Quaternion Group/Cayley Table/Coset Decomposition of (e, a^2)

Cayley Table for Quaternion Group $Q$
The Cayley table for the quaternion group $Q$, with respect to the coset decomposition of the normal subgroup $\gen a^2$, is:

can be presented as:
 * $\begin{array}{r|rr|rr|rr|rr}

& e    & a^2   & a     & a^3   & b     & a^2 b & a b   & a^3 b \\ \hline e    & e     & a^2   & a     & a^3   & b     & a^2 b & a b   & a^3 b \\ a^2  & a^2   & e     & a^3   & a     & a^2 b & b     & a^3 b & a b   \\ \hline a    & a     & a^3   & a^2   & e     & a b   & a^3 b & a^2 b & b     \\ a^3  & a^3   & a     & e     & a^2   & a^3 b & a b   & b     & a^2 b \\ \hline b    & b     & a^2 b & a^3 b & a b   & a^2   & e     & a     & a^3   \\ a^2 b & a^2 b & b    & a b   & a^3 b & e     & a^2   & a^3   & a     \\ \hline a b  & a b   & a^3 b & b     & a^2 b & a^3   & a     & a^2   & e     \\ a^3 b & a^3 b & a b  & a^2 b & b     & a     & a^3   & e     & a^2 \end{array}$

which is seen to be an example of the Klein $4$-group.