Subgroup ($H$) information
| Description: | $C_{30}.D_{10}$ |
| Order: | \(600\)\(\medspace = 2^{3} \cdot 3 \cdot 5^{2} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\left(\begin{array}{rrrr}
6 & 8 & 8 & 0 \\
6 & 3 & 8 & 8 \\
8 & 3 & 9 & 3 \\
8 & 8 & 5 & 6
\end{array}\right), \left(\begin{array}{rrrr}
0 & 7 & 0 & 9 \\
8 & 7 & 4 & 0 \\
8 & 10 & 4 & 4 \\
5 & 8 & 3 & 0
\end{array}\right), \left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
1 & 10 & 0 & 0 \\
10 & 0 & 10 & 0 \\
0 & 1 & 1 & 1
\end{array}\right), \left(\begin{array}{rrrr}
9 & 0 & 0 & 0 \\
0 & 9 & 0 & 0 \\
0 & 0 & 9 & 0 \\
0 & 0 & 0 & 9
\end{array}\right), \left(\begin{array}{rrrr}
3 & 6 & 6 & 1 \\
8 & 1 & 3 & 6 \\
9 & 3 & 2 & 5 \\
7 & 9 & 3 & 0
\end{array}\right), \left(\begin{array}{rrrr}
10 & 0 & 0 & 0 \\
0 & 10 & 0 & 0 \\
0 & 0 & 10 & 0 \\
0 & 0 & 0 & 10
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2\times A_4\times F_5).C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_{15}:(C_2^3\times C_4^2)$, of order \(1920\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \) |
| $\operatorname{res}(S)$ | $C_{15} \rtimes (C_2^3\times C_4^2)$, of order \(1920\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $S_3\times D_{10}$, of order \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $4$ |
| Möbius function | $1$ |
| Projective image | $D_{10}\times S_4$ |