Subgroup ($H$) information
| Description: | $\Unitary(2,3)$ |
| Order: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Index: | \(43740\)\(\medspace = 2^{2} \cdot 3^{7} \cdot 5 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\langle(1,61,73,22)(2,44,75,42)(3,27,74,59)(4,9,79,77)(5,70,81,13)(6,53,80,33) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is nonabelian and solvable.
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 \(8398080\)\(\medspace = 2^{8} \cdot 3^{8} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_2^2\times S_4$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| $W$ | $C_2\times S_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_4$ | ||
| Normalizer: | $D_4.S_4$ | ||
| Normal closure: | $C_3^4:\Sp(4,3)$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $C_3^4:\Unitary(2,3)$ | $D_4.S_4$ | |
| Maximal under-subgroups: | $\SL(2,3):C_2$ | $C_4\wr C_2$ | $C_3:C_8$ |
Other information
| Number of subgroups in this autjugacy class | $21870$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^4:\Sp(4,3)$ |