Subgroup ($H$) information
| Description: | $C_2\times C_6^4.S_3^2$ |
| Order: | \(93312\)\(\medspace = 2^{7} \cdot 3^{6} \) |
| Index: | \(9\)\(\medspace = 3^{2} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$\langle(1,18,5,13,2,8)(3,6,12)(4,14,7,16,11,9)(10,17,15)(19,23,20,22)(21,26,25,24) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_2^5.C_3\wr S_3^2$ |
| Order: | \(839808\)\(\medspace = 2^{7} \cdot 3^{8} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2\times C_6^3).C_3^3.C_6^3.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2^8.D_5^2.C_2^3$, of order \(2239488\)\(\medspace = 2^{10} \cdot 3^{7} \) |
| $W$ | $C_6^4.S_3^2$, of order \(46656\)\(\medspace = 2^{6} \cdot 3^{6} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_6^4.(C_6\times S_3^2)$ |
| Normal closure: | $C_6^4.C_6:S_3^2$ |
| Core: | $(C_3^2\times C_6^3).S_4$ |
Other information
| Number of subgroups in this autjugacy class | $6$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | not computed |