Nontrivial semidirect product of Z9 and Z9

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Definition

This group is defined by the following presentation (here, $e$ denotes the identity element):

$\langle x,y\mid x^{9}=y^{9}=e,yxy^{-1}=x^{4}\rangle$

It is the case $p=3$ of the nontrivial semidirect product of cyclic groups of prime-square order.

Arithmetic functions

Function	Value	Explanation
order	81
prime-base logarithm of order	4
exponent	9
prime-base logarithm of exponent	2
Frattini length	2
derived length	2
nilpotency class	2
minimum size of generating set	2
subgroup rank	2
rank as p-group	2
normal rank	2
characteristic rank	2

Group properties

Property	Satisfied?	Explanation
abelian group	No
metabelian group	Yes
metacylcic group	Yes
group of nilpotency class two	Yes
Frattini-in-center group	Yes
group of prime power order	Yes

GAP implementation

Group ID

This finite group has order 81 and has ID 4 among the groups of order 81 in GAP's SmallGroup library. For context, there are groups of order 81. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(81,4)

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

gap> G := SmallGroup(81,4);

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

IdGroup(G) = [81,4]

or just do:

IdGroup(G)

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

Other descriptions

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