What is Point group, order of a point group?

Example 4.10 (The group of water)

The four operations $E$, $C_2$, $\sigma_v(xz)$, $\sigma_v'(yz)$ of $\ce{H2O}$ form a group of order 4. Its multiplication table (row operation performed after the column one) is

For instance a point $(x, y, z)$ reflected in $xz$ goes to $(x, -y, z)$, then rotated by $\pi$ about $z$ to $(-x, y, z)$: the same as one reflection in $yz$.

Example 4.19 (An achiral molecule without a plane)

Some molecules have neither a mirror plane nor a centre of inversion, yet are achiral because of an $S_4$ axis: the classic case is a spiro compound built from two identically substituted rings at right angles, of point group $S_4$. Chirality cannot be judged by looking for a plane alone.