$a,a^{-1}$ are not conjugate.	This also means that it is not a rational group or rational-representation group.

Elements

Further information: element structure of modular maximal-cyclic group:M16

Conjugacy class structure

Conjugacy class	Size of conjugacy class	Order of elements in conjugacy class	Centralizer of first element of class
$\{e\}$	1	1	whole group
$\{a^{4}\}$	1	2	whole group
$\{a^{2}\}$	1	4	whole group
$\{a^{6}\}$	1	4	whole group
$\{a,a^{5}\}$	2	8	$\langle a\rangle$
$\{a^{3},a^{7}\}$	2	8	$\langle a\rangle$
$\{x,a^{4}x\}$	2	2	$\langle a^{2},x\rangle$
$\{a^{2}x,a^{6}x\}$	2	4	$\langle a^{2},x\rangle$
$\{ax,a^{5}x\}$	2	8	$\langle ax\rangle$
$\{a^{3}x,a^{7}x\}$	2	8	$\langle ax\rangle$

Automorphism class structure

Equivalence class under automorphisms	Size of equivalence class	Number of conjugacy classes in it	Size of each conjugacy class	Characterization(s) up to 1-isomorphism	Characterization(s) involving commutation relationships
$\!\{e\}$	1	1	1	identity element	identity element
$\!\{a^{4}\}$	1	1	1	unique non-identity element that is a fourth power	unique non-identity element that is a commutator
$\!\{a^{2},a^{6}\}$	2	2	1	squares that are not fourth powers	elements in the center but not in the derived subgroup
$\!\{a,a^{3},a^{5},a^{7},ax,a^{3}x,a^{5}x,a^{7}x\}$	8	4	2	elements of order eight
$\!\{x,a^{4}x\}$	2	1	2	elements of order two that are not squares
$\!\{a^{2}x,a^{6}x\}$	2	1	2	elements of order four that are not squares

1-isomorphism

The group is 1-isomorphic to the group direct product of Z8 and Z2. In other words, there is a bijection between the groups that restricts to an isomorphism on all cyclic subgroups on either side. The 1-isomorphism is explained by the cocycle halving generalization of Baer correspondence, where the intermediary is a class two Lie cring.

Subgroups

Further information: subgroup structure of modular maximal-cyclic group:M16

To describe subgroups, we use the defining presentation given at the beginning:

$M_{16}=\langle a,x\mid a^{8}=x^{2}=e,xax=a^{5}\rangle$.

Automorphism class of subgroups	List of subgroups	Isomorphism class	Order of subgroups	Index of subgroups	Number of conjugacy classes(=1 iff automorph-conjugate subgroup)	Size of each conjugacy class(=1 iff normal subgroup)	Total number of subgroups(=1 iff characteristic subgroup)	Isomorphism class of quotient (if exists)	Subnormal depth	Nilpotency class
trivial subgroup	$\{e\}$	trivial group	1	16	1	1	1	M16	1	0
derived subgroup of M16	$\{e,a^{4}\}$	cyclic group:Z2	2	8	1	1	1	direct product of Z4 and Z2	1	1
other subgroups of order two	$\{e,x\}$, $\{e,a^{4}x\}$	cyclic group:Z2	2	8	1	2	2	--	2	1
center of M16	$\{e,a^{2},a^{4},a^{6}\}$	cyclic group:Z4	4	4	1	1	1	Klein four-group	1	1
other cyclic subgroup of order 4	$\{e,a^{2}x,a^{4},a^{6}x\}$	cyclic group:Z4	4	4	1	1	1	cyclic group:Z4	1	1
V4 in M16	$\{e,x,a^{4},a^{4}x\}$	Klein four-group	4	4	1	1	1	cyclic group:Z4	1	1
Z8 in M16	$\{e,a,a^{2},a^{3},a^{4},a^{5},a^{6},a^{7}\}$ $\{e,ax,a^{6},a^{7}x,a^{4},a^{5}x,a^{2},a^{3}x\}$	cyclic group:Z8	8	2	2	1	2	cyclic group:Z2	1	1
direct product of Z4 and Z2 in M16	$\{e,a^{2},a^{4},a^{6},x,a^{2}x,a^{4}x,a^{6}x\}$	direct product of Z4 and Z2	8	2	1	1	1	cyclic group:Z2	1	1
whole group	all elements	M16	16	1	1	1	1	trivial group	1	1
Total (9 rows)	--	--	--	--	10	--	11	--	--	--

Distinguishing features

Smallest of its kind

• This is a minimum order example of a non-abelian finite group that is 1-isomorphic to an abelian group -- it is 1-isomorphic to direct product of Z8 and Z2. It is, however, not the only such example: the other example is central product of D8 and Z4. See element structure of groups of order 16#1-isomorphism for more details.
• This is a minimum order example of a nilpotent group that is not a UL-equivalent group, i.e., the upper central series and lower central series are not the same. However, it is not the only example. In fact, all groups of order 16 and class two share this property with it. The other examples are SmallGroup(16,3), nontrivial semidirect product of Z4 and Z4, direct product of D8 and Z2, direct product of Q8 and Z2, and central product of D8 and Z4.
• This is a minimum order example of a situation where a group has two characteristic subgroups that are both isomorphic to each other but are distinct. Both these are cyclic subgroups of order four. It is, however, not the only example of this order; there are other examples where a similar behavior occurs, albeit for different orders of subgroups -- for instance, order 2 (in nontrivial semidirect product of Z4 and Z4).

Linear representation theory

Further information: linear representation theory of modular maximal-cyclic group:M16

GAP implementation

Group ID

This finite group has order 16 and has ID 6 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(16,6)

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

gap> G := SmallGroup(16,6);

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

IdGroup(G) = [16,6]

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

Gap3CatalogueGroup(16,11)

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(16,11);

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

Gap3CatalogueIdGroup(G) = [16,11]

or just do:

Gap3CatalogueIdGroup(G)

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

Description by presentation

The group can be defined using a presentation as follows:

gap> F := FreeGroup(2);;
gap> G := F/[F.1^8,F.2^2,F.2*F.1*F.2*F.1^(-5)];
<fp group on the generators [ f1, f2 ]>
gap> IdGroup(G);
[ 16, 6 ]