Find all left cosets $H = \{(1,2), id \}$ - subgroup of $S_3$. Find all left cosets of the subgroup $H$ of $S_3$.
I know left cosets should exhaust $S_3$ and left coset looks like $ \{ \tau \cdot (1 2), \tau \cdot id \}$. 
At first, i thought i need just evaluate all possible multiplications $ \{ \tau \cdot (1 2), \tau \cdot id \}$, where $\tau \in S_3$, but $\tau \notin H$. However, i'm assuming i don't need to take all possible multiplications, because $S_3$ will exhaust earlier. 
The second concerning thing is that multiple evaluation of permutation multiplications is very exhaustive action and it's definitely not the right thing to do, so is there any special notation, which helps write down the answer?

You're right, you don't need to take all of them as different $\tau_1, \tau_2 \in S$ may be in the same left coset and thus $\tau_1\in \tau_2 H$ and equivalently $\tau_2 \in \tau_1 H$. For example the cosets $(1,2)H$ is just $H$ as is $id H$. 
Permutation multiplication can be annoying, but in this case, $S_3$ is small enough to do everything out by hand. Are you using two-line notation or cyclic notation? I find the latter works well in small cases like this. 
See: Permutation Notations
