 Direct product of cyclic groups of orders four and two

This abelian group has eight elements $c^r{a}^s$, with $r$ taken modulo four and $s$ modulo two. Its group operation adds these exponent pairs separately. The elements of order two are $c^2,a,c^{2}a$, and the other four nonidentity elements $c,c^3,ca,c^{3}a$ have order four. It is a direct product of groups rather than a cyclic group, since no element has order eight.