Subgroup ($H$) information
| Description: | $C_8.D_{16}$ |
| Order: | \(256\)\(\medspace = 2^{8} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(32\)\(\medspace = 2^{5} \) |
| Generators: |
$\left(\begin{array}{rr}
240 & 0 \\
0 & 223
\end{array}\right), \left(\begin{array}{rr}
16 & 0 \\
0 & 241
\end{array}\right), \left(\begin{array}{rr}
0 & 9 \\
200 & 0
\end{array}\right), \left(\begin{array}{rr}
253 & 0 \\
0 & 64
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 256
\end{array}\right), \left(\begin{array}{rr}
32 & 0 \\
0 & 249
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 241
\end{array}\right), \left(\begin{array}{rr}
256 & 0 \\
0 & 256
\end{array}\right)$
|
| Nilpotency class: | $5$ |
| 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: | $D_{128}:C_8$ |
| Order: | \(2048\)\(\medspace = 2^{11} \) |
| Exponent: | \(128\)\(\medspace = 2^{7} \) |
| Nilpotency class: | $7$ |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{64}.C_8.C_2^3.C_2^4$ |
| $\operatorname{Aut}(H)$ | $(C_4\times C_8).C_2^5$ |
| $\card{\operatorname{res}(S)}$ | \(1024\)\(\medspace = 2^{10} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(8\)\(\medspace = 2^{3} \) |
| $W$ | $D_{16}$, of order \(32\)\(\medspace = 2^{5} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $4$ |
| Möbius function | $0$ |
| Projective image | $C_2\times D_{64}$ |