Are we guaranteed a Lie group by taking the algebraic closure of finitely many one-parameter groups acting (each periodically) on a Euclidean space?

We are given a Euclidean space $E$ and a finite subset ${A_{1}, ldots, A_{k}}$ of $GL(E)$. Consider the one-parameter groups $g_{k}^{t} : t mapsto e^{t A_{k}}$. The operators $A_{k}$ are such that:

1. Each $g_{k}^{t}$ preserves the Euclidean norm.
2. The action of each $g_{k}^{t}$ on each $x in E$ is periodic: there exists a positive time $T$ (dependent on $k$ and on $x$) such that $g_{k}^{t + T}x = g_{k}^{t} x$.

Since the groups $g_{k}^{t}$ are subgroups of $GL(E)$, so is the group, $G$, generated by their union. Are we guaranteed that $G$ is Lie and that the orbit $Gx$ of an $x in E$ under the action of $G$ is a submanifold of $E$? If so, am I correct in thinking that the dimension of this manifold equals the dimension of the image of $x$ under the Lie algebra of $G$? Any sources to study would be appreciated as well.