  We study the functional calculus properties of generators of $C_{0}$-groups
under type and cotype assumptions on the underlying Banach space. In
particular, we show the following.
  Let $-iA$ generate a $C_{0}$-group on a Banach space $X$ with type
$p\in[1,2]$ and cotype $q\in[2,\infty)$. Then $A$ has a bounded
$\mathcal{H}^{\infty}$-calculus from
$\mathrm{D}_{A}(\tfrac{1}{p}-\tfrac{1}{q},1)$ to $X$, i.e.
$f(A):\mathrm{D}_{A}(\tfrac{1}{p}-\tfrac{1}{q},1)\to X$ is bounded for each
bounded holomorphic function $f$ on a sufficiently large strip.
  As a corollary of our main theorem, for sectorial operators we quantify the
gap between bounded imaginary powers and a bounded
$\mathcal{H}^{\infty}$-calculus in terms of the type and cotype of the
underlying Banach space. For cosine functions we obtain similar results as for
$C_{0}$-groups. We extend our results to $R$-bounded operator-valued calculi,
and we give an application to the theory of rational approximation of
$C_{0}$-groups.
