Subgroup ($H$) information
| Description: | $C_9:C_{36}$ |
| Order: | \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| Index: | \(4096\)\(\medspace = 2^{12} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$\langle(2,8,18)(3,23,22,13,14,9,19,5,17)(7,15,12)(10,16,21), (3,13,19)(5,23,14) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, and metacyclic (hence solvable, supersolvable, monomial, and metabelian).
Ambient group ($G$) information
| Description: | $C_2^{12}.C_9:C_{36}$ |
| Order: | \(1327104\)\(\medspace = 2^{14} \cdot 3^{4} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_2^{12}.C_9^2:C_6$, of order \(3981312\)\(\medspace = 2^{14} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | $C_9:C_6^2$, of order \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| $W$ | $C_9:C_6$, of order \(54\)\(\medspace = 2 \cdot 3^{3} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_9:C_{36}$ |
| Normal closure: | $C_2^{12}.C_9:C_{36}$ |
| Core: | $C_2$ |
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 | $C_2^{12}.C_9:C_{18}$ |