Subgroup ($H$) information
| Description: | $D_8:C_2$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$\left(\begin{array}{rr}
50 & 0 \\
0 & 35
\end{array}\right), \left(\begin{array}{rr}
4 & 34 \\
0 & 38
\end{array}\right), \left(\begin{array}{rr}
22 & 6 \\
24 & 46
\end{array}\right)$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $\GL(2,3):D_4$ |
| Order: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^6\times S_4$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $C_8:C_2^3$, of order \(64\)\(\medspace = 2^{6} \) |
| $\operatorname{res}(S)$ | $C_2^2\times D_4$, of order \(32\)\(\medspace = 2^{5} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $6$ |
| Möbius function | $0$ |
| Projective image | $D_4\times S_4$ |