Let $p>4$ be prime, and let $G=GL_2(\mathbb{F}_p)$, $H=O_3(\mathbb{F}_p)$, and $K=Sp_4(\mathbb{F}_p)$.
We know that $|G|=p(p-1)^2(p+1)$, so that a Sylow $p$-subgroup of $G$ is isomorphic to $\mathbb{Z}/p\mathbb{Z}$. In fact, there are $p+1$ such subgroups. Can we write down a generator for each one?
We also know that $|H|=2p(p+1)(p-1)$, so that a Sylow $p$-subgroup of $H$ is isomorphic to $\mathbb{Z}/p\mathbb{Z}$. Are there also $p+1$ such subgroups, and can we write down generators?
Finally, we know that $|K|=p^4(p-1)^2(p+1)^2(p^2+1)$. What is the isomorphism class of a Sylow $p$-subgroup of $K$, how many are there, and can we write down generators and relations?
I remember I solved the first question, involving $G$, but the solution escapes me at the moment. The other two are standard extensions of the first problem that I am also interested in.
Any reference and/or partial solution is appreciated. Thanks!
Edit: To address the comments, I would like to be able to explicitly write down matrices that generate the subgroups, one for each subgroup. As noted by Tobias, the matrix:
$$\begin{pmatrix}1&a\\0&1\end{pmatrix}$$
for $a\ne 0$, generates one Sylow $p$-subgroup of $G$, so I'd like to find $p$ more matrices of order $p$ that generate distinct subgroups. These matrices should be indexed by our field $\mathbb{F}_p$. They are certain to be conjugates of the above matrix, but two arbitrary conjugates may generate the same subgroup.
Edit 2: I've just solved the question for $G$. Here is a list of generators for the $p+1$ Sylow $p$-subgroups of $G$:
$$\begin{pmatrix}1&1\\0&1\end{pmatrix},\;\begin{pmatrix}1&0\\1&1\end{pmatrix},\;\begin{pmatrix}2&a\\-a^{-1}&0\end{pmatrix}$$
where $a=1,\ldots,p-1$. Each matrix has order $p$, and generates a distinct subgroup.
Best Answer
Symplectic group
I won't yet address identifying all maximal unipotent subgroups, but will just describe the standard ones.
The maximal unipotent subgroups of the symplectic group in four dimensions over the field $K$ have a nice description in terms of certain nice elements. $\newcommand{\m}[1]{\left[\begin{smallmatrix}#1\end{smallmatrix}\right]}$
Let $$G=\left\{ g \in \operatorname{GL}(4,K) : gxg^T = x \right\}, \quad x=\m{0&0&0&1\\ 0&0&1&0\\ 0&-1&0&0\\ -1&0&0&0}$$
be a symplectic group. It has two standard maximal unipotent subgroups, the upper and lower, which differ only by being transposes of each other. The lower on is: $$P=\left\{ \m{ 1 & 0 & 0 & 0 \\ x & 1 & 0 & 0 \\ y & w & 1 & 0 \\ z & y-xw & -x & 1 } : x,y,z,w \in K \right\}$$
Define $$ x_1(t) = \m{1&0&0&0\\t&1&0&0\\0&0&1&0\\0&0&-t&1}, \quad x_2(t) = \m{1&0&0&0\\0&1&0&0\\0&t&1&0\\0&0&0&1}, \quad x_3(t) = \m{1&0&0&0\\0&1&0&0\\t&0&1&0\\0&t&0&1}, \quad x_4(t) = \m{1&0&0&0\\0&1&0&0\\0&0&1&0\\t&0&0&1}, \quad $$
Note that $$\begin{array}{lrl} (1) & x_i(s)x_i(t) &= x_i(s+t) \\ (2) & x_2(t)x_1(s) &= x_1(s) x_2(t) x_3(st) x_4(s^2t) \\ (3) & x_3(t)x_1(s) &= x_1(s) x_3(t) x_4(2st) \\ (4) & x_4(t)x_1(s) &= x_1(s) x_4(t) \\ (5) & x_3(t)x_2(s) &= x_2(s) x_3(t) \\ (6) & x_4(t)x_2(s) &= x_2(s) x_3(t) \\ (7) & x_4(t)x_3(s) &= x_3(s) x_4(t) \\ \end{array}$$
If $K$ has characteristic not $2$, then $P$ has rank $2k$, generated by $x_1(s)$, and $x_2(t)$ where $s,t$ range over a generating set of the additive group of $K$. $[P,P]$ is isomorphic to the additive group of a two dimensional vector space $K^2$ over $K$, and is generated as a group by $x_3(s)$ and $x_4(t)$ with $s,t$ from a generating set of the additive group of $K$. $[P,P,P]$ is isomorphic to the additive group of $K$, and is generated as a group by $x_4(t)$ where $t$ ranges over a generating set of the additive group of $K$.
When $K$ has characteristic 2, things make less sense to me. The nilpotency class is 2. When $|K|>2$, $P/[P,P] \cong [P,P] \cong (K^+)^2$, but when $|K|=2$, $P\cong C_2 \times D_8$ behaves differently.
The normalizer of $P$ is the semidirect product of $P$ and a maximally split maximal torus $$H=\left\{ \m{ s & 0 & 0 & 0 \\ 0 & t & 0 & 0 \\ 0 & 0 & t^{-1} & 0 \\ 0 & 0 & 0 & s^{-1} } : s,t \in K^\times \right\}$$
In particular, the collection of maximal unipotent subgroups is in bijection with $(|K|^2+1)(|K|+1)^2$.