Subgroup ($H$) information
| Description: | $C_2^9.A_4^3:S_4$ |
| Order: | \(21233664\)\(\medspace = 2^{18} \cdot 3^{4} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Generators: |
$\langle(29,30)(31,32)(33,34)(35,36), (1,7,11)(2,8,12)(3,6,9)(4,5,10)(13,22,20,15,23,18) \!\cdots\! \rangle$
|
| Derived length: | $5$ |
The subgroup is nonabelian and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_2^{12}.A_4^3:S_4$ |
| Order: | \(169869312\)\(\medspace = 2^{21} \cdot 3^{4} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $5$ |
The ambient group is nonabelian and solvable. 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(5435817984\)\(\medspace = 2^{26} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | $C_2^{12}.C_3^3.C_2^4.C_6.C_2^3$, of order \(84934656\)\(\medspace = 2^{20} \cdot 3^{4} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Normal closure: | not computed |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $4$ |
| Möbius function | not computed |
| Projective image | not computed |