Supply finite groups \(N\triangleleft G\), elementary abelian quotient \(A=G/N\), and irreducibles \(\Pi\) and \(\rho\), with \(\rho\) fixed up to isomorphism by \(G\) and \(\Pi|_N\) isotypic. The multiplicity space is
For a section \(s:A\to G\) and intertwiners \(I_a\) of \(\rho\), write
The operators \(T_a\) give a projective action on \(M_\rho\). Section and intertwiner changes alter it only by scalar changes and conjugation. Its commutator, radical, and symplectic quotient are intrinsic to this supplied datum.
If the commutator is nondegenerate, \(\dim M_\rho=\sqrt{|A|}\). A tame stabilizer acting by automorphisms of the datum preserves the induced symplectic form.
Scope
The normal group, isotypic constituent, and one-break interpretation are assumptions of the restricted construction. They have not been constructed for every wild atom.