= Solution
If an element has order six, it generates $G$, so $G\cong C_6$. Otherwise choose an element $x$ of order three. The preceding result excludes any further order-three elements outside $\{e,x,x^2\}$, and there are no elements of order six by assumption. Each of the other three elements must therefore have order two. Thus
$$
\boxed{G\cong C_6,\qquad\text{or}\qquad G=\{e,x,x^2,h_1,h_2,h_3\},\quad\operatorname{ord}(x)=3,\ \operatorname{ord}(h_i)=2.}
$$
The noncyclic case is the <symmetric group on three points>, as the classification of a <group of order six> further identifies.
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