Subgroup ($H$) information
| Description: | $Q_{64}$ |
| Order: | \(64\)\(\medspace = 2^{6} \) |
| Index: | \(2\) |
| Exponent: | \(32\)\(\medspace = 2^{5} \) |
| Generators: |
$\left(\begin{array}{rr}
47 & 0 \\
0 & 64
\end{array}\right), \left(\begin{array}{rr}
28 & 0 \\
0 & 52
\end{array}\right), \left(\begin{array}{rr}
75 & 0 \\
0 & 22
\end{array}\right), \left(\begin{array}{rr}
96 & 0 \\
0 & 96
\end{array}\right), \left(\begin{array}{rr}
85 & 0 \\
0 & 8
\end{array}\right), \left(\begin{array}{rr}
0 & 1 \\
96 & 0
\end{array}\right)$
|
| Nilpotency class: | $5$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metacyclic (hence metabelian).
Ambient group ($G$) information
| Description: | $D_{32}:C_2$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| Exponent: | \(32\)\(\medspace = 2^{5} \) |
| Nilpotency class: | $5$ |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times D_{32}:C_8$, of order \(1024\)\(\medspace = 2^{10} \) |
| $\operatorname{Aut}(H)$ | $D_{32}:C_8$, of order \(512\)\(\medspace = 2^{9} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $D_{32}:C_8$, of order \(512\)\(\medspace = 2^{9} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $D_{16}$, of order \(32\)\(\medspace = 2^{5} \) |
Related subgroups
| Centralizer: | $C_4$ | ||
| Normalizer: | $D_{32}:C_2$ | ||
| Complements: | $C_2$ $C_2$ $C_2$ | ||
| Minimal over-subgroups: | $D_{32}:C_2$ | ||
| Maximal under-subgroups: | $Q_{32}$ | $Q_{32}$ | $C_{32}$ |
Other information
| Möbius function | $-1$ |
| Projective image | $C_2\times D_{16}$ |