In non-relativistic physics, physical quantities $Q$ are characterized by how they transform under a Galilean transformation $g \in \mathcal{G}$.
$$ Q \rightarrow Q' = D[g]Q$$
where $D[g]$ is the linear representation of g.

Let $r$ be a rotation, $a$ be a spacial translation, $t$ be a time translation and $b$ be a boost. 

Scalars take the trivial representation for everything:
\begin{align\*}
D[r] = 1, D[a] = 1, D[t] = 1, D[b] = 1
\end{align\*}

If we accept force $\vec{F}$ to be the model example of a vector, then it transforms under the following representations:
$$D[a] = 1, D[t] = 1, D[b] = 1\\
D[r] = R, \text{for some $R \in \mathcal{O}(3)$} $$

But 3-velocity does not transform trivially under boosts; the boost velocity just adds up. Does that mean that velocity is not a vector?

Is there a representation where we can see that force and velocity are both the same kind of object? (vectors), or are they just different?