Subgroup ($H$) information
| Description: | $C_{32}\times C_{256}$ |
| Order: | \(8192\)\(\medspace = 2^{13} \) |
| Index: | \(2\) |
| Exponent: | \(256\)\(\medspace = 2^{8} \) |
| Generators: |
$\left(\begin{array}{rr}
64 & 0 \\
0 & 253
\end{array}\right), \left(\begin{array}{rr}
3 & 0 \\
0 & 27
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 64
\end{array}\right), \left(\begin{array}{rr}
249 & 0 \\
0 & 32
\end{array}\right), \left(\begin{array}{rr}
81 & 0 \\
0 & 165
\end{array}\right), \left(\begin{array}{rr}
9 & 0 \\
0 & 200
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 249
\end{array}\right), \left(\begin{array}{rr}
136 & 0 \\
0 & 240
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 241
\end{array}\right), \left(\begin{array}{rr}
241 & 0 \\
0 & 16
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 256
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 136
\end{array}\right), \left(\begin{array}{rr}
256 & 0 \\
0 & 256
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, central (hence abelian, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $C_{64}.D_{128}$ |
| Order: | \(16384\)\(\medspace = 2^{14} \) |
| 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: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| 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), a $p$-group, simple, and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{64}.C_8.C_8^2.C_2^4$, of order \(524288\)\(\medspace = 2^{19} \) |
| $\operatorname{Aut}(H)$ | $(C_4^2\times C_8).C_4^3.C_8.C_2^5$, of order \(2097152\)\(\medspace = 2^{21} \) |
| $\card{W}$ | \(2\) |
Related subgroups
| Centralizer: | $C_{32}\times C_{256}$ | |
| Normalizer: | $C_{64}.D_{128}$ | |
| Complements: | $C_2$ | |
| Minimal over-subgroups: | $C_{64}.D_{128}$ | |
| Maximal under-subgroups: | $C_{32}\times C_{128}$ | $C_{16}\times C_{256}$ |
Other information
| Möbius function | not computed |
| Projective image | not computed |