Exercise 4, § 5

*Let $G_1 = \mathbf{SL}_3(\mathbf{F}_2)$. a) Show that $G_1$ is a non-abelian simple group of order 168, containing 21 elements of order 2. b) Show that the degrees of the irreducible complex representations of $G_1$ are 1, 3, 3, 6, 7, 8. c) Let $\rho : G_1 \to \mathbf{GL}_3(\mathbf{C})$ be an irreducible representation $^{10}$ of $G_1$ of degree 3. If $y \in G_1$ is of order 2, show that $\mathrm{Tr}(\rho(y)) = -1$. Deduce that $-\rho(y)$ is a reflection. d) Let $G$ be the subgroup of $\mathbf{GL}_3(\mathbf{C})$ generated by the elements $-\rho(y)$, for $y$ of order 2 in $G_1$. Show that $G$ is isomorphic to $G_1 \times \{1, -1\}$, and is thus of order 336.

$^{10}$ A detailed study of such a representation can be found in H. WEBER, Lehrbuch der Algebra, Bd. II, Abschn. 15.

e) Show that the characteristic degrees $k_1, k_2, k_3$ of the algebra of symmetric invariants of $G$ are equal to 4, 6 and 14. (Use the relations $\prod_i k_i = 336$ and $\sum_i (k_i - 1) = 21.$)

f) Show that $G$ is not a Coxeter group.*