SmallGroup(32,24): Difference between revisions

Latest revision as of 18:10, 31 January 2021

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
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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

This group is a semidirect product $(Z_{4} \times Z_{4}) ⋊ Z_{2}$ , and can be defined by the following presentation:

$⟨ a, b, c ∣ a^{4} = b^{4} = c^{2} = e, a b = b a, a c = c a, c b c^{- 1} = a^{2} b ⟩$

It can alternatively be defined in the following equivalent ways:

It is the central product of SmallGroup(16,3) and cyclic group:Z4 over a common cyclic central subgroup of order two.
It is the central product of nontrivial semidirect product of Z4 and Z4 (ID: (16,4)) and cyclic group:Z4 over a common cyclic central subgroup of order two.

Arithmetic functions

Function	Value	Similar groups
underlying prime of p-group	2
order (number of elements, equivalently, cardinality or size of underlying set)	32	groups with same order
prime-base logarithm of order	5	groups with same prime-base logarithm of order
exponent of a group	4	groups with same order and exponent of a group \| groups with same exponent of a group
prime-base logarithm of exponent	2	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
Frattini length	2	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	3	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	3	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	3	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
normal rank of a p-group	3	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
characteristic rank of a p-group	3	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

Group properties

Template:Compare and contrast group property

Property	Satisfied?	Explanation
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
metabelian group	Yes
finite group that is 1-isomorphic to an abelian group	Yes	via cocycle halving generalization of Baer correspondence

GAP implementation

Group ID

This finite group has order 32 and has ID 24 among the groups of order 32 in GAP's SmallGroup library. For context, there are groups of order 32. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(32,24)

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

gap> G := SmallGroup(32,24);

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

IdGroup(G) = [32,24]

or just do:

IdGroup(G)

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

Description by presentation

Here is the GAP code to define this group using a presentation:

gap> F := FreeGroup(3);
<free group on the generators [ f1, f2, f3 ]>
gap> G := F/[F.1^4,F.2^4, F.1*F.2*F.1^(-1)*F.2^(-1), F.3^2, F.1*F.3*F.1^(-1)*F.3^(-1),F.3*F.2*F.3^(-1)*F.2^(-1)*F.1^2 ];
<fp group on the generators [ f1, f2, f3 ]>

@@ Line 3: / Line 3: @@
 ==Definition==
-This group is defined by the following [[presentation of a group|presentation]]:
+This group is a [[semidirect product]] <math>(Z_4 \times Z_4) \rtimes Z_2</math>, and can be defined by the following [[presentation of a group|presentation]]:
 <math>\langle a,b,c \mid a^4 = b^4 = c^2 = e, ab = ba, ac = ca, cbc^{-1} = a^2b \rangle</math>
-==Group properties==
+It can alternatively be defined in the following equivalent ways:
+* It is the [[central product]] of [[SmallGroup(16,3)]] and [[cyclic group:Z4]] over a common cyclic central subgroup of order two.
+* It is the [[central product]] of [[nontrivial semidirect product of Z4 and Z4]] (ID: (16,4)) and [[cyclic group:Z4]] over a common cyclic central subgroup of order two.
+==Arithmetic functions==
 {| class="sortable" border="1"
-! Property !! Satisfied? !! Explanation
+! Function !! Value !! Similar groups !! Explanation
 |-
-| [[dissatisfies property::cyclic group]] || No ||
+| [[underlying prime of p-group]] || [[arithmetic function value::underlying prime of p-group;2|2]] || ||
 |-
-| [[dissatisfies property::abelian group]] || No ||
+| {{arithmetic function value order|32}}
 |-
-| [[dissatisfies property::metacyclic group]] No ||
+| {{arithmetic function value order p-log|5}}
 |-
-| [[satisfies property::group of nilpotency class two]] || Yes ||
+| {{arithmetic function value given order|exponent of a group|4|32}}
 |-
-| [[satisfies property::metabelian group]] || Yes ||
+| {{arithmetic function value given order and p-log|prime-base logarithm of exponent|2|32|5}} ||
-|}
+|-
+| {{arithmetic function value given order and p-log|nilpotency class|2|32|5}} ||
-==Arithmetic functions==
+|-
+| {{arithmetic function value given order and p-log|derived length|2|32|5}} ||
-{| class="sortable" border="1"
-! Function !! Value !! Explanation
 |-
-| [[order of a group|order]] || [[arithmetic function value::order of a group;32|32]] ||
+| {{arithmetic function value given order and p-log|Frattini length|2|32|5}} ||
 |-
-| [[exponent of a group|exponent]] || [[arithmetic function value::exponent of a group;4|4]] ||
+| {{arithmetic function value given order and p-log|minimum size of generating set|3|32|5}} ||
 |-
-| [[nilpotency class]] || [[arithmetic function value::nilpotency class;2|2]] ||
+| {{arithmetic function value given order and p-log|subgroup rank of a group|3|32|5}} ||
 |-
-| [[derived length]] || [[arithmetic function value::derived length;2|2]] ||
+| {{arithmetic function value given order and p-log|rank of a p-group|3|32|5}} ||
 |-
-| [[Frattini length]] || [[arithmetic function value::Frattini length;2|2]] ||
+| {{arithmetic function value given order and p-log|normal rank of a p-group|3|32|5}} ||
 |-
-| [[minimum size of generating set]] || [[arithmetic function value::minimum size of generating set;3|3]] ||
+| {{arithmetic function value given order and p-log|characteristic rank of a p-group|3|32|5}} ||
+|}
+==Group properties==
+{{compare and contrast group property|order = 32}}
+{| class="sortable" border="1"
+! Property !! Satisfied? !! Explanation !! Comment
 |-
-| [[subgroup rank of a group|subgroup rank]] || [[arithmetic function value::subgroup rank of a group;3|3]] ||
+| {{group properties because p-group}}
 |-
-| [[rank of a p-group|rank as p-group]] || [[arithmetic function value::rank of a p-group;3|3]] ||
+| [[dissatisfies property::abelian group]] || No ||  ||
 |-
-| [[normal rank of a p-group|normal rank as p-group]] || [[arithmetic function value::normal rank of a p-group;3|3]] ||
+| [[satisfies property::metabelian group]] || Yes || ||
 |-
-| [[characteristic rank of a p-group|characteristic rank as p-group]] || [[arithmetic function value::characteristic rank of a p-group;3|3]] ||
+| [[satisfies property::finite group that is 1-isomorphic to an abelian group]] || Yes || via [[cocycle halving generalization of Baer correspondence]] ||
 |}
 ==GAP implementation==