On account of a homework question
$$ \text{does there exist an injective homomorphism } D_6 \to S_5 \text{ ?}$$
which I solved positively, I raised the question
$$ \text{what does the complement } C = S_5 \setminus \langle (45),(12)(345) \rangle \text{ look like?} $$
If one interprets the subgroup $H = \langle (45),(12)(345) \rangle$ of $S_5$ as $D_6$, the symmetries of the hexagon, then what interpretation can we give to $C$?