Subgroup ($H$) information
| Description: | $C_2^3\times C_8$ |
| Order: | \(64\)\(\medspace = 2^{6} \) |
| Index: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$\left(\begin{array}{rr}
31 & 16 \\
16 & 31
\end{array}\right), \left(\begin{array}{rr}
31 & 0 \\
0 & 31
\end{array}\right), \left(\begin{array}{rr}
1 & 16 \\
0 & 17
\end{array}\right), \left(\begin{array}{rr}
1 & 4 \\
0 & 25
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and a $p$-group (hence elementary and hyperelementary).
Ambient group ($G$) information
| Description: | $C_8^2.C_2^5$ |
| Order: | \(2048\)\(\medspace = 2^{11} \) |
| 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.
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 \(67108864\)\(\medspace = 2^{26} \) |
| $\operatorname{Aut}(H)$ | $C_2.C_2^7:\GL(3,2)$, of order \(43008\)\(\medspace = 2^{11} \cdot 3 \cdot 7 \) |
| $\operatorname{res}(S)$ | $C_2^5:D_4$, of order \(256\)\(\medspace = 2^{8} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(8192\)\(\medspace = 2^{13} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $32$ |
| Number of conjugacy classes in this autjugacy class | $16$ |
| Möbius function | $0$ |
| Projective image | $C_8.C_2^4$ |