Subgroup ($H$) information
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(64\)\(\medspace = 2^{6} \) |
| Exponent: | \(2\) |
| Generators: |
$\left(\begin{array}{rr}
31 & 0 \\
0 & 31
\end{array}\right), \left(\begin{array}{rr}
17 & 0 \\
0 & 17
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, central (hence abelian, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
Ambient group ($G$) information
| Description: | $C_2\times Q_8\times C_{16}$ |
| Order: | \(256\)\(\medspace = 2^{8} \) |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Nilpotency class: | $2$ |
| 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: | $C_2^2\times C_{16}$ |
| Order: | \(64\)\(\medspace = 2^{6} \) |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Automorphism Group: | $(C_2^3\times C_4):S_4$, of order \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| Outer Automorphisms: | $(C_2^3\times C_4):S_4$, of order \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and a $p$-group (hence elementary and hyperelementary).
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^6.C_6.C_2.C_2^6$ |
| $\operatorname{Aut}(H)$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\operatorname{res}(S)$ | $C_2$, of order \(2\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(12288\)\(\medspace = 2^{12} \cdot 3 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_2\times Q_8\times C_{16}$ | |
| Normalizer: | $C_2\times Q_8\times C_{16}$ | |
| Minimal over-subgroups: | $C_2\times C_4$ | $C_2^3$ |
| Maximal under-subgroups: | $C_2$ | $C_2$ |
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $C_2^2\times C_{16}$ |