Shape whose group of isometries has order 7

Question

I have recently been tasked with drawing a shape whose symmetry group of isometries have an order of 7.

Though I am unsure of how to go about this as I have only ever drawn shapes whose symmetry groups are of even order such as regular squares and hexagons. My thinking could be completely wrong, so any pointers would be greatly appreciated.

Answer 1

The book by Katok and Climenhaga shows the following results:

Proposition: Every cylic group of order $n$ can be represented by isometries of $\Bbb R^2$.

For the proof, see page $104$, one takes a regular $n$-gon and markes each side asymmetrically by adding a small triangle, so that there is no reflective symmetry. There is a nice picture given there.

In the book, also the following result is proved:

Corollary $2.16$: Any finite group of order $n\le 7$ can be represented by isometries of $\Bbb R^2$.

Answer 2

A group of order $7$ must be the cyclic group $C_7$. Now each linear isometry of the plane is a rotation or a reflection. Since each reflection is of order two, the group must consist of the multiples of a rotation of order $7$, therefore having a $7$-th root of unity as angle.