Subgroup ($H$) information
| Description: | $C_{16}.C_8$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Generators: |
$\left(\begin{array}{rr}
16 & 0 \\
0 & 241
\end{array}\right), \left(\begin{array}{rr}
253 & 0 \\
0 & 64
\end{array}\right), \left(\begin{array}{rr}
225 & 0 \\
0 & 8
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 256
\end{array}\right), \left(\begin{array}{rr}
0 & 9 \\
207 & 0
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 241
\end{array}\right), \left(\begin{array}{rr}
256 & 0 \\
0 & 256
\end{array}\right)$
|
| Nilpotency class: | $4$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metacyclic (hence metabelian).
Ambient group ($G$) information
| Description: | $C_8.D_{64}$ |
| Order: | \(1024\)\(\medspace = 2^{10} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{32}.C_{32}.C_2^4$ |
| $\operatorname{Aut}(H)$ | $\OD_{32}.C_2^4$, of order \(512\)\(\medspace = 2^{9} \) |
| $\operatorname{res}(S)$ | $\OD_{32}.C_2^4$, of order \(512\)\(\medspace = 2^{9} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(8\)\(\medspace = 2^{3} \) |
| $W$ | $D_{16}$, of order \(32\)\(\medspace = 2^{5} \) |
Related subgroups
| Centralizer: | $C_8$ | |
| Normalizer: | $C_{32}.C_8$ | |
| Normal closure: | $C_{64}.C_8$ | |
| Core: | $C_4\times C_{16}$ | |
| Minimal over-subgroups: | $C_{32}.C_8$ | |
| Maximal under-subgroups: | $C_4\times C_{16}$ | $C_8.C_8$ |
Other information
| Number of subgroups in this conjugacy class | $4$ |
| Möbius function | $0$ |
| Projective image | $D_{64}$ |