Subgroup ($H$) information
| Description: | $(C_2\times C_4).D_8$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| Index: | \(3\) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$a, b, d^{3}$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, a $2$-Sylow subgroup (hence nilpotent, solvable, supersolvable, a Hall subgroup, and monomial), a $p$-group (hence elementary and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_2^2.(S_3\times D_8)$ |
| Order: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3:(C_2^7.C_2^5)$ |
| $\operatorname{Aut}(H)$ | $(C_2^6\times C_4).C_2^4$, of order \(4096\)\(\medspace = 2^{12} \) |
| $\card{\operatorname{res}(S)}$ | \(2048\)\(\medspace = 2^{11} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $C_2\times D_4$, of order \(16\)\(\medspace = 2^{4} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $-1$ |
| Projective image | $S_3\times D_4$ |