Complete set coset representatives of a subgroup $H$ in a group $G$

Question

Let $G$ be a group and $H$ be a finite subgroup of index $n$ in $G$. Is there any systematic way of finding a complete set of coset representatives(That is a set of coset representatives for all conjugates of $H$ in $G$) for $H$ in $G$?

I have a group $G$ of order 24 inside $S_6$ and a subgroup $H$ of order 4 inside $G$ whose conjugacy class of subgroups has 3 elements. I want to find a complete set of coset representatives for it. Is there any way to do it?

Answer 1

Generally, the way for computing all the (right) cosets of a (finite) group $G$ is as follows:

To compute all the right costs of a subgroup $H$ in a finite group $G$, first write $H$, then choose any $a \in G$ such that $a \notin H$, and compute $Ha$. Next, choose any $b \in G$ such that $b \notin H \cup Ha$, and compute $Hb$. Continue in this way until all elements of $G$ have been exhausted.

Hope this helps you.

Answer 2

It is not true in general that there exists a single set of size $n$ that forms a complete set of (right) coset representatives for $g^{-1}Hg$ in $G$ for all $g \in G$. For example this is not possible in a subgroup $H$ of order $2$ in $G = A_4$.