Subgroup ($H$) information
| Description: | $C_9^4.C_6^3.C_2^2$ |
| Order: | \(5668704\)\(\medspace = 2^{5} \cdot 3^{11} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$\langle(1,15,26,2,13,27,3,14,25)(7,9,8)(10,34,24,11,35,22,12,36,23)(19,21,20)(31,33,32) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_3^6.(C_3\times S_3\wr D_4)$ |
| Order: | \(22674816\)\(\medspace = 2^{7} \cdot 3^{11} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.(C_3\times S_3\wr D_4)$, of order \(22674816\)\(\medspace = 2^{7} \cdot 3^{11} \) |
| $\operatorname{Aut}(H)$ | $C_3^7.S_3\wr C_2^2$, of order \(11337408\)\(\medspace = 2^{6} \cdot 3^{11} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_3^7.S_3\wr C_2^2$ |
| Normal closure: | $C_3^7.S_3\wr C_2^2$ |
| Core: | $C_9^4.C_6^3.C_2$ |
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 | not computed |