Subgroup ($H$) information
| Description: | not computed |
| Order: | \(1024\)\(\medspace = 2^{10} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | not computed |
| Generators: |
$\langle(13,14)(15,16)(21,22)(23,24), (1,4)(2,3)(5,7)(6,8)(9,14)(10,13)(11,16)(12,15) \!\cdots\! \rangle$
|
| Nilpotency class: | not computed |
| Derived length: | not computed |
The subgroup is nonabelian and a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary). Whether it is elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_2^8.S_4$ |
| Order: | \(6144\)\(\medspace = 2^{11} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian, solvable, and rational. 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_5^3.C_2^2$, of order \(196608\)\(\medspace = 2^{16} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | not computed |
| $W$ | $C_2^6:D_4$, of order \(512\)\(\medspace = 2^{9} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_2^5.C_2^5.C_2$ |
| Normal closure: | $C_2^8.S_4$ |
| Core: | $C_4^2:C_2^3$ |
Other information
| Number of subgroups in this autjugacy class | $12$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | not computed |
| Projective image | not computed |