6 The group \(( G , \boldsymbol { A } )\) has the elements \(e , r , r ^ { 2 } , q , q r\) and \(q r ^ { 2 }\), where \(r ^ { 2 } = r \boldsymbol { \Delta } r , q r = q \boldsymbol { \Delta } r , q r ^ { 2 } = q \boldsymbol { \Delta } r ^ { 2 }\) and \(e\) is the identity element of \(G\). The elements \(q\) and \(r\) have the following properties: $$\begin{aligned} & r \boldsymbol { \Delta } r \boldsymbol { \Delta } r = e
& q \boldsymbol { \Delta } q = e
& r ^ { 2 } \boldsymbol { \Delta } q = q \boldsymbol { \Delta } r \end{aligned}$$ 6

1. 1. State the order of \(G\). 6
2. (ii) Prove that the inverse of \(q r\) is \(q r\).
   6
3. Complete the Cayley table for elements of \(G\). 6
4. Complete the Cayley table for elements of \(G\).

   A	\(e\)	\(r\)	\(r ^ { 2 }\)	\(q\)	\(q r\)	\(q r ^ { 2 }\)
   \(e\)	\(e\)	\(r\)	\(r ^ { 2 }\)	\(q\)	\(q r\)	\(q r ^ { 2 }\)
   \(r\)	\(r\)	\(r ^ { 2 }\)	\(e\)
   \(r ^ { 2 }\)	\(r ^ { 2 }\)	\(e\)	\(r\)
   \(q\)	\(q\)	\(q r\)	\(q r ^ { 2 }\)	\(e\)
   \(q r\)	\(q r\)	\(q r ^ { 2 }\)	\(q\)	\(r ^ { 2 }\)
   \(q r ^ { 2 }\)	\(q r ^ { 2 }\)	\(q\)	\(q r\)	\(r\)	\(r ^ { 2 }\)	\(e\)

   6
5. State the name of a group which is isomorphic to \(G\).