# Groups of prime-cube order

This article is about the groups of prime-cube order for an odd prime number, i.e., the groups of order $p^3$ where $p$ is an odd prime. The special case $p = 2$ is somewhat different -- see groups of order 8 for a summary of information on these groups.

Want to know how this list of groups is obtained? See classification of groups of prime-cube order

## Statistics at a glance

Quantity Value case $p = 2$ Value case $p$ odd Explanation
Total number of groups 5 5 See classification of groups of prime-cube order
Number of abelian groups 3 3 See classification of finite abelian groups and structure theorem for finitely generated abelian groups. In this case, the number of unordered integer partitions of 3 is 3, so that is the number of abelian groups.
Number of groups of class exactly two 2 2 See classification of groups of prime-cube order

## Particular cases

Prime $p$ $p^3$ Information on groups of order $p^3$
2 8 groups of order 8
3 27 groups of order 27
5 125 groups of order 125
7 343 groups of order 343

## The list

The list below is valid for odd primes. The list is somewhat different for $p = 2$; see groups of order 8.

Common name for group Second part of GAP ID (GAP ID is (p^3, second part)) Case $p = 3$ Nilpotency class Probability in cohomology tree probability distribution
cyclic group of prime-cube order 1 cyclic group:Z27 1 $1 - 2/p + 1/p^2$
direct product of cyclic group of prime-square order and cyclic group of prime order 2 direct product of Z9 and Z3 1 $1/p - 1/p^4$
prime-cube order group:U(3,p) 3 prime-cube order group:U(3,3) 2 $1/p^3 - 1/p^4$
semidirect product of cyclic group of prime-square order and cyclic group of prime order (also denoted $M_{p^3}$) 4 semidirect product of Z9 and Z3, also denoted $M_{27}$ 2 $1/p - 1/p^2 - 1/p^3 + 1/p^4$
elementary abelian group of prime-cube order 5 elementary abelian group:E27 1 $1/p^4$

## Presentations

Further information: presentations for groups of prime-cube order

Group Second part of GAP ID (GAP ID is (p^3,2nd part) Nilpotency class Minimum size of generating set Prime-base logarithm of exponent $\beta(1,2)$ $\beta(1,3)$ $\beta(2,3)$ $\beta(1,2,3)$ full power-commutator presentation
cyclic group of prime-cube order 1 1 1 3 1 0 1 0 [SHOW MORE]
direct product of cyclic group of prime-square order and cyclic group of prime order 2 1 2 2 0 1 0 0 [SHOW MORE]
prime-cube order group:U(3,p) 3 2 2 1 0 0 0 1 [SHOW MORE]
semidirect product of cyclic group of prime-square order and cyclic group of prime order 4 2 2 2 0 1 0 1 [SHOW MORE]
elementary abelian group of prime-cube order 5 1 3 1 0 0 0 0 [SHOW MORE]

## Arithmetic functions

### Functions taking values between 0 and 3

These arithmetic function values are the same for all $p \ne 2$ for the corresponding groups. For $p = 2$, the behavior for the abelian groups is exactly the same, but the two non-abelian groups behave a little differently.

Group GAP ID (second part) prime-base logarithm of exponent nilpotency class derived length Frattini length minimum size of generating set subgroup rank rank as p-group normal rank characteristic rank prime-base logarithm of order of derived subgroup prime-base logarithm of order of inner automorphism group
cyclic 1 3 1 1 3 1 1 1 1 1 0 0
direct product of... 2 2 1 1 2 2 2 2 2 2 0 0
prime-cube order group:U(3,p) 3 1 2 2 2 2 2 2 2 1 1 2
semidirect product of... 4 2 2 2 2 2 2 2 2 2 1 2
elementary abelian 5 1 1 1 1 3 3 3 3 3 0 0
mean (with equal weight on all groups) -- 1.8 1.4 1.4 2 2 2 2 2 1.8 0.4 0.8
mean (weighting by cohomology tree probability distribution) --
Here now is the same table with various measures of averages and deviations: [SHOW MORE]

Same, with rows and columns interchanged:

Function Cyclic group of prime-cube order Direct product of cyclic group of prime-square order and cyclic group of prime order Prime-cube order group:U(3,p) Semidirect product of cyclic group of prime-square order and cyclic group of prime order Elementary abelian group of prime-cube order
prime-base logarithm of exponent 3 2 1 2 1
nilpotency class 1 1 2 2 1
derived length 1 1 2 2 1
Frattini length 3 2 2 2 1
minimum size of generating set 1 2 2 2 3
subgroup rank 1 2 2 2 3
rank 1 2 2 2 3
normal rank 1 2 2 2 3
characteristic rank 1 2 1 2 3
prime-base logarithm of order of derived subgroup 0 0 1 1 0
prime-base logarithm of order of inner automorphism group 0 0 2 2 0
Here are the correlations between the arithmetic function values for the groups of order $p^3$: [SHOW MORE]

### Arithmetic function values of a counting nature

Group GAP ID (second part) number of conjugacy classes number of subgroups number of conjugacy classes of subgroups number of normal subgroups number of automorphism classes of subgroups number of characteristic subgroups
cyclic group of prime-cube order 1 $p^3$ 4 4 4 4 4
direct product of ... 2 $p^3$ $2p + 4$ $2p + 4$ $2p + 4$ 6 4
prime-cube order group:U(3,p) 3 $p^2 + p - 1$ $p^2 + 2p + 4$ $2p + 5$ $p + 4$ 5 3
semidirect product of ... 4 $p^2 + p - 1$ $2p + 4$ $p + 5$ $p + 4$ 6 4
elementary abelian group of prime-cube order 5 $p^3$ $2p^2 + 2p + 4$ $2p^2 + 2p + 4$ $2p^2 + 2p + 4$ 4 2

Same, with rows and columns interchanged:

Function cyclic group of prime-cube order direct product of ... prime-cube order group:U(3,p) semidirect product of ... elementary abelian group of prime-cube order
number of conjugacy classes $p^3$ $p^3$ $p^2 + p - 1$ $p^2 + p - 1$ $p^3$
number of subgroups 4 $2p + 4$ $p^2 + 2p + 4$ $2p + 4$ $2p^2 + 2p + 4$
number of conjugacy classes of subgroups 4 $2p + 4$ $2p + 5$ $p + 5$ $2p^2 + 2p + 4$
number of normal subgroups 4 $2p + 4$ $p + 4$ $p + 4$ $2p^2 + 2p + 4$
number of automorphism classes of subgroups 4 6 5 6 4
number of characteristic subgroups 4 4 3 4 2

## Group properties

Property cyclic group of prime-cube order direct product of ... prime-cube order group:U(3,p) semidirect product of ... elementary abelian group of prime-cube order
cyclic group Yes No No No No
elementary abelian group No No No No Yes
abelian group Yes Yes No No Yes
homocyclic group Yes No No No Yes
metacyclic group Yes Yes Yes Yes No
metabelian group Yes Yes Yes Yes Yes
group of nilpotency class two Yes Yes Yes Yes Yes
maximal class group No No Yes Yes No
ambivalent group No No No No No
rational group No No No No No
rational-representation group No No No No No
group in which every element is automorphic to its inverse Yes Yes Yes No Yes
group in which any two elements generating the same cyclic subgroup are automorphic Yes Yes Yes No Yes
T-group Yes Yes No No Yes
C-group No No No No Yes
SC-group No No No No Yes
UL-equivalent group Yes Yes Yes Yes Yes
algebra group No Yes Yes No Yes

## Element structure

Further information: element structure of groups of prime-cube order

### Order statistics

Group Second part of GAP ID Number of elements of order $1$ Number of elements of order $p$ Number of elements of order $p^2$ Number of elements of order $p^3$
cyclic group of prime-cube order 1 1 $p - 1$ $p^2 - p = p(p-1)$ $p^3 - p^2 = p^2(p - 1)$
direct product of ... 2 1 $p^2 - 1 = (p - 1)(p + 1)$ $p^3 - p^2 = p^2(p - 1)$ 0
prime-cube order group:U(3,p) 3 1 $p^3 - 1 = (p - 1)(p^2 + p + 1)$ 0 0
semidirect product of ... 4 1 $p^2 - 1 = (p - 1)(p + 1)$ $p^3 - p^2 = p^2(p - 1)$ 0
elementary abelian group of prime-cube order 5 1 $p^3 - 1 = (p - 1)(p^2 + p + 1)$ 0 0

Here are the cumulative order statistics, where the number of $n^{th}$ roots is the number of elements whose order divides $n$.

Group Second part of GAP ID Number of elements of order $1$ Number of $p^{th}$ roots Number of $(p^2)^{th}$ roots Number of $(p^3)^{th}$ roots
cyclic group of prime-cube order 1 1 $p$ $p^2$ $p^3$
direct product of ... 2 1 $p^2$ $p^3$ $p^3$
prime-cube order group:U(3,p) 3 1 $p^3$ $p^3$ $p^3$
semidirect product of ... 4 1 $p^2$ $p^3$ $p^3$
elementary abelian group of prime-cube order 5 1 $p^3$ $p^3$ $p^3$

### Equivalence classes

Up to order statistics-equivalence, there are three equivalence classes. Moreover, these are the same as the eqiuvalence classes up to 1-isomorphism. A deeper explanation of this is that all the group in a given equivalence class actually have the same additive group of their respective Lazard Lie rings.

Order statistics Abelian group with those order statistics Non-abelian group with those order statistics $1,p-1,p^2-p,p^3-p^2$ cyclic group of prime-cube order None $1,p^2-1,p^3 - p^2$ direct product of cyclic group of prime-square order and cyclic group of prime order semidirect product of cyclic group of prime-square order and cyclic group of prime order $1,p^3 - 1, 0,0$ elementary abelian group of prime-cube order prime-cube order group:U(3,p)

## Subgroup structure

Further information: subgroup structure of groups of prime-cube order