Let $G$ and $H$ be two finite groups, where $H$ acts on $G$ trivially, so that $\eta_h(g)=g$ for all $g\in G$ and $h\in H$, and $G$ acts on $H$ by conjugation. We want to construct the semi direct product of $G$ and $H$, namely $H\rtimes G$. Note that we have the multiplication rule $$(h_1, g_1)(h_2, g_2)= (h_1g_1h_2g_1^{-1}, g_1g_2).$$ Is there a "formula" or a way to determine the order of a typical element $(h,g)$? In particular, is there a way to determine the order of an element of the form $g_1h_2g_1^{-1}$?
As I stated below, the second question is a silly question. For the order of the element in the semi direct product, $(h,g)$, if you raise it to the power $|h|n$ where $|h|$ divides $n-1$ and $|g|$ divides $n$, then we have that $(h,g)^{|h|n}=(e_H, e_G)$. So this begs the questions, is the order of the element $(h,g)$ $lcm(|h|, |g|)$? If not, can anyone shed some light on the situation.
Thanks in advance.