Subgroup ($H$) information
| Description: | $\He_3^2:(C_4\times S_3)$ |
| Order: | \(17496\)\(\medspace = 2^{3} \cdot 3^{7} \) |
| Index: | \(3\) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$\langle(19,21)(20,22), (1,9,14,2,15,18,7,11,13)(3,10,16,4,17,6,12,8,5), (3,12,4) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, and supersolvable (hence solvable and monomial).
Ambient group ($G$) information
| Description: | $(C_2\times \He_3^2).S_3^2$ |
| Order: | \(52488\)\(\medspace = 2^{3} \cdot 3^{8} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and supersolvable (hence solvable and monomial).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.C_3^2.C_6^2.C_2^4$, of order \(3779136\)\(\medspace = 2^{6} \cdot 3^{10} \) |
| $\operatorname{Aut}(H)$ | $C_3^5.C_3^3.C_2^5$, of order \(209952\)\(\medspace = 2^{5} \cdot 3^{8} \) |
| $\card{W}$ | 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 | $3$ |
| Möbius function | not computed |
| Projective image | not computed |