Groups of order 2187
This article gives information about, and links to more details on, groups of order 2187
See pages on algebraic structures of order 2187| See pages on groups of a particular order
Statistics at a glance
To understand these in a broader context, see: groups of order 3^n|groups of prime-seventh order
Since is a prime power, and prime power order implies nilpotent, all groups of this order are nilpotent groups.
Quantity | Value | Explanation |
---|---|---|
Total number of groups up to isomorphism | 9310 | |
Number of abelian groups | 15 | equal to the number of unordered integer partitions of 7. See classification of finite abelian groups |
Number of groups of nilpotency class exactly two | 1757 | |
Number of groups of nilpotency class exactly three | 6050 | |
Number of groups of nilpotency class exactly four | 1309 | |
Number of groups of nilpotency class exactly five | 173 | |
Number of groups of nilpotency class exactly six (i.e., maximal class groups) | 6 |
References
- The groups with order p^7 for odd prime p by E. A. O'Brien and M. R. Vaughan-Lee, : ^{Preprint on author webpage}^{More info}
GAP implementation
The order 2187 is part of GAP's SmallGroup library. Hence, any group of order 2187 can be constructed using the SmallGroup function by specifying its group ID. Unfortunately, IdGroup is not available for this order, i.e., given a group of this order, it is not possible to directly query GAP to find its GAP ID.
Further, the collection of all groups of order 2187 can be accessed as a list using GAP's AllSmallGroups function. However, the list size may be too large relative to the memory allocation given in typical GAP installations. To overcome this problem, use the IdsOfAllSmallGroups function which stores and manipulates only the group IDs, not the groups themselves.
Here is GAP's summary information about how it stores groups of this order, accessed using GAP's SmallGroupsInformation function:
gap> SmallGroupsInformation(2187); There are 9310 groups of order 2187. E.A. O'Brien and M.R. Vaughan-Lee determined presentations of the groups with order p^7. A preprint of their paper is available at http://www.math.auckland.ac.nz/%7Eobrien/research/p7/paper-p7.pdf For p in { 3, 5, 7, 11 } explicit lists of groups of order p^7 have been produced and stored into the database. Giving the power commutator presentations of any of these groups using a standard notation they might be reduced to 35 elements of the group or a 245 p-digit number. Only 56 of these digits may be unlike 0 for any group and even these 56 digits are mostly like 0. Further on these digits are often quite likely for sequences of subsequent groups. Thus storage of groups was done by finding a so called head group and a so called tail. Along the tail only the different digits compared to the head are relevant. Even the tails occur more or less often and this is used to improve storage too. Since p^7 is too big the data is stored into some remaing holes of SMALL_GROUP_LIB at Primes[ p + 10 ]. This size belongs to layer 11 of the SmallGroups library. IdSmallGroup is not available for this size.