Subgroup ($H$) information
| Description: | $C_3^3:S_4$ |
| Order: | \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
| Index: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$\left(\begin{array}{llll}0 & 0 & \alpha^{2} & 0 \\ \alpha & 0 & 1 & \alpha \\ \alpha & 0 & 0 & 0 \\ 1 & \alpha^{2} & \alpha^{2} & 0 \\ \end{array}\right), \left(\begin{array}{llll}\alpha^{2} & \alpha^{2} & \alpha & \alpha \\ 0 & 0 & \alpha & \alpha \\ 0 & \alpha & \alpha & 1 \\ 0 & 0 & 0 & \alpha^{2} \\ \end{array}\right), \left(\begin{array}{llll}1 & 0 & 0 & 0 \\ \alpha & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & \alpha^{2} & 1 \\ \end{array}\right), \left(\begin{array}{llll}1 & \alpha^{2} & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & 1 & \alpha \\ 0 & 0 & 0 & 1 \\ \end{array}\right), \left(\begin{array}{llll}0 & 0 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & \alpha^{2} & \alpha \\ \alpha^{2} & \alpha & \alpha & \alpha^{2} \\ \end{array}\right), \left(\begin{array}{llll}0 & \alpha^{2} & \alpha^{2} & 1 \\ 0 & \alpha & \alpha & 1 \\ \alpha & 1 & 0 & \alpha \\ \alpha^{2} & 1 & 0 & \alpha^{2} \\ \end{array}\right), \left(\begin{array}{llll}\alpha^{2} & \alpha^{2} & \alpha^{2} & 0 \\ 0 & \alpha & 0 & \alpha \\ 0 & 0 & \alpha & \alpha \\ 0 & 0 & 0 & \alpha^{2} \\ \end{array}\right)$
|
| Derived length: | $4$ |
The subgroup is maximal, nonabelian, and monomial (hence solvable).
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).
Quotient set structure
Since this subgroup has trivial core, the ambient group $G$ acts faithfully and transitively on the set of cosets of $H$. The resulting permutation representation is isomorphic to 40T14345.
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 \(51840\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $S_3\wr S_3$, of order \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| $W$ | $C_3^3:S_4$, of order \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
Related subgroups
| Centralizer: | $C_1$ | |||
| Normalizer: | $C_3^3:S_4$ | |||
| Normal closure: | $\SU(4,2)$ | |||
| Core: | $C_1$ | |||
| Minimal over-subgroups: | $\SU(4,2)$ | |||
| Maximal under-subgroups: | $C_3^3:A_4$ | $S_3^2:S_3$ | $C_3\wr S_3$ | $S_4$ |
Other information
| Number of subgroups in this conjugacy class | $40$ |
| Möbius function | $-1$ |
| Projective image | $\SU(4,2)$ |