Subgroup ($H$) information
| Description: | $C_2\wr C_4$ |
| Order: | \(64\)\(\medspace = 2^{6} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$\langle(1,10,3,12)(2,9,4,11)(5,14,8,15)(6,13,7,16), (1,5,2,6)(3,7,4,8)(9,13,10,14) \!\cdots\! \rangle$
|
| 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: | $D_4^2:D_4$ |
| Order: | \(512\)\(\medspace = 2^{9} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $4$ |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^4.(C_2^2\times D_4^2)$, of order \(4096\)\(\medspace = 2^{12} \) |
| $\operatorname{Aut}(H)$ | $C_2^4:D_4$, of order \(128\)\(\medspace = 2^{7} \) |
| $\card{W}$ | \(128\)\(\medspace = 2^{7} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | not computed |