= Solution
Act on vectors in the plane of the regular hexagon using its actual <rotations> and <reflections>. Choose the first reflection axis as the horizontal axis. A <faithful representation> is generated by
$$
\boxed{D_2(R)=\begin{pmatrix}1/2&-\sqrt3/2\\\sqrt3/2&1/2\end{pmatrix},\qquad D_2(m_1)=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.}
$$
They are real <orthogonal matrices> obeying $D_2(R)^6=I$, $D_2(m_1)^2=I$ and $D_2(m_1)D_2(R)D_2(m_1)=D_2(R)^{-1}$. Hence they reproduce the group relations. Since $m_2=Rm_1$,
$$
\boxed{D_2(m_2)=\begin{pmatrix}1/2&\sqrt3/2\\\sqrt3/2&-1/2\end{pmatrix}.}
$$
This is reflection in the line at angle $\pi/6$. More generally $D_2(m_j)=D_2(R)^{j-1}D_2(m_1)$. The six rotation matrices are distinct, as are the six reflection matrices; their determinant signs distinguish the two sets. Thus no two group elements have the same matrix, proving faithfulness rather than merely the representation property.
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