Subgroup ($H$) information
| Description: | $C_2\times C_{11}^3:C_{10}$ |
| Order: | \(26620\)\(\medspace = 2^{2} \cdot 5 \cdot 11^{3} \) |
| Index: | \(300\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \) |
| Exponent: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
3 & 3 & 10 & 0 \\
5 & 3 & 0 & 10
\end{array}\right), \left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
1 & 2 & 3 & 0 \\
7 & 7 & 0 & 0 \\
7 & 7 & 10 & 1
\end{array}\right), \left(\begin{array}{rrrr}
9 & 1 & 10 & 4 \\
5 & 5 & 2 & 10 \\
3 & 2 & 8 & 10 \\
0 & 3 & 6 & 4
\end{array}\right), \left(\begin{array}{rrrr}
2 & 2 & 5 & 1 \\
2 & 8 & 5 & 5 \\
10 & 8 & 5 & 9 \\
0 & 10 & 9 & 0
\end{array}\right), \left(\begin{array}{rrrr}
10 & 0 & 0 & 0 \\
4 & 3 & 1 & 0 \\
3 & 3 & 8 & 0 \\
1 & 3 & 7 & 1
\end{array}\right), \left(\begin{array}{rrrr}
6 & 3 & 5 & 2 \\
0 & 7 & 5 & 7 \\
3 & 6 & 2 & 5 \\
7 & 7 & 1 & 7
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_{11}^3:C_{10}^2.A_5$ |
| Order: | \(7986000\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{3} \cdot 11^{3} \) |
| Exponent: | \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable. Whether it is almost simple 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}^3.(C_2\times C_{10}\times F_5).A_5$ |
| $\operatorname{Aut}(H)$ | $C_{11}^3.C_5.C_{10}^2.C_2^3$ |
| $\card{W}$ | 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 | $12$ |
| Möbius function | not computed |
| Projective image | not computed |