The dihedral group $D_{6}$ of order $12$ is generated by two elements $a, b$ satisfying the relations $a^{6} = b^2 = 1, ba = a^{-1}b.$. For each of the following subgroups $H$ of $D_{6}$, determine if each of the following pairs of elements represent the same left $H$-coset:

$H$	Element $\#1$	Element $\#2$	Answer (Y/N)
$\langle a^{3} \rangle$	$a^{1}$	$a^{4}$
$\langle a^{3} \rangle$	$a^{1} b$	$a^{4} b$
$\langle a^{3} b\rangle$	$a^{1}$	$a^{4}$
$\langle a^{3} b\rangle$	$a^{1} b$	$a^{4} b$