The character tables of the irreducible representations of $T_d$ and $C_{3v}$ are linked. In the notation on those pages, $A_1$ and $A_2$ are irreducible representations of degree 1, $E$ is degree 2 and $T$ is degree 3. All the representation are in characteristic 0. Refer only to the character tables, not the product tables.
$T_d$ is isomorphic to the symmetric group $S_4$ . In the representations of $T_d$, identify the conjugacy class $E$ with identity element, the conjugacy class $8C_3$ with $8$ elements which are $3-$cycles, the class $3C_2$ as the $3$ elements which are product of $2$ $2-$cycles, the class $6\sigma_d$ with 6 elements which are $2-$cycles, and the class $6S_4$ with 6 elements which are $4-$cycles.
$C_{3v}$ is isomorphic to $S_3$. In the representations of $C_{3v}$, identify the conjugacy class $E$ with identity element, the conjugacy class $2C_3$ with $2$ elements which are $3-$cycles and the class $3\sigma_v$ with 3 elements which are $2-$cycles.
If $V_N=\{(), (1 2)(3 4), (13)(24),(14)(23)\}$ is the normal Kelin-4 subgroup of $S_4$, then $S_4=S_3\ltimes V_N$. In the isomorphism $$S_4/V_N\cong S_3,$$ it is readily verified that the $3-$cycles in $S_4$ map to $3-$cycles in $S_3$, the $2-$cycles as well as the $4-$cycles in $S_4$ map to $2-$cycles in $S_3$, whereas the elements which are products of 2 $2-$cycles in $S_4$ map to the identity in $S_3$.
The point that comes across is that the characters of all the classes in $T_d$ in representations which are identified by the same symbol in $C_{3v}$ are exactly equal to the characters of the classes in $C_{3v}$ to which those classes in $T_d$ are mapped. For example, in the representations $E$, the characters of $6\sigma_d$ and the $6S_4$ classes in the group $T_d$ are the same as that of the class $3\sigma_v$ of the group $C_{3v}$. Similarly for other classes and characters.
I know how to calculate characters of irreducible representations of such a group which is a direct products of its subgroups (whose irreducible characters are known). Is there a general way to do this for groups which are semi-direct products as above? Can the observation made in the previous paragraph be generalized in some way, or these observations are specific to this particular example?
