Subgroup ($H$) information
| Description: | $C_3^5:(C_2\times S_4)$ |
| Order: | \(11664\)\(\medspace = 2^{4} \cdot 3^{6} \) |
| Index: | \(64\)\(\medspace = 2^{6} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$\langle(13,19,18)(15,20,16), (15,20,16), (17,21)(18,19), (4,10)(7,12)(8,9), (2,10) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is maximal, nonabelian, and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $(C_2^3\times C_6^3).C_6^2:D_6$ |
| Order: | \(746496\)\(\medspace = 2^{10} \cdot 3^{6} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $4$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^6.(C_3^2\times S_3^3):D_6$, of order \(1492992\)\(\medspace = 2^{11} \cdot 3^{6} \) |
| $\operatorname{Aut}(H)$ | $(C_3^2\times S_3^3):D_6$, of order \(23328\)\(\medspace = 2^{5} \cdot 3^{6} \) |
| $W$ | $C_3^5:(C_2\times S_4)$, of order \(11664\)\(\medspace = 2^{4} \cdot 3^{6} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_3^5:(C_2\times S_4)$ |
| Normal closure: | $(C_2^3\times C_6^3).C_6^2:D_6$ |
| Core: | $C_3:S_3^2$ |
Other information
| Number of subgroups in this autjugacy class | $64$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $(C_2^3\times C_6^3).C_6^2:D_6$ |