Subgroup ($H$) information
| Description: | $(C_2^2\times C_6^2).S_3^3$ |
| Order: | \(31104\)\(\medspace = 2^{7} \cdot 3^{5} \) |
| Index: | \(729\)\(\medspace = 3^{6} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,3,2)(4,16,30)(5,17,28)(6,18,29)(7,20,31)(8,21,32)(9,19,33)(22,24,23) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_3^8.C_2^4:S_3^3$ |
| Order: | \(22674816\)\(\medspace = 2^{7} \cdot 3^{11} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^8.(D_6\times S_4\wr C_2)$, of order \(90699264\)\(\medspace = 2^{9} \cdot 3^{11} \) |
| $\operatorname{Aut}(H)$ | $C_3^2:S_4^2:D_6$, of order \(62208\)\(\medspace = 2^{8} \cdot 3^{5} \) |
| $W$ | $(C_2^2\times C_6^2).S_3^3$, of order \(31104\)\(\medspace = 2^{7} \cdot 3^{5} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $(C_2^2\times C_6^2).S_3^3$ |
| Normal closure: | $C_3^8.C_2^4:S_3^3$ |
| Core: | $C_3^2:S_3$ |
Other information
| Number of subgroups in this autjugacy class | $729$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^8.C_2^4:S_3^3$ |