Subgroup ($H$) information
| Description: | $C_2\times D_4$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Index: | \(64\)\(\medspace = 2^{6} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$\langle(1,13)(2,14)(3,16)(4,15)(5,12)(6,11)(7,9)(8,10), (5,12)(6,11)(7,9)(8,10) \!\cdots\! \rangle$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metabelian, and rational.
Ambient group ($G$) information
| Description: | $(C_2\times D_4^2):D_4$ |
| Order: | \(1024\)\(\medspace = 2^{10} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $4$ |
| Derived length: | $3$ |
The ambient group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^9.C_2\wr D_4$, of order \(65536\)\(\medspace = 2^{16} \) |
| $\operatorname{Aut}(H)$ | $C_2\wr C_2^2$, of order \(64\)\(\medspace = 2^{6} \) |
| $\card{W}$ | \(4\)\(\medspace = 2^{2} \) |
Related subgroups
| Centralizer: | $C_2^3$ | ||||
| Normalizer: | $C_2^2\times D_4$ | ||||
| Normal closure: | $C_2^5:C_2^3$ | ||||
| Core: | $C_1$ | ||||
| Minimal over-subgroups: | $C_2^2\times D_4$ | ||||
| Maximal under-subgroups: | $C_2^3$ | $C_2^3$ | $C_2\times C_4$ | $D_4$ | $D_4$ |
Other information
| Number of subgroups in this autjugacy class | $256$ |
| Number of conjugacy classes in this autjugacy class | $8$ |
| Möbius function | not computed |
| Projective image | not computed |