Let $Q$ denote the quaternian group of order 8 and let $A = \{-1,1\}$ Let $ G=\{(x,y)|x \in Q, y \in A\} $ and let $H$ denote the subgroup of $G$ generated by $(i,-1)$. What are the cosets of $H$. One coset of $H$ is just $H$ itself so $H_1 = \{e,(i,-1),(-1,1),(-i,-1)\} $. To find the other cosets do I just multiply $H$ by elements in $G$? Something like $(j,1)H$, $(j,-1)H$, $(-j,1)H$,... and so on?