Subgroup ($H$) information
| Description: | $C_{55}:(Q_8\times F_{11})$ |
| Order: | \(48400\)\(\medspace = 2^{4} \cdot 5^{2} \cdot 11^{2} \) |
| Index: | \(33\)\(\medspace = 3 \cdot 11 \) |
| Exponent: | \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
8 & 8 & 3 & 0 \\
6 & 8 & 0 & 3
\end{array}\right), \left(\begin{array}{rrrr}
3 & 0 & 2 & 9 \\
3 & 6 & 2 & 2 \\
2 & 2 & 7 & 0 \\
4 & 2 & 8 & 10
\end{array}\right), \left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
4 & 4 & 2 & 0 \\
3 & 4 & 0 & 2
\end{array}\right), \left(\begin{array}{rrrr}
10 & 5 & 8 & 7 \\
10 & 8 & 2 & 8 \\
1 & 7 & 5 & 6 \\
10 & 1 & 1 & 3
\end{array}\right), \left(\begin{array}{rrrr}
9 & 1 & 7 & 4 \\
3 & 8 & 6 & 2 \\
1 & 1 & 5 & 8 \\
8 & 10 & 4 & 0
\end{array}\right), \left(\begin{array}{rrrr}
9 & 0 & 0 & 0 \\
0 & 9 & 0 & 0 \\
2 & 2 & 4 & 0 \\
7 & 2 & 0 & 4
\end{array}\right), \left(\begin{array}{rrrr}
8 & 3 & 5 & 3 \\
3 & 2 & 4 & 5 \\
3 & 1 & 9 & 8 \\
7 & 3 & 8 & 3
\end{array}\right), \left(\begin{array}{rrrr}
10 & 0 & 0 & 0 \\
0 & 10 & 0 & 0 \\
0 & 0 & 10 & 0 \\
0 & 0 & 0 & 10
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_{10}.D_{11}^3:C_{15}$ |
| Order: | \(1597200\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \cdot 11^{3} \) |
| Exponent: | \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}^3.(C_5\times A_4).C_{10}.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_{11}^2.C_{10}^2.C_{10}.C_2^6$ |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Normal closure: | not computed |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $33$ |
| Möbius function | not computed |
| Projective image | not computed |