Let $R$ be the orthogonal matrix corresponding to an operation in $O(3)$. If

R is a proper rotation, then both vectors $\vec{V}$ and axial vectors $\vec{A}$ are transformed in the same way

$$ \vec{V} \rightarrow \vec{V}' = R\vec{V}$$

$$ \vec{A} \rightarrow \vec{A}' = R\vec{A}$$

if $R$ is an improper rotations, then,

$$ \vec{V} \rightarrow \vec{V}' = R\vec{V}$$

$$ \vec{A} \rightarrow \vec{A}' = -R\vec{A}$$

And let $\mathcal{V}$ be the vector representation of O(3) and $\mathcal{P}$ be the axial-vector representation of O(3). Thus we can write,

$$\vec{V} \rightarrow \vec{V}' = \mathcal{V}(R)\vec{V}$$

$$\vec{A} \rightarrow \vec{A}' = \mathcal{P}(R)\vec{A}$$

My question is, considering $\mathcal{V}$ and $\mathcal{P}$ as representations of $C_{4v}$ group, how can I find their composition in terms of irreps ?

One can use the character tablegiven in this link http://symmetry.jacobs-university.de/cgi-bin/group.cgi?group=404&option=4