Subgroup ($H$) information
| Description: | $C_2.C_2^6.C_2^2$ |
| Order: | \(512\)\(\medspace = 2^{9} \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$\langle(5,6)(7,8)(13,14)(15,16)(21,22)(23,24), (13,14)(15,16)(17,19,18,20)(21,24,22,23) \!\cdots\! \rangle$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian. Whether it is metacyclic, monomial, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_4^3:C_2^2:S_4$ |
| Order: | \(6144\)\(\medspace = 2^{11} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_4^2:A_4.C_2^4.C_2^4$, of order \(49152\)\(\medspace = 2^{14} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | Group of order \(1048576\)\(\medspace = 2^{20} \) |
| $W$ | $D_4^2:C_2$, of order \(128\)\(\medspace = 2^{7} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $6$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |