Subgroup ($H$) information
| Description: | $C_2^4.D_4$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| Index: | \(6561\)\(\medspace = 3^{8} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$\langle(1,11,8,6)(2,12,9,5)(3,10,7,4)(13,20,14,21)(15,19)(16,17), (1,8)(2,7)(3,9) \!\cdots\! \rangle$
|
| Nilpotency class: | $4$ |
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, a $2$-Sylow subgroup (hence nilpotent, solvable, supersolvable, a Hall subgroup, and monomial), a $p$-group (hence elementary and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_3^8:C_2^4.D_4$ |
| Order: | \(839808\)\(\medspace = 2^{7} \cdot 3^{8} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^8:(C_2^2\times D_4).D_4^2$, of order \(13436928\)\(\medspace = 2^{11} \cdot 3^{8} \) |
| $\operatorname{Aut}(H)$ | $C_2^4.D_4^2$, of order \(1024\)\(\medspace = 2^{10} \) |
| $W$ | $C_2^4:C_4$, of order \(64\)\(\medspace = 2^{6} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_2^4.D_4$ |
| Normal closure: | $C_3^8:C_2^4.D_4$ |
| Core: | $C_1$ |
Other information
| Number of subgroups in this autjugacy class | $6561$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^8:C_2^4.D_4$ |