Subgroup ($H$) information
| Description: | $D_4$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$\langle(1,5)(2,3)(4,8)(6,7)(9,11), (1,6)(2,7)(3,8)(4,5)(9,10)(11,12), (1,3)(2,5)(4,7)(6,8)(9,11)(10,12)\rangle$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metacyclic (hence metabelian), and rational.
Ambient group ($G$) information
| Description: | $D_4^2:C_2$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The ambient group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metabelian, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^8.D_4^2$, of order \(16384\)\(\medspace = 2^{14} \) |
| $\operatorname{Aut}(H)$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| $\operatorname{res}(S)$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(64\)\(\medspace = 2^{6} \) |
| $W$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
| Centralizer: | $C_2^3$ | ||
| Normalizer: | $C_2^2\times D_4$ | ||
| Normal closure: | $C_2\times D_4$ | ||
| Core: | $C_2$ | ||
| Minimal over-subgroups: | $C_2\times D_4$ | $C_2\times D_4$ | |
| Maximal under-subgroups: | $C_2^2$ | $C_2^2$ | $C_4$ |
Other information
| Number of subgroups in this autjugacy class | $64$ |
| Number of conjugacy classes in this autjugacy class | $16$ |
| Möbius function | $0$ |
| Projective image | $D_4:C_2^3$ |