Chapter 4.4, Problem 7E

In Exercises 7 and 8, let $G$ be the multiplicative group of permutation matrices $\{I_3,P_3,P_3{}^2,P_1,P_4,P_2\}$ in Example 6 of Section 3.5

Let $H$ be the subgroup of $G$ given by

$H=\{I_3,P_4\}=\left\{\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right),\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)\right\}$.

Find the distinct left cosets of $H$ in $G$, write out their elements, partition $G$ into left cosets of $H$, and give $[G:H]$.

Find the distinct right cosets of $H$ in $G$, write out their elements, and partition $G$ into right cosets of $H$.