Subgroup ($H$) information
| Description: | $C_2^2\times D_4$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$\langle(1,3)(2,6)(4,7)(5,8), (1,4)(2,5)(3,7)(6,8), (2,5), (9,10), (2,5)(6,8)\rangle$
|
| 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: | $C_2^5:C_4$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $4$ |
| Derived length: | $2$ |
The ambient group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^4:D_4^2$, of order \(1024\)\(\medspace = 2^{10} \) |
| $\operatorname{Aut}(H)$ | $C_2^6:(C_2\times S_4)$, of order \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
| $\operatorname{res}(S)$ | $D_4\times C_2^3$, of order \(64\)\(\medspace = 2^{6} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(8\)\(\medspace = 2^{3} \) |
| $W$ | $C_2^3$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $C_2^3:C_4$ |