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


Following [this article](http://www.pvamu.edu/PDFFiles/edu/Include/Math/MDTRS/DMTRS11_b.pdf) 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 $\langle,\rangle$.



Representations of the $3\times 3$ Heisenberg group
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
$$((w\_1,w\_2),a) \mapsto \begin{pmatrix} 1 & w\_1 & \tfrac{1}{2}(a+w\_1 w\_2)\\ 0 & 1 & w\_2 \\ 0 & 0 & 1 \end{pmatrix}.$$


