Subgroup ($H$) information
| Description: | not computed |
| Order: | \(8192\)\(\medspace = 2^{13} \) |
| Index: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | not computed |
| Generators: |
$\langle(1,4)(2,3)(5,7)(6,8)(9,12,10,11)(13,15,14,16)(17,18)(21,22)(29,30)(31,32) \!\cdots\! \rangle$
|
| Nilpotency class: | not computed |
| Derived length: | not computed |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian. Whether it is elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_2^5.C_2^8:C_{10}$ |
| Order: | \(81920\)\(\medspace = 2^{14} \cdot 5 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_6^2:S_4$, of order \(10485760\)\(\medspace = 2^{21} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | not computed |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_2^6.C_2^6.C_2^2$ |
| Normal closure: | $C_2^6.C_2^6.C_2^2$ |
| Core: | $C_2^6.C_2^4$ |
Other information
| Number of subgroups in this autjugacy class | $5$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |