# Direct product of cyclic group of prime-square order and cyclic group of prime order

View other such prime-parametrized groups

## Definition

Let $p$ be a prime number. This group of order $p^3$ is defined as the external direct product of the cyclic group of prime-square order and the cyclic group of prime order.

## Particular cases

Value of prime number $p$ Corresponding group
2 direct product of Z4 and Z2
3 direct product of Z9 and Z3
5 direct product of Z25 and Z5

## Arithmetic functions

Compare and contrast arithmetic function values with other groups of prime-cube order at Groups of prime-cube order#Arithmetic functions

### Arithmetic functions taking values between 0 and 3

Function Value Similar groups Explanation for function value
prime-base logarithm of order 3 groups with same prime-base logarithm of order
max-length of a group 3 max-length of a group equals prime-base logarithm of order for group of prime power order
chief length 3 chief length equals prime-base logarithm of order for group of prime power order
composition length 3 composition length equals prime-base logarithm of order for group of prime power order
prime-base logarithm of exponent 2 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 prime-base logarithm of order and Frattini length | groups with same Frattini length Frattini length equals prime-base logarithm of exponent for abelian group of prime power order
minimum size of generating set 2 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 2 groups with same prime-base logarithm of order and subgroup rank of a group | groups with same subgroup rank of a group same as minimum size of generating set since it is an abelian group of prime power order
rank of a p-group 2 groups with same prime-base logarithm of order and rank of a p-group | groups with same rank of a p-group same as minimum size of generating set since it is an abelian group of prime power order
normal rank of a p-group 2 groups with same prime-base logarithm of order and normal rank of a p-group | groups with same normal rank of a p-group same as minimum size of generating set since it is an abelian group of prime power order
characteristic rank of a p-group 2 groups with same prime-base logarithm of order and characteristic rank of a p-group | groups with same characteristic rank of a p-group same as minimum size of generating set since it is an abelian group of prime power order
nilpotency class 1 The group is a nontrivial abelian group
derived length 1 The group is a nontrivial abelian group
Fitting length 1 The group is a nontrivial abelian group

### Arithmetic functions of a counting nature

Note that since the group is abelian, the number of subgroups equals the number of conjugacy classes of subgroups as well as the number of normal subgroups.

Function Value Explanation
number of subgroups $2p + 4$
number of automorphism classes of subgroups 6
number of characteristic subgroups 4

## GAP implementation

### Group ID

This finite group has order p^3 and has ID 2 among the group of order p^3 in GAP's SmallGroup library. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(p^3,2)

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

gap> G := SmallGroup(p^3,2);

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

IdGroup(G) = [p^3,2]

or just do:

IdGroup(G)

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