Subgroup ($H$) information
| Description: | $D_{128}:C_8$ |
| Order: | \(2048\)\(\medspace = 2^{11} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(128\)\(\medspace = 2^{7} \) |
| Generators: |
$\left(\begin{array}{rr}
1 & 0 \\
0 & 256
\end{array}\right), \left(\begin{array}{rr}
222 & 0 \\
0 & 22
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 64
\end{array}\right), \left(\begin{array}{rr}
197 & 0 \\
0 & 227
\end{array}\right), \left(\begin{array}{rr}
4 & 0 \\
0 & 193
\end{array}\right), \left(\begin{array}{rr}
2 & 0 \\
0 & 129
\end{array}\right), \left(\begin{array}{rr}
0 & 1 \\
1 & 0
\end{array}\right), \left(\begin{array}{rr}
215 & 0 \\
0 & 104
\end{array}\right), \left(\begin{array}{rr}
16 & 0 \\
0 & 241
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 241
\end{array}\right), \left(\begin{array}{rr}
256 & 0 \\
0 & 256
\end{array}\right)$
|
| Nilpotency class: | $7$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_{32}.D_{128}$ |
| Order: | \(8192\)\(\medspace = 2^{13} \) |
| Exponent: | \(256\)\(\medspace = 2^{8} \) |
| Nilpotency class: | $8$ |
| 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
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{64}.C_8.C_4^2.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_{64}.C_8.C_2^3.C_2^4$ |
| $W$ | $D_{128}$, of order \(256\)\(\medspace = 2^{8} \) |
Related subgroups
| Centralizer: | $C_{32}$ | ||||
| Normalizer: | $C_{32}.D_{128}$ | ||||
| Minimal over-subgroups: | $D_{128}:C_{16}$ | ||||
| Maximal under-subgroups: | $D_{128}:C_4$ | $C_8\times C_{128}$ | $C_{128}.C_8$ | $D_{64}:C_8$ | $C_8.D_{64}$ |
Other information
| Möbius function | $0$ |
| Projective image | $D_{64}:C_4$ |