Sketch a figure which has a group of symmetries of order 5.

Question

I am trying to draw a shape which has only 5 symmetries

I know Square has 8

Rectangle/parallelogram has 4

Triangle has 6

Circle has infinite

how do i know which shape has only 5

I know that a regular polygon with $n$ sides has $2n$ symmetries so there is no regular polygon with order of symmetry 5

so i came up with this one which has only rotational symmetry and order 5

my question is are there any shapes with symmetries of order 5 and which are not entirely rotational symmetries

also how do i generally find them for a given an odd order?

Thanks for any help.

Answer 1

Note that any reflection has order $2$, so any group of symmetries which includes a reflection must have even order.

You can generalise your solution for $5$ to any odd order.