Subgroup ($H$) information
| Description: | $C_{32}\times C_{128}$ |
| Order: | \(4096\)\(\medspace = 2^{12} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(128\)\(\medspace = 2^{7} \) |
| Generators: |
$\left(\begin{array}{rr}
1 & 0 \\
0 & 256
\end{array}\right), \left(\begin{array}{rr}
81 & 0 \\
0 & 165
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 136
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 64
\end{array}\right), \left(\begin{array}{rr}
64 & 0 \\
0 & 253
\end{array}\right), \left(\begin{array}{rr}
241 & 0 \\
0 & 16
\end{array}\right), \left(\begin{array}{rr}
249 & 0 \\
0 & 32
\end{array}\right), \left(\begin{array}{rr}
9 & 0 \\
0 & 200
\end{array}\right), \left(\begin{array}{rr}
136 & 0 \\
0 & 240
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 241
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 249
\end{array}\right), \left(\begin{array}{rr}
256 & 0 \\
0 & 256
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Frattini subgroup (hence characteristic and normal), abelian (hence 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: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, 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 \(524288\)\(\medspace = 2^{19} \) |
| $\operatorname{Aut}(H)$ | $(C_4^2\times C_8).C_4^3.C_8.C_2^4$, of order \(1048576\)\(\medspace = 2^{20} \) |
| $\card{W}$ | \(2\) |
Related subgroups
Other information
| Möbius function | not computed |
| Projective image | not computed |