Subgroup ($H$) information
| Description: | $C_4^2.D_4$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| Index: | \(6561\)\(\medspace = 3^{8} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$\langle(1,5)(2,6)(3,4)(7,11)(8,12)(9,10)(13,18)(14,17)(15,16)(19,23)(20,24)(21,22) \!\cdots\! \rangle$
|
| Nilpotency class: | $4$ |
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, a $2$-Sylow subgroup (hence nilpotent, solvable, supersolvable, a Hall subgroup, and monomial), and a $p$-group (hence elementary and hyperelementary).
Ambient group ($G$) information
| Description: | $C_3^8:(C_4^2.D_4)$ |
| Order: | \(839808\)\(\medspace = 2^{7} \cdot 3^{8} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^8:\SD_{16}.C_2\wr D_4$, of order \(13436928\)\(\medspace = 2^{11} \cdot 3^{8} \) |
| $\operatorname{Aut}(H)$ | $C_2^6:D_4$, of order \(512\)\(\medspace = 2^{9} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Normal closure: | $C_3^8:(C_4^2.D_4)$ |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $6561$ |
| Möbius function | not computed |
| Projective image | not computed |