# Groups of order 823543

## Contents

See pages on algebraic structures of order 823543| See pages on groups of a particular order

## Statistics at a glance

Since $823543 = 7^7$ 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 113147
Number of abelian groups 15 Equals the number of unordered integer partitions of $7$ (the exponent in $5^7$). See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Number of groups of nilpotency class exactly two 26709
Number of groups of nilpotency class exactly three 76238
Number of groups of nilpotency class exactly four 7890
Number of groups of nilpotency class exactly five 2097
Number of groups of nilpotency class exactly six (i.e., maximal class groups) 198

## GAP implementation

The order 823543 is part of GAP's SmallGroup library. Hence, any group of order 823543 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 823543 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(823543);

There are 113147 groups of order 823543.

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.```