Subgroup ($H$) information
| Description: | $Q_{256}$ |
| Order: | \(256\)\(\medspace = 2^{8} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(128\)\(\medspace = 2^{7} \) |
| Generators: |
$\left(\begin{array}{rr}
165 & 0 \\
0 & 81
\end{array}\right), \left(\begin{array}{rr}
16 & 0 \\
0 & 241
\end{array}\right), \left(\begin{array}{rr}
253 & 0 \\
0 & 64
\end{array}\right), \left(\begin{array}{rr}
9 & 0 \\
0 & 200
\end{array}\right), \left(\begin{array}{rr}
0 & 256 \\
1 & 0
\end{array}\right), \left(\begin{array}{rr}
32 & 0 \\
0 & 249
\end{array}\right), \left(\begin{array}{rr}
256 & 0 \\
0 & 256
\end{array}\right), \left(\begin{array}{rr}
240 & 0 \\
0 & 136
\end{array}\right)$
|
| Nilpotency class: | $7$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metacyclic (hence 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.
Quotient group ($Q$) structure
| Description: | $C_8$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Automorphism Group: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{64}.C_8.C_2^3.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_{64}.C_{32}.C_2^2$ |
| $\card{\operatorname{res}(\operatorname{Aut}(G))}$ | \(8192\)\(\medspace = 2^{13} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(8\)\(\medspace = 2^{3} \) |
| $W$ | $D_{64}$, of order \(128\)\(\medspace = 2^{7} \) |
Related subgroups
| Centralizer: | $C_{16}$ | ||
| Normalizer: | $D_{128}:C_8$ | ||
| Complements: | $C_8$ $C_8$ $C_8$ $C_8$ | ||
| Minimal over-subgroups: | $D_{128}:C_2$ | ||
| Maximal under-subgroups: | $Q_{128}$ | $Q_{128}$ | $C_{128}$ |
Other information
| Möbius function | $0$ |
| Projective image | not computed |