input file:

F:=FiniteField(2);
G:=GeneralLinearGroup(3, F);
Classes(G);
quit;

output file:

Magma V2.14-11    Wed Feb 11 2009 12:31:50 on altair1  [Seed = 2488194614]

Conjugacy Classes of group G
----------------------------
[1]     Order 1       Length 1      
        Rep [1 0 0]
            [0 1 0]
            [0 0 1]

[2]     Order 2       Length 21        
        Rep [1 0 0]
            [0 1 1]                [char poly (1-x)(1-x)^2 = x^3+x^2+x+1,
            [0 0 1]                   divides x^4-1, min poly (1-x)^2]

[3]     Order 3       Length 56     
        Rep [1 0 0]
            [0 0 1]                [char poly (1-x)(x^2+x+1) = x^3 -1 (mod 2)]
            [0 1 1]

[4]     Order 4       Length 42     
        Rep [1 1 0]
            [0 1 1]               [char poly (1-x)^3 = x^3+x^2+x+1 = min poly,
            [0 0 1]                  divides x^4-1]  

[5]     Order 7       Length 24     
        Rep [0 1 0]
            [0 0 1]               [char poly x^3+x^2+1, divides x^7-1 mod 2]
            [1 0 1]

[6]     Order 7       Length 24     
        Rep [0 1 0]
            [0 0 1]               [char poly x^3+x+1, divides x^7-1 mod 2]
            [1 1 0]

Total time: 7.710 seconds, Total memory usage: 7.02MB
---

note 1: A from class [5], then A^3 is in class [6] and B from class [6],
  then B^3 in class [5].

note 2: C from class [4], C^2 of order 2 in class [2].