Subgroup ($H$) information
| Description: | $C_2\times C_4^2$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(256\)\(\medspace = 2^{8} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$\left(\begin{array}{rr}
17 & 16 \\
24 & 25
\end{array}\right), \left(\begin{array}{rr}
31 & 16 \\
0 & 31
\end{array}\right), \left(\begin{array}{rr}
9 & 0 \\
16 & 25
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), and a $p$-group (hence elementary and hyperelementary). Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $(C_2\times C_4^3).D_4^2$ |
| Order: | \(8192\)\(\medspace = 2^{13} \) |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Nilpotency class: | $4$ |
| 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: | $C_4.D_4^2$ |
| Order: | \(256\)\(\medspace = 2^{8} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Automorphism Group: | $C_2^4.C_2^6.C_2^3$ |
| Outer Automorphisms: | $C_2^4:D_4$, of order \(128\)\(\medspace = 2^{7} \) |
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metabelian, and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(67108864\)\(\medspace = 2^{26} \) |
| $\operatorname{Aut}(H)$ | $C_2^6:S_4$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Möbius function | not computed |
| Projective image | not computed |