Are there many ways to construct quotient group?
For example
Suppose
$(Z/13Z)^*=\{1,2,3,4,5,6,7,8,9,10,11,12\}$
$\langle 3 \rangle = \{1,3,9\} \mod 13$
then, what is $(Z/13Z)^*/\langle 3 \rangle?$
At least one coset is $\{1,3,9\}$. But, how can we divide $(Z/13Z)^*$?
My answer is $\{\{1,3,9\},\{5,2,6\},\{12,10,4\},\{8,11,7\}\} =\{\langle 3 \rangle,5\langle 3 \rangle, 5^2\langle 3 \rangle, 5^3\langle 3 \rangle \} $.
But there exist another answer : $\{\{1,3,9\},\{2,6,5\},\{4,12,10\},\{8,11,7\}\}$
Which is correct?