Subgroup ($H$) information
| Description: | $S_4$ |
| Order: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Index: | \(1080\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 5 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\left(\begin{array}{llll}1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 \\ \end{array}\right), \left(\begin{array}{llll}0 & \alpha & 0 & 1 \\ \alpha^{2} & \alpha^{2} & 1 & \alpha \\ 0 & \alpha^{2} & 0 & 1 \\ \alpha & 0 & 1 & \alpha \\ \end{array}\right), \left(\begin{array}{llll}0 & \alpha & 0 & 1 \\ \alpha^{2} & 1 & 1 & 0 \\ \alpha^{2} & 1 & 1 & \alpha^{2} \\ 0 & \alpha & \alpha & 0 \\ \end{array}\right), \left(\begin{array}{llll}1 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ 1 & 1 & 1 & 1 \\ \end{array}\right)$
|
| Derived length: | $3$ |
The subgroup is nonabelian, monomial (hence solvable), 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 \(51840\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $W$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_2$ | ||
| Normalizer: | $C_2\times S_4$ | ||
| Normal closure: | $\SU(4,2)$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $C_3^3:S_4$ | $S_5$ | $C_2\times S_4$ |
| Maximal under-subgroups: | $A_4$ | $D_4$ | $S_3$ |
Other information
| Number of subgroups in this conjugacy class | $540$ |
| Möbius function | $2$ |
| Projective image | $\SU(4,2)$ |