Function permutation_character_decomposition 

pub fn permutation_character_decomposition(
    degree: usize,
    gens: &[Perm],
    cap: usize,
) -> Option<(Vec<u128>, Vec<u128>, Vec<Vec<u128>>)>

The permutation-character decomposition — the bridge between the table of marks and the character table. M[i][s] is the multiplicity of the irreducible χ_s in the permutation representation of G on the cosets G/H_i, i.e. M[i][s] = ⟨Ind_{H_i}^G 1, χ_s⟩ = (1/|H_i|)·Σ_{h ∈ H_i} χ_s(h) (Frobenius reciprocity = the dimension of the H_i-fixed subspace of χ_s). Rows are subgroup conjugacy classes (as in table_of_marks), columns are irreducibles (as in character_table). Returns (subgroup_orders, irreducible_degrees, M).

This is the “linearization” map from the Burnside ring to the representation ring made explicit. The character values are read off the GF(p) character table; each multiplicity is a small non-negative integer (≤ d_s), so it decodes uniquely. FAIL-CLOSED: None unless M[1] = the degrees (the regular representation contains each χ_s with multiplicity d_s), M[i][trivial] = 1 (every transitive action contains the trivial character once), M[G] = e_trivial, and Σ_s M[i][s]·d_s = [G:H_i] for every row.