(* The character table. *)
Definition irr_class i := enum_val (cast_ord (NirrE G) i).
Definition class_Iirr xG :=
cast_ord (esym (NirrE G)) (enum_rank_in (classes1 G) xG).
Local Notation c := irr_class.
Local Notation g i := (repr (c i)).
Local Notation iC := class_Iirr.
Definition character_table := \matrix_(i, j) 'chi[G]_i (g j).
Local Notation X := character_table.

(* Specifications: *)
Lemma irr_classP i : c i \in classes G.
Let X' := \matrix_(i, j) (#|'C_G[g i]|%:R^-1 * ('chi[G]_j (g i))^*).