  In this paper we study the Lie theoretic properties of a class of topological
groups which carry a Banach manifold structure but whose multiplication is not
smooth. If $G$ and $N$ are Banach-Lie groups and $\pi : G \to \mathrm{Aut}(N)$
is a homomorphism defining a continuous action of $G$ on $N$, then $H := N
\rtimes_\pi G$ is a Banach manifold with a topological group structure for
which the left multiplication maps are smooth, but the right multiplication
maps need not to be. We show that these groups share surprisingly many
properties with Banach-Lie groups: (a) for every regulated function $\xi :
[0,1] \to T_1H$ the initial value problem $\dot\gamma(t) = \gamma(t)\xi(t)$,
$\gamma(0)= 1_H$, has a solution and the corresponding evolution map from
curves in $T_1H$ to curves in $H$ is continuous; (b) every $C^1$-curve $\gamma$
with $\gamma(0) = 1$ and $\gamma'(0) = x$ satisfies $\lim_{n \to \infty}
\gamma(t/n)^n = \exp(tx)$; (c) the Trotter formula holds for $C^1$
one-parameter groups in $H$; (d) the subgroup $N^\infty$ of elements with
smooth $G$-orbit maps in $N$ carries a natural Fr\'echet-Lie group structure
for which the $G$-action is smooth; (e) the resulting Fr\'echet-Lie group
$H^\infty := N^\infty \rtimes G$ is also regular in the sense of (a).
