Subgroup ($H$) information
| Description: | $C_2^6.\SD_{16}$ |
| Order: | \(1024\)\(\medspace = 2^{10} \) |
| Index: | \(390625\)\(\medspace = 5^{8} \) |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Generators: |
$\langle(1,31,21,11)(2,32,24,13,5,35,23,14)(3,33,22,15,4,34,25,12)(6,38,26,16)(7,37,29,19,10,39,28,18) \!\cdots\! \rangle$
|
| Nilpotency class: | $7$ |
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, a $2$-Sylow subgroup (hence nilpotent, solvable, supersolvable, a Hall subgroup, and monomial), and a $p$-group (hence elementary and hyperelementary).
Ambient group ($G$) information
| Description: | $C_5^7.(C_2^2\times F_5).C_2\wr C_4$ |
| Order: | \(400000000\)\(\medspace = 2^{10} \cdot 5^{8} \) |
| Exponent: | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(3200000000\)\(\medspace = 2^{13} \cdot 5^{8} \) |
| $\operatorname{Aut}(H)$ | $C_2^5:C_4.D_4^2$, of order \(8192\)\(\medspace = 2^{13} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Normal closure: | $C_5^7.(C_2^2\times F_5).C_2\wr C_4$ |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $390625$ |
| Möbius function | not computed |
| Projective image | not computed |