Extend Clifford class docs (#12177)

* extend clifford docs

* mimor fix

* minor fix

* update docs following comment

* update following review

qiskit/quantum_info/operators/symplectic/clifford.py

@@ -37,7 +37,23 @@
 
 
 class Clifford(BaseOperator, AdjointMixin, Operation):
-    """An N-qubit unitary operator from the Clifford group.
+    r"""
+    An N-qubit unitary operator from the Clifford group.
+
+    An N-qubit Clifford operator takes Paulis to Paulis via conjugation
+    (up to a global phase). More precisely, the Clifford group :math:`\mathcal{C}_N`
+    is defined as
+
+     .. math::
+
+        \mathcal{C}_N = \{ U \in U(2^N) | U \mathcal{P}_N U^{\dagger} = \mathcal{P}_N \} / U(1)
+
+     where :math:`\mathcal{P}_N` is the Pauli group on :math:`N` qubits
+     that is generated by single-qubit Pauli operators,
+     and :math:`U` is a unitary operator in the unitary group
+     :math:`U(2^N)` representing operations on :math:`N` qubits.
+     :math:`\mathcal{C}_N` is the quotient group by the subgroup of
+     scalar unitary matrices :math:`U(1)`.
 
     **Representation**
 
@@ -91,7 +107,7 @@ class Clifford(BaseOperator, AdjointMixin, Operation):
     :class:`~qiskit.circuit.library.SGate`, :class:`~qiskit.circuit.library.SdgGate`,
     :class:`~qiskit.circuit.library.SXGate`, :class:`~qiskit.circuit.library.SXdgGate`,
     :class:`~qiskit.circuit.library.CXGate`, :class:`~qiskit.circuit.library.CZGate`,
-    :class:`~qiskit.circuit.library.CYGate`, :class:`~qiskit.circuit.library.DXGate`,
+    :class:`~qiskit.circuit.library.CYGate`, :class:`~qiskit.circuit.library.DCXGate`,
     :class:`~qiskit.circuit.library.SwapGate`, :class:`~qiskit.circuit.library.iSwapGate`,
     :class:`~qiskit.circuit.library.ECRGate`, :class:`~qiskit.circuit.library.LinearFunction`,
     :class:`~qiskit.circuit.library.PermutationGate`.