Subgroup ($H$) information
| Description: | $C_6^4.\SOPlus(4,2)$ |
| Order: | \(93312\)\(\medspace = 2^{7} \cdot 3^{6} \) |
| Index: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Generators: |
$\langle(20,23)(22,25), (1,2,4,8,14,17,3,5,10,11,16,13)(6,9,12,7)(15,18)(19,20,21,23,24,25,26,22) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is nonabelian and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_3^6.S_4^2:C_2^2$ |
| Order: | \(1679616\)\(\medspace = 2^{8} \cdot 3^{8} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_6^4.C_3^4.(C_6\times D_4).C_2$ |
| $\operatorname{Aut}(H)$ | $C_3^3.A_4^2.C_6^2.C_2$ |
| $W$ | $C_6^4.\SOPlus(4,2)$, of order \(93312\)\(\medspace = 2^{7} \cdot 3^{6} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_6^4.\SOPlus(4,2)$ |
| Normal closure: | $C_3^6.S_4\wr C_2$ |
| Core: | $C_6^4$ |
Other information
| Number of subgroups in this autjugacy class | $36$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | $C_3^6.S_4^2:C_2^2$ |