Affine and linear groups : Conjugacy Classes in small dimensions

The page provides the conjugacy classes of the Affine Group AG(n,2) and General Linear Group GL(n,2) for the dimension less or equal to 12. In particular the sequence of class numbers in the affine case is :


a(n) := 1, 2, 5, 11, 25, 52, 112, 229, 475, 965, 1967, 3965, 8018.
It is not on OEIS yet while the analogue for the linear group is well known
http://oeis.org/A006951: b(n):= 1, 1, 3, 6, 14, 27, 60, 117, 246, 490, 1002, 1998, 4053.
Xiang-Dong Hou gave me the nice relation :
a(n) = b(0) +b(1) + ... + b(n)
The a(n) representatives of AG(n,2) are given by Xiang-Dong Hou in his Lectures on Finite Field.

Description of the data files

In the file agl-conj-5.txt, one finds the entry : 
---
line 1 : class 9:
line 2 : [10000 00100 00010 00001 01000]00000
line 3 : centralizer=32
line 4 : [10000 00100 00010 00001 01000]00000
         [10000 01110 00111 01011 01101]00000
         [10000 11011 11101 11110 10111]00000
         [11111 11110 10111 11011 11101]00000
line 8 : (00000) x 8
line 9 : (10000) x 8
line 10: (01000) x 16
describing the conjugacy classes of the elements (A,u) whose linear part A presented on line 2 is a member of GL(5,2). The centralizer of A in GL(5,2) has order 32, it is generated the 4 elements presented on line 4-5-6-7. In n the affine group, there are 3 conjugacy classes (A,u) with u = (00000), (10000) or (01000). The first and second conjugacy class in the AG(5,2) contain : GL(5,2) / Cent(A) x 8 members, and the double in the third.

files

Philippe Langevin, Last modification April 2020.