Subgroup ($H$) information
| Description: | $\OD_{16}$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Index: | \(378\)\(\medspace = 2 \cdot 3^{3} \cdot 7 \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$\left(\begin{array}{lll}0 & 0 & \alpha \\ 0 & \alpha^{6} & 0 \\ \alpha^{5} & 0 & \alpha^{3} \\ \end{array}\right), \left(\begin{array}{lll}\alpha^{2} & \alpha^{6} & \alpha^{7} \\ \alpha^{7} & \alpha^{5} & 1 \\ \alpha^{3} & \alpha^{3} & \alpha^{3} \\ \end{array}\right), \left(\begin{array}{lll}\alpha^{6} & 0 & \alpha^{4} \\ 0 & \alpha^{4} & 0 \\ 1 & 0 & \alpha^{2} \\ \end{array}\right), \left(\begin{array}{lll}\alpha^{5} & \alpha^{3} & \alpha^{4} \\ 1 & 0 & \alpha \\ 1 & 1 & \alpha^{7} \\ \end{array}\right)$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metacyclic (hence metabelian).
Ambient group ($G$) information
| Description: | $\SU(3,3)$ |
| Order: | \(6048\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 7 \) |
| Exponent: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Derived length: | $0$ |
The ambient group is nonabelian and simple (hence nonsolvable, perfect, quasisimple, and almost simple).
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $G(2,2)$, of order \(12096\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_2\times D_4$, of order \(16\)\(\medspace = 2^{4} \) |
| $W$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
| Centralizer: | $C_4$ | |
| Normalizer: | $C_4\wr C_2$ | |
| Normal closure: | $\SU(3,3)$ | |
| Core: | $C_1$ | |
| Minimal over-subgroups: | $C_4\wr C_2$ | |
| Maximal under-subgroups: | $C_8$ | $C_2\times C_4$ |
Other information
| Number of subgroups in this conjugacy class | $189$ |
| Möbius function | $0$ |
| Projective image | $\SU(3,3)$ |