I am reading "Topics in Algebra 2nd Edition" by I. N. Herstein.
The following problem is Problem 21 on p.81 in this book:
Let $G$ be the group $\{e,\theta,a,b,c,\theta a,\theta b,\theta c\}$, where $a^2=b^2=c^2=\theta$, $\theta^2=e$, $ab=\theta ba=c$, $bc=\theta cb=a$, $ca=\theta ac=b$.
(a) Show that $\theta$ is in the center $Z$ of $G$, and that $Z=\{e,\theta\}$.
(b) Find the commutator subgroup of $G$.
(c) Show that every subgroup of $G$ is normal.
(d) Find the permutation representation of $G$.
(Note: $G$ is often called the group of quaternion units; it, and algebraic systems constructed from it, will reappear in the book.)
I tried to fill the Cayley table for this $G$.
But I was not able to fill the Cayley table for this $G$.
\begin{array}{c|cccccccc}
\cdot & e & \theta & a & b & c &\theta a&\theta b&\theta c\\
\hline
e & e & \theta & a & b & c &\theta a&\theta b&\theta c\\
\theta & \theta & e & \theta a & \theta b & \theta c &a&b&c\\
a & a & ? & \theta & c & \theta b &?&?&?\\
b & b & ? & \theta c & \theta & a &?&?&?\\
c & c & ? & b & \theta a & \theta &?&?&?\\
\theta a & \theta a & ? & e & \theta c & b &?&?&?\\
\theta b & \theta b & ? & c & e & \theta a &?&?&?\\
\theta c & \theta c & ? & \theta b & a & e &?&?&?\\
\end{array}
For example, how to calculate the value of $a\theta$?
postmortes, Thank you very much for your comment.
I was able to fill the Cayley table for this $G$.
\begin{array}{c|cccccccc}
\cdot & e & \theta & a & b & c &\theta a&\theta b&\theta c\\
\hline
e & e & \theta & a & b & c &\theta a&\theta b&\theta c\\
\theta & \theta & e & \theta a & \theta b & \theta c &a&b&c\\
a & a & \theta a & \theta & c & \theta b &e&\theta c&b\\
b & b & \theta b & \theta c & \theta & a &c&e&\theta a\\
c & c & \theta c & b & \theta a & \theta &\theta b&a&e\\
\theta a & \theta a & a & e & \theta c & b &\theta&c&\theta b\\
\theta b & \theta b & b & c & e & \theta a &\theta c&\theta&a\\
\theta c & \theta c & c & \theta b & a & e &b&\theta a&\theta\\
\end{array}