Subgroup ($H$) information
| Description: | $C_4\wr C_2$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(9720\)\(\medspace = 2^{3} \cdot 3^{5} \cdot 5 \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$\langle(1,72)(2,51)(3,68)(4,20)(5,73)(6,75)(7,59)(8,48)(9,66)(10,65)(11,46)(12,67) \!\cdots\! \rangle$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_3^3:S_3.C_2^4:S_5$ |
| Order: | \(311040\)\(\medspace = 2^{8} \cdot 3^{5} \cdot 5 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3:S_3.C_2^4:S_5$, of order \(311040\)\(\medspace = 2^{8} \cdot 3^{5} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_2^2\times D_4$, of order \(32\)\(\medspace = 2^{5} \) |
| $W$ | $C_2\times D_4$, of order \(16\)\(\medspace = 2^{4} \) |
Related subgroups
| Centralizer: | $C_4$ | ||
| Normalizer: | $D_4.D_4$ | ||
| Normal closure: | $C_3^3:S_3.C_2^4:S_5$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $C_3^4:C_4\wr C_2$ | $D_4.D_4$ | |
| Maximal under-subgroups: | $D_4:C_2$ | $C_4^2$ | $\OD_{16}$ |
Other information
| Number of subgroups in this conjugacy class | $4860$ |
| Möbius function | $0$ |
| Projective image | $C_3^3:S_3.C_2^4:S_5$ |