Subgroup ($H$) information
| Description: | $C_2^2\times C_8^2$ |
| Order: | \(256\)\(\medspace = 2^{8} \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$\left(\begin{array}{rr}
7 & 16 \\
12 & 27
\end{array}\right), \left(\begin{array}{rr}
5 & 16 \\
24 & 29
\end{array}\right), \left(\begin{array}{rr}
17 & 16 \\
16 & 17
\end{array}\right), \left(\begin{array}{rr}
17 & 16 \\
0 & 17
\end{array}\right)$
|
| 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).
Ambient group ($G$) information
| Description: | $C_4^4.C_2^4$ |
| Order: | \(4096\)\(\medspace = 2^{12} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $3$ |
| 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: | $C_2\times D_4$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2\wr C_2^2$, of order \(64\)\(\medspace = 2^{6} \) |
| Outer Automorphisms: | $C_2\times D_4$, of order \(16\)\(\medspace = 2^{4} \) |
| Nilpotency class: | $2$ |
| 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 \(2147483648\)\(\medspace = 2^{31} \) |
| $\operatorname{Aut}(H)$ | $C_2^4.C_2^6.A_4^2.C_2^3.C_2$ |
| $\card{W}$ | \(2\) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | not computed |