Subgroup ($H$) information
| Description: | $C_{11}^2:D_{22}$ |
| Order: | \(5324\)\(\medspace = 2^{2} \cdot 11^{3} \) |
| Index: | \(275\)\(\medspace = 5^{2} \cdot 11 \) |
| Exponent: | \(22\)\(\medspace = 2 \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
3 & 1 & 5 & 0 \\
0 & 0 & 1 & 0 \\
7 & 0 & 8 & 1
\end{array}\right), \left(\begin{array}{rrrr}
10 & 0 & 0 & 0 \\
10 & 10 & 2 & 0 \\
0 & 0 & 10 & 0 \\
5 & 0 & 1 & 10
\end{array}\right), \left(\begin{array}{rrrr}
8 & 6 & 8 & 10 \\
10 & 8 & 2 & 8 \\
9 & 3 & 5 & 5 \\
5 & 9 & 1 & 5
\end{array}\right), \left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
1 & 0 & 10 & 0 \\
0 & 1 & 0 & 10
\end{array}\right), \left(\begin{array}{rrrr}
10 & 7 & 4 & 8 \\
2 & 10 & 7 & 4 \\
10 & 9 & 3 & 4 \\
1 & 10 & 9 & 3
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence 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: | $C_{55}:C_5$ |
| Order: | \(275\)\(\medspace = 5^{2} \cdot 11 \) |
| Exponent: | \(55\)\(\medspace = 5 \cdot 11 \) |
| Automorphism Group: | $F_5\times F_{11}$, of order \(2200\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 11 \) |
| Outer Automorphisms: | $C_2\times F_5$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 5$, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_{11}^2.F_{11}^2$, of order \(2928200\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 11^{4} \) |
| $\operatorname{Aut}(H)$ | $C_2\times C_{11}^3.C_{10}.\PSL(3,11)$ |
| $\card{W}$ | 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 |