Subgroup ($H$) information
| Description: | $C_2^5.C_2^8:C_{10}$ |
| Order: | \(81920\)\(\medspace = 2^{14} \cdot 5 \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$\langle(1,6,2,5)(3,7,4,8)(9,15,10,16)(11,14,12,13)(27,28)(29,30)(33,34)(39,40) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_2^6.C_2^{12}:C_5$ |
| Order: | \(1310720\)\(\medspace = 2^{18} \cdot 5 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Derived length: | $3$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | not computed |
| $\operatorname{Aut}(H)$ | $D_6^2:S_4$, of order \(10485760\)\(\medspace = 2^{21} \cdot 5 \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_2^5.C_2^8:C_{10}$ |
| Normal closure: | $C_2^6.C_2^{12}:C_5$ |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $16$ |
| Möbius function | not computed |
| Projective image | not computed |