Subgroup ($H$) information
| Description: | $D_4:C_2$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Index: | \(300\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$\left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
3 & 8 & 2 & 0 \\
3 & 7 & 3 & 0 \\
5 & 3 & 8 & 10
\end{array}\right), \left(\begin{array}{rrrr}
9 & 3 & 3 & 6 \\
4 & 8 & 7 & 3 \\
10 & 7 & 3 & 8 \\
9 & 10 & 7 & 2
\end{array}\right), \left(\begin{array}{rrrr}
7 & 4 & 7 & 0 \\
6 & 0 & 7 & 7 \\
2 & 7 & 0 & 7 \\
3 & 2 & 5 & 4
\end{array}\right), \left(\begin{array}{rrrr}
10 & 0 & 0 & 0 \\
0 & 10 & 0 & 0 \\
0 & 0 & 10 & 0 \\
0 & 0 & 0 & 10
\end{array}\right)$
|
| Nilpotency class: | $2$ |
| 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: | $C_5\times \GL(2,3):D_{10}$ |
| Order: | \(4800\)\(\medspace = 2^{6} \cdot 3 \cdot 5^{2} \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2\times A_4\times F_5).C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_2\times S_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| $\operatorname{res}(S)$ | $C_2^3$, of order \(8\)\(\medspace = 2^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(128\)\(\medspace = 2^{7} \) |
| $W$ | $C_2^3$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $15$ |
| Möbius function | $0$ |
| Projective image | not computed |