Subgroup ($H$) information
| Description: | $Q_{16}:C_{10}$ |
| Order: | \(160\)\(\medspace = 2^{5} \cdot 5 \) |
| Index: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Exponent: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Generators: |
$\left(\begin{array}{rrrr}
6 & 9 & 5 & 9 \\
0 & 6 & 0 & 5 \\
8 & 8 & 5 & 2 \\
0 & 8 & 0 & 5
\end{array}\right), \left(\begin{array}{rrrr}
0 & 7 & 0 & 8 \\
8 & 8 & 6 & 6 \\
8 & 2 & 6 & 4 \\
1 & 4 & 4 & 8
\end{array}\right), \left(\begin{array}{rrrr}
5 & 9 & 9 & 7 \\
1 & 2 & 10 & 9 \\
8 & 10 & 9 & 2 \\
5 & 8 & 10 & 6
\end{array}\right), \left(\begin{array}{rrrr}
9 & 0 & 0 & 0 \\
0 & 9 & 0 & 0 \\
0 & 0 & 9 & 0 \\
0 & 0 & 0 & 9
\end{array}\right), \left(\begin{array}{rrrr}
7 & 4 & 7 & 0 \\
6 & 0 & 7 & 7 \\
2 & 7 & 0 & 7 \\
3 & 2 & 5 & 4
\end{array}\right), \left(\begin{array}{rrrr}
10 & 0 & 0 & 0 \\
0 & 10 & 0 & 0 \\
0 & 0 & 10 & 0 \\
0 & 0 & 0 & 10
\end{array}\right)$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The subgroup is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2\times A_4\times F_5).C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_4\times D_4^2$, of order \(256\)\(\medspace = 2^{8} \) |
| $\operatorname{res}(S)$ | $C_4^2:C_2^3$, of order \(128\)\(\medspace = 2^{7} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(8\)\(\medspace = 2^{3} \) |
| $W$ | $C_2^2\times D_4$, of order \(32\)\(\medspace = 2^{5} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $15$ |
| Möbius function | $-1$ |
| Projective image | $D_{10}\times S_4$ |