Subgroup ($H$) information
| Description: | not computed |
| Order: | \(3456\)\(\medspace = 2^{7} \cdot 3^{3} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | not computed |
| Generators: |
$\langle(5,9)(10,15), (10,15)(16,17), (4,6)(5,9), (3,4,6), (1,8,7)(3,4,6), (16,17) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is nonabelian and solvable. Whether it is elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_2^2\times S_3^3:S_4$ |
| Order: | \(20736\)\(\medspace = 2^{8} \cdot 3^{4} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $4$ |
The ambient group is nonabelian, solvable, and rational. Whether it is monomial has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^4:(C_2\times C_4)$, of order \(995328\)\(\medspace = 2^{12} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $W$ | $S_3^3:C_2^2$, of order \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_2^2\times C_3^3.C_2^5.C_2$ |
| Normal closure: | $C_2^2\times S_3^3:S_4$ |
| Core: | $C_3:D_6^2$ |
Other information
| Number of subgroups in this autjugacy class | $6$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | $S_3^3:S_4$ |