Subgroup ($H$) information
| Description: | $(C_2\times D_4^2):D_4$ |
| Order: | \(1024\)\(\medspace = 2^{10} \) |
| Index: | \(390625\)\(\medspace = 5^{8} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$\langle(1,39)(2,40)(3,36)(4,37)(5,38)(6,14)(7,15)(8,11)(9,12)(10,13), (1,38,5,39) \!\cdots\! \rangle$
|
| Nilpotency class: | $4$ |
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, a $2$-Sylow subgroup (hence nilpotent, solvable, supersolvable, a Hall subgroup, and monomial), a $p$-group (hence elementary and hyperelementary), and rational.
Ambient group ($G$) information
| Description: | $C_5^8.D_4^2.(C_2\times D_4)$ |
| Order: | \(400000000\)\(\medspace = 2^{10} \cdot 5^{8} \) |
| Exponent: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| 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 \(12800000000\)\(\medspace = 2^{15} \cdot 5^{8} \) |
| $\operatorname{Aut}(H)$ | $C_2^9.C_2\wr D_4$, of order \(65536\)\(\medspace = 2^{16} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Normal closure: | $C_5^8.D_4^2.(C_2\times D_4)$ |
| 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 |