Subgroup ($H$) information
| Description: | $C_2^4.C_8$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Generators: |
$\langle(1,4,2,3)(5,8,6,7)(9,12,10,11)(13,16,14,15), (1,7,4,5,2,8,3,6)(9,13,12,16,10,14,11,15) \!\cdots\! \rangle$
|
| Nilpotency class: | $4$ |
| 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: | $C_2^4.C_2\wr C_4$ |
| Order: | \(1024\)\(\medspace = 2^{10} \) |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Nilpotency class: | $7$ |
| Derived length: | $3$ |
The ambient group is nonabelian and a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^7.C_2\wr C_2^2$, of order \(8192\)\(\medspace = 2^{13} \) |
| $\operatorname{Aut}(H)$ | $D_4^2:C_2^3$, of order \(512\)\(\medspace = 2^{9} \) |
| $\operatorname{res}(S)$ | $D_4^2:C_2^2$, of order \(256\)\(\medspace = 2^{8} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(8\)\(\medspace = 2^{3} \) |
| $W$ | $C_2\wr C_4$, of order \(64\)\(\medspace = 2^{6} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_2^5.D_8$ |