Subgroup ($H$) information
| Description: | $C_{11}^2:C_8$ |
| Order: | \(968\)\(\medspace = 2^{3} \cdot 11^{2} \) |
| Index: | \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
| Exponent: | \(88\)\(\medspace = 2^{3} \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rrrr}
2 & 4 & 1 & 6 \\
5 & 3 & 5 & 1 \\
8 & 10 & 0 & 7 \\
5 & 8 & 6 & 1
\end{array}\right), \left(\begin{array}{rrrr}
0 & 3 & 1 & 1 \\
7 & 5 & 4 & 1 \\
3 & 4 & 6 & 8 \\
8 & 3 & 4 & 0
\end{array}\right), \left(\begin{array}{rrrr}
6 & 2 & 9 & 8 \\
3 & 2 & 5 & 9 \\
1 & 10 & 0 & 9 \\
6 & 1 & 8 & 7
\end{array}\right), \left(\begin{array}{rrrr}
7 & 7 & 9 & 1 \\
4 & 3 & 8 & 9 \\
4 & 8 & 10 & 4 \\
10 & 4 & 7 & 6
\end{array}\right), \left(\begin{array}{rrrr}
10 & 0 & 0 & 0 \\
0 & 10 & 0 & 0 \\
0 & 0 & 10 & 0 \\
0 & 0 & 0 & 10
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, monomial (hence solvable), metabelian, and an A-group. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_{11}^3:C_{20}.D_{12}$ |
| Order: | \(638880\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 11^{3} \) |
| Exponent: | \(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $S_3\times F_{11}$ |
| Order: | \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
| Exponent: | \(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \) |
| Automorphism Group: | $S_3\times F_{11}$, of order \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}^3.C_{30}.C_5.C_2^6$ |
| $\operatorname{Aut}(H)$ | $C_{11}^2.C_{60}.C_2^3$ |
| $\card{W}$ | \(14520\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11^{2} \) |
Related subgroups
| Centralizer: | $C_{11}:C_4$ |
| Normalizer: | $C_{11}^3:C_{20}.D_{12}$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |