Subgroup ($H$) information
| Description: | $C_5\times C_{11}^3:C_{10}^2$ |
| Order: | \(665500\)\(\medspace = 2^{2} \cdot 5^{3} \cdot 11^{3} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rrrr}
7 & 2 & 7 & 7 \\
3 & 9 & 3 & 8 \\
9 & 4 & 6 & 5 \\
5 & 5 & 0 & 0
\end{array}\right), \left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
10 & 10 & 9 & 0 \\
2 & 10 & 0 & 9
\end{array}\right), \left(\begin{array}{rrrr}
4 & 0 & 0 & 0 \\
8 & 1 & 2 & 0 \\
5 & 5 & 8 & 0 \\
4 & 5 & 3 & 5
\end{array}\right), \left(\begin{array}{rrrr}
10 & 0 & 0 & 0 \\
3 & 2 & 9 & 0 \\
10 & 10 & 7 & 0 \\
10 & 10 & 8 & 10
\end{array}\right), \left(\begin{array}{rrrr}
0 & 6 & 1 & 3 \\
9 & 9 & 6 & 6 \\
8 & 5 & 3 & 7 \\
1 & 10 & 6 & 5
\end{array}\right), \left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
4 & 5 & 1 & 0 \\
6 & 6 & 8 & 0 \\
6 & 6 & 7 & 1
\end{array}\right), \left(\begin{array}{rrrr}
4 & 5 & 2 & 2 \\
7 & 10 & 3 & 2 \\
4 & 10 & 3 & 6 \\
7 & 4 & 4 & 9
\end{array}\right), \left(\begin{array}{rrrr}
4 & 2 & 7 & 10 \\
8 & 5 & 5 & 7 \\
0 & 9 & 8 & 9 \\
3 & 0 & 3 & 9
\end{array}\right)$
|
| 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: | $C_{22}:F_{11}^2.C_{10}$ |
| Order: | \(2662000\)\(\medspace = 2^{4} \cdot 5^{3} \cdot 11^{3} \) |
| Exponent: | \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(212960000\)\(\medspace = 2^{8} \cdot 5^{4} \cdot 11^{3} \) |
| $\operatorname{Aut}(H)$ | Group of order \(532400000\)\(\medspace = 2^{7} \cdot 5^{5} \cdot 11^{3} \) |
| $\card{W}$ | 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 |