Subgroup ($H$) information
| Description: | $D_6$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Index: | \(2160\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 5 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\left(\begin{array}{llll}1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 \\ \end{array}\right), \left(\begin{array}{llll}0 & 1 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 \\ 1 & 0 & 0 & 1 \\ \end{array}\right), \left(\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ \end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, an A-group, and rational.
Ambient group ($G$) information
| Description: | $\SU(4,2)$ |
| Order: | \(25920\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5 \) |
| Exponent: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\SO(5,3)$, of order \(51840\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $W$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
| Centralizer: | $C_6$ | |||
| Normalizer: | $C_6\times S_3$ | |||
| Normal closure: | $\SU(4,2)$ | |||
| Core: | $C_1$ | |||
| Minimal over-subgroups: | $S_5$ | $C_2\times S_4$ | $C_6\times S_3$ | $S_3^2$ |
| Maximal under-subgroups: | $C_6$ | $S_3$ | $S_3$ | $C_2^2$ |
Other information
| Number of subgroups in this conjugacy class | $720$ |
| Möbius function | $-3$ |
| Projective image | $\SU(4,2)$ |