Subgroup ($H$) information
| Description: | not computed |
| Order: | \(839808\)\(\medspace = 2^{7} \cdot 3^{8} \) |
| Index: | \(256\)\(\medspace = 2^{8} \) |
| Exponent: | not computed |
| Generators: |
$\langle(1,2,8,7)(3,5,9,4)(19,26,22,27)(20,23,24,21), (10,15,17)(11,13,18)(12,14,16) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is normal, nonabelian, metabelian (hence solvable), and an A-group. Whether it is a direct factor, a semidirect factor, elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_3^8:C_8.D_8^2.\SD_{16}$ |
| Order: | \(214990848\)\(\medspace = 2^{15} \cdot 3^{8} \) |
| Exponent: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $(C_2^2\times C_8):C_2^3$ |
| Order: | \(256\)\(\medspace = 2^{8} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Automorphism Group: | $C_2\times D_4^3.D_6$, of order \(12288\)\(\medspace = 2^{12} \cdot 3 \) |
| Outer Automorphisms: | $C_2^2\times S_4$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^8:C_8.D_8^2.\SD_{16}.C_2^2$, of order \(859963392\)\(\medspace = 2^{17} \cdot 3^{8} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Möbius function | not computed |
| Projective image | not computed |