Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 3 / 19G / i / Solution

Every group element has a unique form $a^r{b}^s$, with $r\mathrm{mod}7$ and $s\mathrm{mod}3$. Conjugation by $b$ doubles the exponent of $a$, giving the two orbits $\{1,2,4\}$ and $\{3,6,5\}$. Conjugation by powers of $a$ runs through all seven elements of each nonidentity coset of $⟨a⟩$. The classes are thereforeTheir sizes $1,3,3,7,7$ sum to twenty-one.