Subgroup ($H$) information
| Description: | $C_6^2:S_3^2$ |
| Order: | \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| Index: | \(4096\)\(\medspace = 2^{12} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,3,16,2,18,11)(4,9,6,21,5,24)(7,8,19,10,14,22)(12,17,13,15,23,20)(26,28) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_2^{12}.C_6^2:S_3^2$ |
| Order: | \(5308416\)\(\medspace = 2^{16} \cdot 3^{4} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^{12}.(C_3^2:D_6\times S_4)$, of order \(10616832\)\(\medspace = 2^{17} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | $C_3^2:D_6\times S_4$, of order \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
| $W$ | $C_6^2:S_3^2$, of order \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_6^2:S_3^2$ |
| Normal closure: | $C_2^{12}.C_6^2:S_3^2$ |
| Core: | $C_2^2$ |
Other information
| Number of subgroups in this autjugacy class | $4096$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_2^{12}.C_6^2:S_3^2$ |