Subgroup ($H$) information
| Description: | $C_8.D_{64}$ |
| Order: | \(1024\)\(\medspace = 2^{10} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(128\)\(\medspace = 2^{7} \) |
| Generators: |
$\left(\begin{array}{rr}
165 & 0 \\
0 & 81
\end{array}\right), \left(\begin{array}{rr}
253 & 0 \\
0 & 64
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 256
\end{array}\right), \left(\begin{array}{rr}
32 & 0 \\
0 & 249
\end{array}\right), \left(\begin{array}{rr}
240 & 0 \\
0 & 136
\end{array}\right), \left(\begin{array}{rr}
0 & 1 \\
1 & 0
\end{array}\right), \left(\begin{array}{rr}
16 & 0 \\
0 & 241
\end{array}\right), \left(\begin{array}{rr}
9 & 0 \\
0 & 207
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 241
\end{array}\right), \left(\begin{array}{rr}
256 & 0 \\
0 & 256
\end{array}\right)$
|
| Nilpotency class: | $7$ |
| Derived length: | $2$ |
The subgroup is normal, nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $D_{128}:C_{16}$ |
| Order: | \(4096\)\(\medspace = 2^{12} \) |
| 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.
Quotient group ($Q$) structure
| Description: | $C_4$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group) and a $p$-group.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{64}.C_4.C_4^2.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_{32}.C_{32}.C_2^4$ |
| $\card{\operatorname{res}(S)}$ | \(16384\)\(\medspace = 2^{14} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $D_{64}$, of order \(128\)\(\medspace = 2^{7} \) |
Related subgroups
| Centralizer: | $C_{32}$ | ||
| Normalizer: | $D_{128}:C_{16}$ | ||
| Minimal over-subgroups: | $D_{128}:C_8$ | ||
| Maximal under-subgroups: | $D_{64}:C_4$ | $C_{64}.C_8$ | $C_4\times C_{128}$ |
| Autjugate subgroups: | 4096.boc.4.b1.b1 |
Other information
| Möbius function | not computed |
| Projective image | $C_{32} . (C_2^2\times C_4)$ |