SmallGroup(32,35)

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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 quaternion group-like variant of the generalized dihedral group of direct product of Z4 and Z4. It can also be described as the semidirect product $Z_4 \rtimes Q_8$. It is given by the presentation:

$\langle a,i,j \mid a^4 = i^4 = j^4 = e, i^2 = j^2, ja = aj, iji^{-1} = j^{-1}, iai^{-1} = a^{-1} \rangle$.

The group can also be described (up to isomorphism) as a subgroup of the direct product of Q8 and Q8 that is not isomorphic to the direct product of Q8 and Z4.

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)	32	groups with same 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 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	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
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
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

Group properties

Property	Satisfied?	Explanation
Cyclic group	No
Abelian group	No
Metacyclic group	No
Group of nilpotency class two	Yes
Frattini-in-center group	Yes
Special group	Yes
Extraspecial group	No

GAP implementation

Group ID

This finite group has order 32 and has ID 35 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(32,35)

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

gap> G := SmallGroup(32,35);

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

IdGroup(G) = [32,35]

or just do:

IdGroup(G)

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

Other descriptions

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