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 where
is an odd prime. The special case
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
Contents
Statistics at a glance
Quantity | Value case ![]() |
Value case ![]() |
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 ![]() |
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Information on groups of order ![]() |
---|---|---|
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 ; see groups of order 8.
Common name for group | Second part of GAP ID (GAP ID is (p^3, second part)) | Case ![]() |
Nilpotency class | Probability in cohomology tree probability distribution |
---|---|---|---|---|
cyclic group of prime-cube order | 1 | cyclic group:Z27 | 1 | ![]() |
direct product of cyclic group of prime-square order and cyclic group of prime order | 2 | direct product of Z9 and Z3 | 1 | ![]() |
prime-cube order group:U(3,p) | 3 | prime-cube order group:U(3,3) | 2 | ![]() |
semidirect product of cyclic group of prime-square order and cyclic group of prime order (also denoted ![]() |
4 | semidirect product of Z9 and Z3, also denoted ![]() |
2 | ![]() |
elementary abelian group of prime-cube order | 5 | elementary abelian group:E27 | 1 | ![]() |
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 | ![]() |
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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 for the corresponding groups. For
, 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) | -- |
Same, with rows and columns interchanged:

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 | ![]() |
4 | 4 | 4 | 4 | 4 |
direct product of ... | 2 | ![]() |
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6 | 4 |
prime-cube order group:U(3,p) | 3 | ![]() |
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5 | 3 |
semidirect product of ... | 4 | ![]() |
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6 | 4 |
elementary abelian group of prime-cube order | 5 | ![]() |
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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 | ![]() |
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number of subgroups | 4 | ![]() |
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number of conjugacy classes of subgroups | 4 | ![]() |
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number of normal subgroups | 4 | ![]() |
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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 ![]() |
Number of elements of order ![]() |
Number of elements of order ![]() |
Number of elements of order ![]() |
---|---|---|---|---|---|
cyclic group of prime-cube order | 1 | 1 | ![]() |
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direct product of ... | 2 | 1 | ![]() |
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0 |
prime-cube order group:U(3,p) | 3 | 1 | ![]() |
0 | 0 |
semidirect product of ... | 4 | 1 | ![]() |
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0 |
elementary abelian group of prime-cube order | 5 | 1 | ![]() |
0 | 0 |
Here are the cumulative order statistics, where the number of roots is the number of elements whose order divides
.
Group | Second part of GAP ID | Number of elements of order ![]() |
Number of ![]() |
Number of ![]() |
Number of ![]() |
---|---|---|---|---|---|
cyclic group of prime-cube order | 1 | 1 | ![]() |
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direct product of ... | 2 | 1 | ![]() |
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prime-cube order group:U(3,p) | 3 | 1 | ![]() |
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semidirect product of ... | 4 | 1 | ![]() |
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elementary abelian group of prime-cube order | 5 | 1 | ![]() |
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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 |
---|---|---|
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cyclic group of prime-cube order | None |
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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 |
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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