Subgroup ($H$) information
| Description: | $C_6^4:D_6$ |
| Order: | \(15552\)\(\medspace = 2^{6} \cdot 3^{5} \) |
| Index: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$\langle(19,24)(20,25)(21,26)(22,23), (19,20)(24,25), (2,9,17)(3,4,12)(6,7,18)(8,10,13) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_6^4.C_6:S_3^2$ |
| Order: | \(279936\)\(\medspace = 2^{7} \cdot 3^{7} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times C_6^4.C_3^5.C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_6^4.C_6^2.C_2^2$ |
| $W$ | $C_{11}:D_{11}^2$, of order \(5324\)\(\medspace = 2^{2} \cdot 11^{3} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_2\times C_6^4:D_6$ |
| Normal closure: | $C_3^5.\POPlus(4,3)$ |
| Core: | $C_3\times C_6^2.A_4$ |
Other information
| Number of subgroups in this autjugacy class | $36$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | not computed |
| Projective image | $C_6^4.C_6:S_3^2$ |