Subgroup ($H$) information
| Description: | $C_2^6.C_2\wr S_4$ |
| Order: | \(98304\)\(\medspace = 2^{15} \cdot 3 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\langle(19,20)(23,24), (3,4)(5,6)(9,10)(11,12)(15,16)(17,18)(21,22)(23,24), (13,14) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is maximal, nonabelian, and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_2^8.C_2\wr S_4$ |
| Order: | \(393216\)\(\medspace = 2^{17} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^8.C_2^6.C_6.C_2^4.C_2^4$, of order \(25165824\)\(\medspace = 2^{23} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | Group of order \(50331648\)\(\medspace = 2^{24} \cdot 3 \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_2^6.C_2\wr S_4$ |
| Normal closure: | $C_2^8.C_2\wr S_4$ |
| Core: | $C_2^9.C_2^5$ |
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |