Subgroup ($H$) information
| Description: | $C_3^7.C_3^4:\SL(2,3)$ |
| Order: | \(4251528\)\(\medspace = 2^{3} \cdot 3^{12} \) |
| Index: | \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$\langle(7,8,9)(19,20,21)(31,32,33), (1,4)(2,5)(3,6)(7,22,33,10,8,23,31,11,9,24,32,12) \!\cdots\! \rangle$
|
| Derived length: | $5$ |
The subgroup is nonabelian and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_3^8.(C_3\times S_3\wr S_4)$ |
| Order: | \(612220032\)\(\medspace = 2^{7} \cdot 3^{14} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $6$ |
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 \(1224440064\)\(\medspace = 2^{8} \cdot 3^{14} \) |
| $\operatorname{Aut}(H)$ | $C_3\times C_3^7.C_3:S_3.A_4:S_3^2$, of order \(51018336\)\(\medspace = 2^{5} \cdot 3^{13} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Normal closure: | not computed |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $72$ |
| Möbius function | not computed |
| Projective image | not computed |