TITLE: How group of order $21$ arises naturally?
QUESTION [6 upvotes]: I was explaining following theorem in undergraduates:

"If $|G|=pq$ with $p>q$ and $q\nmid p-1$ then $G$ must be cyclic".

"But what about $|G|=21=7.3$? Here $3\mid (7-1)$". Students stuck to say something about $G$. Then I tried to explain that that there is a non-cyclic (hence non-abelian) group of order $21$. I tried to write the collection of maps $x\mapsto ax+b$ where $b$ is in field of order $7$ and $a$ is a cube root of unity in the field of order $7$. I soon realized that I moved partially out of Group Theory.
How can we illustrate easily the example of non-abelian group of order $21$ to undergraduates with minimum prerequisite from Group Theory?

REPLY [3 votes]: Personally,  I'd probably just give them the group in presentation form that you get from the semidirect product construction of it....and for the more advanced students, explain semidirect products.   
$\langle a,b\mid a^3=b^7=e,aba^{-1}=b^2\rangle$,  and show them that that well defines a group of order 21 that is non abelian.