14. Let $D_6$ be the dihedral group of symmetries of a regular hexagon, so $D_6 = \{e, \sigma, \dots, \sigma^5, \tau_1, \dots, \tau_6\}$ where $\sigma$ is rotation counter-clockwise by $60^\circ$ and $\tau_1, \dots, \tau_6$ are flips around the red lines in the diagram
Using the labeling, we can identify $D_6$ with a subgroup of $S_6$, explicitly $\sigma = (123456)$ and
$\tau_1 = (12)(36)(45)$
$\tau_4 = (15)(24)$
$\tau_2 = (26)(35)$
$\tau_5 = (14)(23)(56)$
Let $H = \langle \sigma^2 \rangle$ be the cyclic subgroup generated $\sigma^2$.
$\tau_3 = (16)(25)(34)$
$\tau_6 = (13)(46)$
(a) How many cosets of $H$ are there in $D_6$?
(b) Is $\tau_2H = \tau_4H$?
(c) Is $\sigma H = \tau_1H$?
(d) Is $\tau_1H = \tau_2H$?
