Subgroup ($H$) information
| Description: | $\Sp(4,3)$ |
| Order: | \(51840\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 5 \) |
| Index: | \(81\)\(\medspace = 3^{4} \) |
| Exponent: | \(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \) |
| Generators: |
$\langle(4,21,17)(5,19,18)(6,20,16)(7,11,24)(8,12,22)(9,10,23)(28,41,54)(29,42,52) \!\cdots\! \rangle$
|
| Derived length: | $0$ |
The subgroup is maximal, nonabelian, and quasisimple (hence nonsolvable and perfect).
Ambient group ($G$) information
| Description: | $C_3^4:\Sp(4,3)$ |
| Order: | \(4199040\)\(\medspace = 2^{7} \cdot 3^{8} \cdot 5 \) |
| Exponent: | \(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \) |
| Derived length: | $0$ |
The ambient group is nonabelian and perfect (hence nonsolvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3:S_3.\SO(5,3)$, of order \(8398080\)\(\medspace = 2^{8} \cdot 3^{8} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $\SO(5,3)$, of order \(51840\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 5 \) |
| $W$ | $\SU(4,2)$, of order \(25920\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $81$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^4:\Sp(4,3)$ |