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] $\langle a_{1},a_{2},a_{3}\mid a_{1}^{p}=a_{2},a_{2}^{p}=a_{3},a_{3}^{p}=e,[a_{1},a_{2}]=e,[a_{1},a_{3}]=e,[a_{2},a_{3}]=e\rangle$
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] $\langle a_{1},a_{2},a_{3}\mid a_{1}^{p}=a_{3},a_{2}^{p}=e,a_{3}^{p}=e,[a_{1},a_{2}]=e,[a_{1},a_{3}]=e,[a_{2},a_{3}]=e\rangle$
prime-cube order group:U(3,p)	3	2	2	1	0	0	0	1	[SHOW MORE] $\langle a_{1},a_{2},a_{3}\mid a_{1}^{p}=e,a_{2}^{p}=e,a_{3}^{p}=e,[a_{1},a_{2}]=a_{3},[a_{1},a_{3}]=e,[a_{2},a_{3}]=e\rangle$
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] $\langle a_{1},a_{2},a_{3}\mid a_{1}^{p}=a_{3},a_{2}^{p}=e,a_{3}^{p}=e,[a_{1},a_{2}]=a_{3},[a_{1},a_{3}]=e,[a_{2},a_{3}]=e\rangle$
elementary abelian group of prime-cube order	5	1	3	1	0	0	0	0	[SHOW MORE] $\langle a_{1},a_{2},a_{3}\mid a_{1}^{p}=e,a_{2}^{p}=e,a_{3}^{p}=e,[a_{1},a_{2}]=e,[a_{1},a_{3}]=e,[a_{2},a_{3}]=e\rangle$

Arithmetic functions

Functions taking values between 0 and 3

These arithmetic function values are the same for all $p\neq 2$ 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]

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	--	1.8	1.4	1.4	2	2	2	2	2	1.8	0.4	0.8
Median	--	2	1	1	2	2	2	2	2	2	0	0
Mode	--	1	1	1	2	2	2	2	2	1	0	0
Standard deviation	--	0.84	0.55	0.55	0.71	0.71	0.71	0.71	0.71	0.84	0.55	1.1
Standard deviation to mean ratio	0.46	0.39	0.39	0.35	0.35	0.35	0.35	0.35	0.46	1.37	1.37

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]

Function	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
prime-base logarithm of exponent	1	-0.33	-0.33	0.85	-0.85	-0.85	-0.85	-0.85	-0.43	-0.33	-0.33
nilpotency class	-0.33	1	1	0	0	0	0	0	-0.33	1	1
derived length	-0.33	1	1	0	0	0	0	0	-0.33	1	1
Frattini length	0.85	0	0	1	-1	-1	-1	-1	-0.85	0	0
minimum size of generating set	-0.85	0	0	-1	1	1	1	1	0.85	0	0
subgroup rank	-0.85	0	0	-1	1	1	1	1	0.85	0	0
rank	-0.85	0	0	-1	1	1	1	1	0.85	0	0
normal rank	-0.85	0	0	-1	1	1	1	1	0.85	0	0
characteristic rank	-0.43	-0.33	-0.33	-0.85	0.85	0.85	0.85	0.85	1	-0.33	-0.33
prime-base logarithm of order of derived subgroup	-0.33	1	1	0	0	0	0	0	-0.33	1	1
prime-base logarithm of order of inner automorphism group	-0.33	1	1	0	0	0	0	0	-0.33	1	1

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.

Further information: order statistics-equivalent not implies 1-isomorphic, Lazard Lie group has the same order statistics as the additive group of its Lazard Lie ring

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