Subgroup ($H$) information
| Description: | $C_{11}^3:C_{10}$ |
| Order: | \(13310\)\(\medspace = 2 \cdot 5 \cdot 11^{3} \) |
| Index: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Exponent: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
3 & 1 & 5 & 0 \\
0 & 0 & 1 & 0 \\
7 & 0 & 8 & 1
\end{array}\right), \left(\begin{array}{rrrr}
8 & 6 & 8 & 10 \\
10 & 8 & 2 & 8 \\
9 & 3 & 5 & 5 \\
5 & 9 & 1 & 5
\end{array}\right), \left(\begin{array}{rrrr}
5 & 0 & 0 & 0 \\
0 & 5 & 0 & 0 \\
6 & 0 & 4 & 0 \\
0 & 6 & 0 & 4
\end{array}\right), \left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
1 & 0 & 10 & 0 \\
0 & 1 & 0 & 10
\end{array}\right), \left(\begin{array}{rrrr}
10 & 7 & 4 & 8 \\
2 & 10 & 7 & 4 \\
10 & 9 & 3 & 4 \\
1 & 10 & 9 & 3
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is normal, nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $\He_{11}.(C_{10}\times F_{11})$ |
| Order: | \(1464100\)\(\medspace = 2^{2} \cdot 5^{2} \cdot 11^{4} \) |
| Exponent: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and supersolvable (hence solvable and monomial).
Quotient group ($Q$) structure
| Description: | $C_{11}:C_{10}$ |
| Order: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Exponent: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Automorphism Group: | $F_{11}$, of order \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 5$.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_{11}^2.F_{11}^2$, of order \(2928200\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 11^{4} \) |
| $\operatorname{Aut}(H)$ | $C_{11}^3.C_{10}.\PSL(3,11)$ |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Möbius function | not computed |
| Projective image | not computed |