Subgroup ($H$) information
| Description: | not computed |
| Order: | \(41472\)\(\medspace = 2^{9} \cdot 3^{4} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | not computed |
| Generators: |
$\langle(11,12)(14,16), (13,15)(14,16), (5,8)(6,7)(9,11)(10,12)(13,16)(14,15), (5,8) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is nonabelian and solvable. Whether it is elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_2^8.C_3^4:\OD_{16}$ |
| Order: | \(331776\)\(\medspace = 2^{12} \cdot 3^{4} \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $A_4^2\wr C_2.C_2^2.D_4$, of order \(1327104\)\(\medspace = 2^{14} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $A_4^2.(C_2\times A_4^2:C_4)$ |
| Normal closure: | $C_2^8.C_3^3.D_6$ |
| Core: | $C_2^8.C_3^4$ |
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_2^8.C_3^4:\OD_{16}$ |