Groups of order 81: Difference between revisions
(Created page with '==The list== {| class="sortable" border="1" ! Common name for group !! Second part of GAP ID (GAP ID is (p^3, second part)) !! Nilpotency class |- | Cyclic group:Z81 || 1 ||...') |
No edit summary |
||
| Line 33: | Line 33: | ||
|- | |- | ||
| [[Elementary abelian group:E81]] || 15 || 1 | | [[Elementary abelian group:E81]] || 15 || 1 | ||
|} | |||
==Arithmetic functions== | |||
===Functions taking values between 0 and 4=== | |||
These measure ranks of subgroups, lengths of series, or the prime-base logarithms of orders of certain subgroups. | |||
{| class="sortable" border="1" | |||
! Group !! GAP ID second part !! [[prime-base logarithm of exponent]] !! [[nilpotency class]] !! [[derived length]] !! [[Frattini length]] !! [[minimum size of generating set]] !! [[subgroup rank of a group|subgroup rank]] !! [[rank of a p-group|rank]] !! [[normal rank of a p-group|normal rank]] !! [[characteristic rank of a p-group|characteristic rank]] !! prime-base logarithm of order of derived subgroup !! prime-base logarithm of order of inner automorphism group | |||
|- | |||
| [[Cyclic group:Z81]] || 1 || 4 || 1 || 1 || 4 || 1 || 1 || 1 || 1 || 1 || 0 || 0 | |||
|- | |||
| [[Direct product of Z9 and Z9]] || 2 || 2 || 1 || 1 || 2 || 2 || 2 || 2 || 2 || 2 || 0 || 0 | |||
|- | |||
| [[SmallGroup(81,3)]] || 3 || 2 || 2 || 2 || 2 || 2 || 3 || 3 || 3 || 3 || 1 || 2 | |||
|- | |||
| [[Nontrivial semidirect product of Z9 and Z9]] || 4 || 2 || 2 || 2 || 2 || 2 || 2 || 2 || 2 || 2 || 1 || 2 | |||
|- | |||
| [[Direct product of Z27 and Z3]] || 5 || 3 || 1 || 1 || 3 || 2 || 2 || 2 || 2 || 2 || 0 || 0 | |||
|- | |||
| [[Semidirect product of Z27 and Z3]] || 6 || 3 || 2 || 2 || 3 || 2 || 2 || 2 || 2 || 2 || 1 || 2 | |||
|- | |||
| [[Wreath product of Z3 and Z3]] || 7 || 2 || 3 || 2 || 2 || 2 || 3 || 3 || 3 || 3 || 2 || 3 | |||
|} | |} | ||
Revision as of 21:10, 29 June 2010
The list
| Common name for group | Second part of GAP ID (GAP ID is (p^3, second part)) | Nilpotency class |
|---|---|---|
| Cyclic group:Z81 | 1 | 1 |
| Direct product of Z9 and Z9 | 2 | 1 |
| SmallGroup(81,3) | 3 | 2 |
| Nontrivial semidirect product of Z9 and Z9 | 4 | 2 |
| Direct product of Z27 and Z3 | 5 | 1 |
| Semidirect product of Z27 and Z3 | 6 | 2 |
| Wreath product of Z3 and Z3 | 7 | 3 |
| SmallGroup(81,8) | 8 | 3 |
| SmallGroup(81,9) | 9 | 3 |
| SmallGroup(81,10) | 10 | 3 |
| Direct product of Z9 and E9 | 11 | 1 |
| Direct product of prime-cube order group:U(3,3) and Z3 | 12 | 2 |
| Direct product of semidirect product of Z9 and Z3 and Z3 | 13 | 2 |
| SmallGroup(81,14) | 14 | 2 |
| Elementary abelian group:E81 | 15 | 1 |
Arithmetic functions
Functions taking values between 0 and 4
These measure ranks of subgroups, lengths of series, or the prime-base logarithms of orders of certain subgroups.
| Group | GAP ID second part | prime-base logarithm of exponent | nilpotency class | derived length | Frattini length | minimum size of generating set | subgroup rank | rank | normal rank | characteristic rank | prime-base logarithm of order of derived subgroup | prime-base logarithm of order of inner automorphism group |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cyclic group:Z81 | 1 | 4 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 0 | 0 |
| Direct product of Z9 and Z9 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 |
| SmallGroup(81,3) | 3 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 1 | 2 |
| Nontrivial semidirect product of Z9 and Z9 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 2 |
| Direct product of Z27 and Z3 | 5 | 3 | 1 | 1 | 3 | 2 | 2 | 2 | 2 | 2 | 0 | 0 |
| Semidirect product of Z27 and Z3 | 6 | 3 | 2 | 2 | 3 | 2 | 2 | 2 | 2 | 2 | 1 | 2 |
| Wreath product of Z3 and Z3 | 7 | 2 | 3 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 2 | 3 |