M16

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
View a complete list of particular groups (this is a very huge list!)[SHOW MORE]

VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
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 group, sometimes denoted $M_{16}$ , is defined as follows:

$M_{16} = ⟨ a, x ∣ a^{8} = x^{2} = e, x a x = a^{5} ⟩$ .

Here, $e$ denotes the identity element.

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., $6^{t h}$ among groups of order 16
Hall-Senior number	12 among groups of order 16
Hall-Senior symbol	$16 Γ_{2} d$

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

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
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

Elements

Further information: element structure of 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	$⟨ a ⟩$
${a^{3}, a^{7}}$	2	8	$⟨ a ⟩$
${x, a^{4} x}$	2	2	$⟨ a^{2}, x ⟩$
${a^{2} x, a^{6} x}$	2	4	$⟨ a^{2}, x ⟩$
${a x, a^{5} x}$	2	8	$⟨ a x ⟩$
${a^{3} x, a^{7} x}$	2	8	$⟨ a x ⟩$

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
${a, a^{3}, a^{5}, a^{7}, a x, a^{3} x, a^{5} x, a^{7} x}$	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 M16

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

$M_{16} = ⟨ a, x ∣ a^{8} = x^{2} = e, x a x = a^{5} ⟩$ .

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

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.

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 ]