Subgroup ($H$) information
| Description: | $C_2\times D_8$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$b, c^{3}, d^{7}$
|
| Nilpotency class: | $3$ |
| 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: | $D_8:C_2\times F_7$ |
| Order: | \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \) |
| Exponent: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7.(C_3\times D_4^2).C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_4.D_4^2$, of order \(256\)\(\medspace = 2^{8} \) |
| $\operatorname{res}(S)$ | $C_2^2\times D_4$, of order \(32\)\(\medspace = 2^{5} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $W$ | $C_2\times D_4$, of order \(16\)\(\medspace = 2^{4} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $7$ |
| Möbius function | $-1$ |
| Projective image | $C_2\times D_4\times F_7$ |