# SmallGroup(81,3)

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## Definition

This group is defined by the following presentation:

$G := \langle a,b,c \mid a^9 = b^3 = c^3 = e, ab = ba, bc = cb, cac^{-1} = ab \rangle$

## Arithmetic functions

Want to compare with other groups of the same order? Check out groups of order 81#Arithmetic functions
Function Value Explanation
order 81
exponent 9
Frattini length 2
Fitting length 1
derived length 2
nilpotency class 2
minimum size of generating set 2
rank as p-group 3
normal rank as p-group 3
characteristic rank as p-group 3
number of conjugacy classes 33
number of subgroups 41
number of conjugacy classes of subgroups 23

## Group properties

Want to compare with other groups of the same order? Check out groups of order 81#Group properties.
Property Satisfied Explanation
group of prime power order Yes
abelian group No
nilpotent group Yes prime power order implies nilpotent
group of nilpotency class two Yes
metabelian group Yes
T-group No
ambivalent group No odd-order and ambivalent implies trivial

## Families

This group is part of the family SmallGroup(p^4,3).

## GAP implementation

### Group ID

This finite group has order 81 and has ID 3 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(81,3)

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

gap> G := SmallGroup(81,3);

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

IdGroup(G) = [81,3]

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.1^9,F.2^3,F.3^3,F.1*F.2*F.1^(-1)*F.2^(-1),F.2*F.3*F.2^(-1)*F.3^(-1),F.3*F.1*F.3^(-1)*F.1^(-1)*F.2^(-1)];
<fp group on the generators [ f1, f2, f3 ]>