Return to Answer

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 ether of the parabolics isomorphic to $S_{4},$ 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 representation 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) irreducible representations, up to equivalence.

Later edit: Note that for the irreducible characters of degree $6,7$ and $8$ the representations above may be explicitly given as real representations. The degree $8$ representation requires a little further thought. If we induce the trivial character from a Sylow $2$-subgroup, the $8$ dimensional irreducible character occurs with multiplicity $1,$ the trivial character occurs once, the $7$-dimensional irreducible character does not occur, the $6$-dimensional character occurs once and the two three dimensional character each occur once. Since the permuation representation is realized over $\mathbb{R},$ it follows that the $8$-dimensional representation may be reaized over the real field.

The two $3$-dimensional representations do not have real characters. To obtain a real representation of their sum, do the last procedure instead with a real irreducible two dimensional orthogonal representation of the Sylow $7$-subgroup (which is just as group of real rotations): two copies of each of the $6,7$ and $8$ degree real orthogonal representations show up, so take the orthogonal complement of their sum, obtaining an orthogonal representation of degree $6$ which is irreducible as a real representation, but not a complex representation.

The irreducible character degrees for this group have degree $1,3,3,6,7,8$.

• 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 $7$ irreducible, induce a non-trivial linear character of either of the maximal parabolics isomorphic to $S_4$.
• To get the degree $6$ irreducible, induce the trivial character of either of the parabolics isomorphic to $S_{4},$ 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 representation 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) irreducible representations, up to equivalence.

Later edit: Note that for the irreducible characters of degree $6,7$ and $8$ the representations above may be explicitly given as real representations. The degree $8$ representation requires a little further thought. If we induce the trivial character from a Sylow $2$-subgroup, the $8$-dimensional irreducible character occurs with multiplicity $1$, the trivial character occurs once, the $7$-dimensional irreducible character does not occur, the $6$-dimensional character occurs once and the two three dimensional character each occur once. Since the permutation representation is realized over $\mathbb{R},$ it follows that the $8$-dimensional representation may be realized over the real field.

The two $3$-dimensional representations do not have real characters. To obtain a real representation of their sum, do the last procedure instead with a real irreducible two dimensional orthogonal representation of the Sylow $7$-subgroup (which is just as group of real rotations): two copies of each of the $6,7$ and $8$ degree real orthogonal representations show up, so take the orthogonal complement of their sum, obtaining an orthogonal representation of degree $6$ which is irreducible as a real representation, but not a complex representation.

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.

Later edit: Note that for the irreducible characters of degree $6,7$ and $8$ the representations above are explicitly given as real representations. The two $3$-dimensional representations do not have real characters. To obtain a real representation of their sum, do the last procedure instead with a real irreducible two dimensional orthogonal representation of the Sylow $7$-subgroup (which is just as group of real rotations): two copies of each of the $6,7$ and $8$ degree real orthogonal representations show up, so take the orthogonal complement of their sum, obtaining an orthogonal representation of degree $6$ which is irreducible as a real representation, but not a complex representation.

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 ether of the parabolics isomorphic to $S_{4},$ 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 representation 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) irreducible representations, up to equivalence.

Later edit: Note that for the irreducible characters of degree $6,7$ and $8$ the representations above may be explicitly given as real representations. The degree $8$ representation requires a little further thought. If we induce the trivial character from a Sylow $2$-subgroup, the $8$ dimensional irreducible character occurs with multiplicity $1,$ the trivial character occurs once, the $7$-dimensional irreducible character does not occur, the $6$-dimensional character occurs once and the two three dimensional character each occur once. Since the permuation representation is realized over $\mathbb{R},$ it follows that the $8$-dimensional representation may be reaized over the real field.

The two $3$-dimensional representations do not have real characters. To obtain a real representation of their sum, do the last procedure instead with a real irreducible two dimensional orthogonal representation of the Sylow $7$-subgroup (which is just as group of real rotations): two copies of each of the $6,7$ and $8$ degree real orthogonal representations show up, so take the orthogonal complement of their sum, obtaining an orthogonal representation of degree $6$ which is irreducible as a real representation, but not a complex representation.

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.

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.

Later edit: Note that for the irreducible characters of degree $6,7$ and $8$ the representations above are explicitly given as real representations. The two $3$-dimensional representations do not have real characters. To obtain a real representation of their sum, do the last procedure instead with a real irreducible two dimensional orthogonal representation of the Sylow $7$-subgroup (which is just as group of real rotations): two copies of each of the $6,7$ and $8$ degree real orthogonal representations show up, so take the orthogonal complement of their sum, obtaining an orthogonal representation of degree $6$ which is irreducible as a real representation, but not a complex representation.