This is to complement the answer given already and to address @Ivan Neretin's comment under OP's question. It is an example that generally (with lots of asterisks) rotations cannot be excluded. This is a copy from J. March's 6ed (p. 158).
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/1eqam.png)
From the text:
Such compounds possess ... an alternating axis of symmetry as in 1
Of course Ivan is correct that you only care for "determinant $-1$" motions or else orientation reversing and in this case we use a rotation by $\pi$ which has determinant 1 (or else belongs to $SO_3$). My group theory is not as good since I am an organic chemist but I think the actual symmetry element is something like $\rho_{\pi/2} \circ r = 1$ whereas the rotation element above is $\rho_\pi=(\rho_{\pi/2} \circ r)\circ (\rho'_{\pi/2} \circ r')=1 \circ 1=1$ and it does have determinant 1 but is not one of the symmetry group elements (in the sense that $x^{34}$ is in $C_4=\{1,x,x^2,x^3 \}$ but only as $x^{34}=x^2$ ).
To make things more complicated March is defining an "alternating axis of symmetry" as:
An alternating axis of symmetry 17 of order n is an axis such that
when an object containing such an axis is rotated by 360/n about the
axis and then reflection is effected across a plane at right angles to
the axis, a new object is obtained that is indistinguishable from the
original one
So it appears that in the example of (1) we talk of $Z_4$ symmetry (if we call $Z$ the alternating axis subgroup) and $Z_4 \sim C_2$
Finally, I think that (1) is an extreme example and almost always for organic compounds chirality (or not) is determined by the lack (or not) of a plane of symmetry