The symmetric group $S_3$ has three conjugacy classes: $\{e\}$, transpositions $\{(12), (13), (23)\}$, and 3-cycles $\{(123), (132)\}$. The character table is:

| Rep | $e$ | $(12)$ | $(123)$ | |-----|-----|--------|---------| | $\chi_{\text{triv}}$ | 1 | 1 | 1 | | $\chi_{\text{sgn}}$ | 1 | -1 | 1 | | $\chi_{\text{std}}$ | 2 | 0 | -1 |

The rows are orthonormal under the inner product $\langle \chi, \psi \rangle = \frac{1}{6}(1 \cdot \overline{\chi(e)}\psi(e) + 3 \cdot \overline{\chi((12))}\psi((12)) + 2 \cdot \overline{\chi((123))}\psi((123)))$.

The dihedral group $D_4$ (symmetries of a square) has order 8 and five conjugacy classes. The dimensions satisfy $1^2 + 1^2 + 1^2 + 1^2 + 2^2 = 8$. The character table:

| Rep | $e$ | $r^2$ | $r, r^3$ | $s, sr^2$ | $sr, sr^3$ | |-----|-----|-------|----------|-----------|------------| | $\chi_1$ | 1 | 1 | 1 | 1 | 1 | | $\chi_2$ | 1 | 1 | 1 | -1 | -1 | | $\chi_3$ | 1 | 1 | -1 | 1 | -1 | | $\chi_4$ | 1 | 1 | -1 | -1 | 1 | | $\chi_5$ | 2 | -2 | 0 | 0 | 0 |

where $r$ is rotation by $90°$ and $s$ is a reflection.

Given the regular representation of $S_3$ with character $\chi_{\text{reg}} = (6, 0, 0)$, we compute: $\langle \chi_{\text{reg}}, \chi_i \rangle = \chi_i(e) = \dim V_i$

Thus $\mathbb{C}[S_3] \cong V_{\text{triv}} \oplus V_{\text{sgn}} \oplus V_{\text{std}}^{\oplus 2}$, confirming that each irreducible appears with multiplicity equal to its dimension.