I'm looking at an example that instructs me to consider the group:
$$ G = \langle(1, 2, 3, 4, 5, 6, 7, 8), (1, 8)(2, 7)(3, 6)(4, 5)\rangle \cong Dih(16) $$
And the subgroup:
$$ H = \langle(1, 3, 5, 7)(2, 4, 6, 8)\rangle $$
I believe these are cycles, typically written without the internal commas from what I can see in my reading of related questions here.
I've managed to decipher that $G$ is the generating set $\langle r, f \rangle$ where $r$ is a rotation and $f$ is a reflection of the octagon. So I know the full set G contains each of the nodes in this diagram and can see how $\langle r, f \rangle$ was turned into cycle notation.
However, I'm not sure how to parse $H$.
Is $\langle(1, 3, 5, 7)(2, 4, 6, 8)\rangle$ a specific rotation?
Should these two cycles be composed (and if so, how)?