Groups of order 2187

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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 2187 = 3^7 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


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

      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.