SmallGroup(32,2)

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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 can be defined by the following presentation:

$G := \langle a_1,a_2,a_3 \mid a_1^4 = a_2^4 = a_3^2 = e, [a_1,a_2] = a_3, [a_1,a_3] = [a_2,a_3] = e \rangle$

where $e$ denotes the identity element and $[ , ]$ stands for the commutator of two elements (the isomorphism type of the group is independent of the choice of commutator map).

See the section #Description by presentation for how to construct the group using this presentation in GAP.

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 32#Arithmetic functions

Function	Value	Similar groups	Explanation for function value	GAP verification (set G := SmallGroup(32,2);) -- see more at #GAP implementation
underlying prime of p-group	2
order (number of elements, equivalently, cardinality or size of underlying set)	32	groups with same order		Order(G); using Order.
prime-base logarithm of order	5	groups with same prime-base logarithm of order
max-length of a group	5		max-length of a group equals prime-base logarithm of order for group of prime power order
chief length	5		chief length equals prime-base logarithm of order for group of prime power order
composition length	5		composition length equals prime-base logarithm of order for group of prime power order
exponent of a group	4	groups with same order and exponent of a group | groups with same prime-base logarithm of order and exponent of a group | groups with same exponent of a group		Exponent(G); using Exponent.
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
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		FrattiniLength(G); using FrattiniLength.
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		NilpotencyClassOfGroup(G); using NilpotencyClassOfGroup.
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		DerivedLength(G); using DerivedLength.
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		Rank(G); using Rank.
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		RankAsPGroup(G); using RankAsPGroup.
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		NormalRank(G); using NormalRank.
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		CharacteristicRank(G); using CharacteristicRank.

Group properties

Want to compare and contrast group properties with other groups of the same order? Check out groups of order 32#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
T-group	No
monolithic group	No
one-headed group	No
group of nilpotency class two	Yes
metabelian group	Yes
finite group that is 1-isomorphic to an abelian group	Yes	via cocycle skew reversal generalization of Baer correspondence

GAP implementation

Group ID

This finite group has order 32 and has ID 2 among the groups of order 32 in GAP's SmallGroup library. For context, there are 51 groups of order 32. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(32,2)

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

gap> G := SmallGroup(32,2);

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

IdGroup(G) = [32,2]

or just do:

IdGroup(G)

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

Description by presentation

gap> F := FreeGroup(3);
<free group on the generators [ f1, f2, f3 ]>
gap> G := F/[F.1^4,F.2^4,F.3^2,Comm(F.1,F.2)*F.3^(-1),Comm(F.1,F.3),Comm(F.2,F.3)];
<fp group on the generators [ f1, f2, f3 ]>
gap> IdGroup(G);
[ 32, 2 ]

GAP verification of function values and group properties

Below is a GAP implementation verifying the various function values and group properties as stated in this page. Before beginning, set G := SmallGroup(32,2); or set it to this group in any other manner:

gap> Order(G);
32
gap> Exponent(G);
4
gap> NilpotencyClassOfGroup(G);
2
gap> DerivedLength(G);
2
gap> FrattiniLength(G);
2
gap> Rank(G);
2
gap> RankAsPGroup(G);
3
gap> NormalRank(G);
3
gap> CharacteristicRank(G);
3