Subgroup ($H$) information
| Description: | $C_3^6.C_3\wr C_4$ |
| Order: | \(236196\)\(\medspace = 2^{2} \cdot 3^{10} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$\langle(1,15,27,3,14,26,2,13,25)(4,28,16,5,29,17,6,30,18)(7,21,32,8,19,33,9,20,31) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_3^6.(C_3^2\times S_3^2):C_4$ |
| Order: | \(944784\)\(\medspace = 2^{4} \cdot 3^{10} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(51018336\)\(\medspace = 2^{5} \cdot 3^{13} \) |
| $\operatorname{Aut}(H)$ | $C_3^5.C_3^5.C_2.C_6^3$, of order \(25509168\)\(\medspace = 2^{4} \cdot 3^{13} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_3^6.C_3^3:(C_4\times S_3)$ |
| Normal closure: | $C_3^6.C_3^3:(C_4\times S_3)$ |
| Core: | $C_3^5.C_3^4.C_6$ |
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^6.(C_3^2\times S_3^2):C_4$ |