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I am trying to understand how the Heisenberg group is defined because I would like to understand the (irreducible) representations.

Following this article given a symplectic bilinear form $\langle, \rangle$ on a finite-dimensional vector space $W$ over a finite field $F$ one can define the Heisenberg group as $W\times F$ with the operation $$ (w, a)(w',a') = (w + w', a+a' + \langle w, w'\rangle). $$ But I also know that one can define the Heisenberg group as a group of matrices $$ \pmatrix{1 & x & t \\ 0 & 1 & y \\ 0 & 0 & 1} $$

My question is what a isomorphism between the two groups looks like.

I assume we can take $W = F^2$, so one might think that $((w_1,w_2),a)$ ($w_i\in F, a\in F$) is sent to $$ \pmatrix{1 & w_1 & a \\ 0 & 1 & w_2 \\ 0 & 0 & 1} $$ but this doesn't work.

I am guessing that my confusion is about the definition of $\langle,\rangle$.

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    $\begingroup$ You need char$(F)\neq 2$. Then try $(x,y,t)\mapsto (x,y,t-\frac{1}{2}xy )$ (from the triangular group to the Heisenberg group, with symplectic form $\langle (x,y),(x',y')\rangle = \frac{1}{2}(xy'-yx')$. $\endgroup$
    – abx
    Commented Apr 5, 2016 at 16:00

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Assuming your symplectic form is $\langle (w_1,w_2), (w'_1, w'_2) \rangle = w_1 w'_2 - w_2 w'_1$, the isomorphism is $$((w_1,w_2),a) \mapsto \begin{pmatrix} 1 & w_1 & \tfrac{1}{2}(a+w_1 w_2)\\ 0 & 1 & w_2 \\ 0 & 0 & 1 \end{pmatrix}.$$

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