Example. semidirect products make Clifford theory concrete [fgap-0020]
Example. semidirect products make Clifford theory concrete [fgap-0020]
Suppose \(G=N\rtimes H\). Conjugation by \(H\) permutes the irreducible constituents of \(V|_N\). Their stabilizers are the inertia subgroups, so the orbit and homogeneity statements in the single orbit in Clifford restriction and Clifford homogeneity and the semidirect route reduce the search for irreducible \(G\)-representations to stabilizer data, projective extension, and induction. Clifford theory thereby gives a character-theoretic route for semidirect products.
For \(Q_8\rtimes C_3\cong 2T\), the action and ordinary skew-Group-Algebra realization are recorded in the central split of the real Group Algebra of 2T and two routes from binary tetrahedral symmetry. They supply the normal-subgroup and algebraic input to this route, but do not give the full character table of \(2T\).