Return to Revisions

The irreducible character degrees for this group have degree 1,3,3,6,7,8. To get the irreducible degree $7$-representation, induce a non trivial linear character of either of the maximal parabolics isomorphic to $S_4$. To get the degree $8$ irreducible, induce a non-trivial linear character of the Sylow $7$-normalizer ( a Frobenius group of order $21$). To get the degree $6$ irreducible, induce the trivial character of the Sylow $7$-normalizer, obtaining an orthogonal representation. Then take the orthogonal complement of the $1$-dimensional fixed-point space. To get one of the two $3$-dimensional representations, induce a non-trivial linear character of the Sylow $7$-subgroup. The already constructed (unitary) $8$-dimensional represntation shows up, as does the $6$-dimensional (unitary) repesentation and the $7$-dimensional unitary irreducible representation. Take the orthogonal complement of the sum of these. This gives a $3$-dimensional unitary representation. Take the dual of that as well, and you have all (non-trivial) representations, up to equivalence.