  A twirling channel is a quantum channel induced by a continuous unitary
representation $\pi = \sum_{i}^{\oplus} m_i\pi_i$, where $\pi_i$ are
inequivalent irreducible representations. Motivated by a recent work
\cite{Twirling} on minimal mixed unitary rank of $\Phi_{\pi}$, we explore the
connections of the independence number, zero error capacity, quantum codes,
orthogonality index and phase retrievability of the quantum channel
$\Phi_{\pi}$ with the irreducible representation multiplicities $m_i$, the
irreducible representation dimensions $\dim H_{\pi_i}$. In particular we show
that the independence number of $\Phi_{\pi}$ is the sum of the multiplicities,
the orthogonal index of $\Phi_{\pi}$ is exactly the sum of those representation
dimensions, and the zero-error capacity is equal to $\log (\sum_{i=1}^{d}m_i)$.
We also present a lower bound for the phase retrievability in terms of the
minimal length of phase retrievable frames for $C^n$.
