Subgroup ($H$) information
| Description: | $\SL(2,3)$ |
| Order: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Index: | \(12960\)\(\medspace = 2^{5} \cdot 3^{4} \cdot 5 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,70,54,22)(2,41,77,3)(5,33,43,45)(6,67,39,66)(7,81,21,78)(8,71,19,74) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and solvable.
Ambient group ($G$) information
| Description: | $C_3^3:S_3.C_2^4:S_5$ |
| Order: | \(311040\)\(\medspace = 2^{8} \cdot 3^{5} \cdot 5 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3:S_3.C_2^4:S_5$, of order \(311040\)\(\medspace = 2^{8} \cdot 3^{5} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $W$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_6$ | ||
| Normalizer: | $C_3\times \GL(2,3)$ | ||
| Normal closure: | $C_3^4.(C_4.C_2^3).A_5$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $\PU(3,2)$ | $C_3\times \SL(2,3)$ | $\GL(2,3)$ |
| Maximal under-subgroups: | $Q_8$ | $C_6$ |
Other information
| Number of subgroups in this conjugacy class | $2160$ |
| Möbius function | $0$ |
| Projective image | $C_3^3:S_3.C_2^4:S_5$ |