Subgroup ($H$) information
| Description: | $C_2^5:C_4$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| Index: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$\langle(1,2)(6,8), (1,4,6,7)(2,3,8,5)(9,14)(10,13)(11,16)(12,15), (1,5,8,3)(2,7,6,4) \!\cdots\! \rangle$
|
| 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: | $C_2^5:(C_4\times S_4)$ |
| Order: | \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^4:C_3.C_2^5.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2^7.(D_4\times S_4)$, of order \(24576\)\(\medspace = 2^{13} \cdot 3 \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed | ||
| Normalizer: | $C_2^6.C_2^4$ | ||
| Normal closure: | $C_2^5:S_4$ | ||
| Core: | $C_2^4$ | ||
| Minimal over-subgroups: | $C_2^6:C_4$ | $C_2^5:D_4$ | $C_2^5:D_4$ |
| Maximal under-subgroups: | $C_2^4$ | $C_2$ | $C_2$ |
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |