Groups of order 16807

This article gives information about, and links to more details on, groups of order 16807
See pages on algebraic structures of order 16807 | See pages on groups of a particular order

Statistics at a glance

To understand these in a broader context, see: groups of order 7^n|groups of prime-fifth order

Since $16807=7^{5}$ 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	83	PORC function for number of groups of order $p^{5}$ for $p\geq 5$ is: $2p+61+2\operatorname {gcd} (p-1,3)+\operatorname {gcd} (p-1,4)$ . Plugging in $p=7$ gives 83.
Number of abelian groups	7	equals the number of unordered integer partitions of $5$ (this is the $5$ appearing in the exponent part of $7^{5}$ ). See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Number of groups of nilpotency class exactly two	32	the general formula for order $p^{5}$ with $p\geq 5$ is $p+25$ .
Number of groups of nilpotency class exactly three	33	the general formula for order $p^{5}$ with $p\geq 5$ is $p+26$ .
Number of groups of nilpotency class exactly four, i.e., maximal class groups	11	the general formula for order $p^{5}$ with $p\geq 5$ is $3+2\operatorname {gcd} (p-1,3)+\operatorname {gcd} (p-1,4)$ .

GAP implementation

The order 16807 is part of GAP's SmallGroup library. Hence, any group of order 16807 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 16807 can be accessed as a list using GAP's AllSmallGroups function.

Here is GAP's summary information about how it stores groups of this order, accessed using GAP's SmallGroupsInformation function:

gap> SmallGroupsInformation(16807);

  There are 83 groups of order 16807.
  They are sorted by their ranks.
     1 is cyclic.
     2 - 42 have rank 2.
     43 - 76 have rank 3.
     77 - 82 have rank 4.
     83 is elementary abelian.

  This size belongs to layer 9 of the SmallGroups library.
  IdSmallGroup is not available for this size.