I'm asked to find out all of the conjugacy classes, their order and their size for $GL(4,2)$.
Finding representatives is possible by looking for all the rational canonical forms over the field and the order, just by taking powers of the representatives.
Now, about calculating the size of each class, I know it can be done by trying to calculate the size of the center for each class, $C_G(x_i)$, where $x_i$ are the representatives, and then $\frac{|GL(4,2)|}{|C_G(x_i)|}$ is the size of the class, but it appears to be very difficult to do it straightforward.
Can someone suggest me a better way to do it?
Thanks
Note: $GL(4,2)$ is all invertible matrices of size $4\times4$ over $F_2$.
http://en.wikipedia.org/wiki/General_linear_group#Over_finite_fields
Best Answer
$GL_2(\mathbb{F}_2)$ has order six. The identity is always its own conjugacy class, so since the size of each conjugacy class divides the order of the group, the other two classes must be of order $2$ and $3$. Conjugate elements have the same order, and every group of even order contains an odd number of elements of order $2$, so we know the size $3$ class corresponds to elements of order $2$ and the size $2$ class corresponds to elements of order $3$. These are easy enough to calculate.
$$\text{Order }3: \left( \begin{array}{cc} 0 & 1 \\ 1 & 1 \end{array} \right) , \left( \begin{array}{cc} 1 & 1 \\ 1 & 0 \end{array} \right)$$ $$\text{Order }2: \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right) , \left( \begin{array}{ccc} 1 & 0 \\ 1 & 1 \end{array} \right) , \left( \begin{array}{ccc} 1 & 1 \\ 0 & 1 \end{array} \right)$$
So, along with the identity, these are the conjugacy classes of $GL_2(\mathbb{F}_2)$.
EDIT: Okay, so evidently OP wanted $\operatorname{GL}_4(\mathbb{F}_2)$ after all. This is a much more difficult question, especially because the field has even characteristic. It has been studied, however, and you can find a paper about it here that includes a table of the conjugacy classes with singer polynomials and explanations behind why everything works the way it does.
The conjugacy classes end up coming out like this: $$\begin{array}{rl}\text{Size} & \text{Order}\\ 1 & 1 \\ 105 & 2 \\ 210 & 2 \\ 112 & 3 \\ 1120 & 3 \\ 1260 & 4 \\ 2520 & 4 \\ 1344 & 5 \\ 1680 & 6 \\ 3360 & 6 \\ 2880 & 7 \\ 2880 & 7 \\ 1344 & 15 \\ 1344 & 15 \end{array}$$