# 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 15 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.